Spiral Out – Using the Golden Ratio and Fibonacci Sequence to Predict Bitcoin Price Cycles – Bitcoin News
The value of the leading crypto asset Bitcoin has fallen more than 53% from its all-time high of over $ 64,000, to a low of $ 30,000 on May 19. While Bitcoin is still down 25% in the past 30 days, it’s hard to predict where Bitcoin’s value will go from here. Although a number of people use indicators like the golden ratio multiplier, the Fibonacci sequence, logarithmic growth curves, and tools like the infamous stock-to-flow pricing model (S2F) to predict future Bitcoin valuations.
Predicting the booms and busts of Bitcoin with the number Phi
Most people cannot predict the future and when it comes to Bitcoin (BTC) and the crypto-economy, busts and booms are generally common. In addition, bankruptcies and booms are often unpredictable except in some cases like certain news that shakes investors. However, there is an abundance of technical analysis tools, charts, and models out there to help a large number of people stay one step ahead of the game.
For example, fans of technical analysis use the golden ratio and Fibonacci sequence perspective to predict future Bitcoin valuations. In essence, a trader will be applying math to things like the value of Bitcoin and the moving averages. The golden ratio is also known as the divine ratio, divine ratio, number phi, extreme and medium ratio and golden number. Essentially, in the “science of quantity” two quantities reach a mathematical divine section if their ratio is equal to the ratio of their sum to the larger of the two quantities.
The golden ratio is equal to 1.618 and is used not only in mathematics, but also occurs in architecture, geometry, and many natural elements. Then there is the classic example of mathematics called the Fibonacci sequence, a sequence of quantities where a number is the addition of the last two quantities, starting with 0 and 1. The Golden Ratio Multiplier hosted on lookintobitcoin.com is there a detailed description of how the multiplier works with the leading crypto asset Bitcoin (BTC).
“Bitcoin adoption curve and market cycles to understand how price may behave over medium to long-term time frames,” notes the website. “This is done using multiples of the 350-day moving average (350DMA) of Bitcoin price to identify areas of potential resistance to price movement.” Lookintobitcoin.com adds:
Multiples are the value of the 350DMA, not the number of days. The multiples refer to the golden ratio (1.6) and the Fibonacci sequence (0, 1, 1, 2, 3, 5, 8, 13, 21). These are important math numbers. These specific multiplications of the 350DMA have been very effective over time in identifying intra-cycle highs for Bitcoin price as well as key market cycle highs.
The Golden Ratio Multiplier as applied to Bitcoin price predictions was conjured up by Philip Swift when he published an article on the subject on June 17, 2019. The article, entitled, “The Golden Ratio Multiplier: Unlocking the Mathematically Organic Nature of Bitcoin Adoption,” helps a trader get a better view of multi-year cycles.
A 2019 Golden Ratio Multiplier chart shared by Philip Swift on Twitter.
“The article begins by noting how the 350-day moving average has acted as the axis for Bitcoin’s major market cycles – as soon as we break out, a new BTC bull run begins,” Swift tweeted two years ago. Swift went on by saying:
The new insight begins when this important moving average is multiplied by key numbers: the golden ratio (1,618) and numbers in the Fibonacci sequence (1,2,3,5,8,13,21). In doing so, we are able to single out almost all of the major intra-cycle price highs in the history of Bitcoin (colored lines) AND every market cycle high (dotted lines). See the 2015-17 bull run as a clear example of the MA multipliers acting as great resistance at the highs within the cycle.
Bitcoin’s sequence of numbers such as Fibonacci poetry and Nautilus shells
Of course, not everyone agrees to use the golden ratio and the Fibonacci sequence to predict Bitcoin busts and booms. For example, Alvaro Fernández of the open insurance platform Nsure Network noted, “Historically it seemed to be respected, but how much could it be trusted? We might as well pass the first accumulation high. ”Other critics believe that using the golden ratio and the Fibonacci sequence is no different from using tarot cards.
Despite the skeptics, Swift’s bitcoin tool, the Golden Ratio Multiplier, is highly respected and used by a wide variety of technical analysts. The golden ratio has been used since the time of the ancient Greeks and many believe that it is deeply connected to the universe and nature. Similar to the golden ratio, Satoshi Nakamoto’s invention is a well-known and irrational technology. Interestingly, Bitcoin’s epic price surge since its first trade has closely followed the patterns of the number-phi and fibonacci sequence.
Just like the golden ratio multiplier, the leading asset followed succinctly with a “natural long-term power law growth corridor,” according to Harold Christopher Burger. Burger has published a comprehensive article discussing the logarithmic growth curves of Bitcoin. Like Swift’s Golden Ratio Multiplier, logarithmic growth curves can give a trader an idea of when to expect busts and booms and specific time frames. But these tools have allegedly been exposed on occasion, and the golden ratio is often viewed as a fairy tale.
Similar to the Nautilus shell, the price of Bitcoin has often been associated with the golden mean and the Fibonacci sequence. The Nautilus bowl is often compared and associated with the golden ratio, but contrary research and reflection suggests that the famous bowl is not a good example of the naturally occurring logarithmic spiral of the golden ratio. Studies show that the Nautilus shell has phi proportions but follows a 4: 3 ratio.
What do you think of using the golden ratio and the Fibonacci sequence to predict future bitcoin prices? Let us know what you think on this matter in the comments below.
Tags in this story
Alvaro Fernández, Bitcoin Math, Charts, Divine Section, Fibonacci Sequence, Golden Ratio, Golden Ratio Multiplier, Harold Christopher Burger, Logarithmic Growth Curves, Math, Models, Multi-Year Cycles, Number Phi, Philip Swift, Science of Quantity, Tools
Photo credit: Shutterstock, Pixabay, Wiki Commons, lookintobitcoin.com,
Disclaimer of liability: This article is for informational purposes only. It is not a direct offer or solicitation of an offer to buy or sell, or a recommendation or endorsement for any product, service, or company. Bitcoin.com does not provide investment, tax, legal, or accounting advice. Neither the company nor the author are directly or indirectly responsible for any damage or loss caused or allegedly caused by or in connection with the use or reliance on any content, goods or services mentioned in this article.


(Written: Wednesday, August 6,
West on Amtrak
Ok. So Marja
and I are traveling by train across the plains of North Dakota on our way from
DC to Seattle to go hiking and to see our West Coast children. We see a large wind farm, the blades
revolving so slowly that Marja wonders out-loud why they sometimes kill
migrating birds. I think, well, the ends
of the blades are probably moving more quickly than they seem. How fast, I wonder?
There being nothing better to do after staring out the
window at corn and grass and sky for a couple of hours, I decide to figure it
out. (For those of you who don’t care
how fast they’re going or don’t follow math easily, you might skip to the last
I google the size of a wind-turbine blade: 116 feet
long (Holy smokes! I thought maybe 40 feet long). I time one revolution of the blades: six
(Five years ago I could have gone from here and
figured the speed of the tips in my head, but I can’t come close now, so I take
a scrap piece of paper and write each step down.)
I’m happy that I remember from my basic geometry that
the circumference of a circle is πr (pi [3.14]
times the radius), so I calculate that the distance the tip travels in one
revolution is 364 feet. Now I need to
translate the feet into miles. I dig out
from my memory that one mile is 5280 feet; so the number of miles traveled in one revolution is 364
divided by 5280 (364/5280). Since there
are 60 seconds to a minute and 60 minutes to an hour, there are 3600 seconds in
every hour. So the time of one
revolution per hour is 6 divided by 3600 (6/3600). Therefore, the speed in miles per hour is the
result of 364/5280 (the distance expressed in miles) divided by the result of 6/3600
(the time expressed as a fraction of an hour).
Now I realize that this may sound complicated when you
read it. If you write out the numbers
and if you know basic geometry and algebra, however, it’s really pretty straightforward.
(It would have been a whole lot simpler, of course, if
I’d done the long division of each of the separate fractions into their decimal
equivalents before proceeding, but in my confusion I didn’t see that until I’m
writing this blog post and checking it
over several times.)
The answer as a complex fraction is 364/5280 divided
by 6/3600. This should not be difficult
for a high-school valedictorian, Yale-graduate physician, whose best subject
was math. But I can’t do it, even on
paper. The source of my problem is a usual
one for me: it’s a multistep process. I should translate the feet traveled
into miles traveled; translate the time of one revolution per second to the
time per hour; make them into a complex fraction; and do the arithmetic. But by the time I finish with the first step
and begin the second, I’m already confused about where I am in the process. I keep flipping each of the fractions,
multiplying and dividing and getting thoroughly confused. My scrap paper is covered with the four
numbers (364, 5280, 6 and 3600) in various combinations plus others I can’t
remember the source of.
So I finally remember to calculate the decimal
equivalents by dividing the fraction in the numerator (364/5280) into its
decimal equivalent, but I get confused even doing that. (Divide
the numerator by the denominator, right? Or is it the other way around? How do I do the long division of 364 divided
by 5280? C’mon, David; long division is
elementary school arithmetic!) I
figure out one of the decimals and now I can’t remember where I am in the
process, which of the fractions on the paper means what? My brain feels parboiled.
Finally, I have either to give up or “cheat” using the
calculator on my phone. I calculate the 364/5280
into a decimal (0.069 of a mile) and write it down on a fresh piece of paper. Then I calculate the 6/3600 into its decimal
form (0.00166) hour and write that down.
Finally, I divide the nominator decimal by the denominator decimal and
get 43 mph.
I then decide that you might be interested in reading the
whole debacle. But as I write the fifth paragraph
above about my calculations, I notice that in my first step I used the wrong
formula: The circumference is supposed to be pi times the diameter and not pi times the radius. Does
that make my result twice as large or half as large? I have to work that out on paper, too. And now I can’t remember what my initial
result was nor can I find it in the jungle of numbers on the papers, so I recalculate
the whole thing on my calculator, getting confused again along the way. I make so many mistakes that it takes me perhaps
twenty minutes just to repeat the simple process. And checking all the calculations again takes
me another half an hour, and I’m still not sure I’m right. So far, I’ve gotten three different answers,
but the final one seems right.
To those of you who wisely jumped here after the
third paragraph or tried and didn’t make it through the preceding paragraphs, I
don’t mean to imply that anyone
should be able to figure this out easily.
The point is that I used to be
able to get an approximate answer to something like this in less than a minute in
my head; With pencil and paper I could get the exact answer in two or three
minutes. And now it takes me well over an
hour and the use of a calculator to work out an answer I’m only shakily
If ever I need clear demonstration of my decline,
something like this is it.
I have no
idea why my decline doesn’t show up
on cognitive testing, but the reality is obvious.
For about a minute I notice myself getting depressed
about it, but that lifts pretty quickly.
I already know that I’m cognitively impaired. Do I really care how fast the tips of the
propellers are moving? No, I don’t. (It’s 83 miles an hour if you’re interested,
probably fast enough to clobber a goose who’s blindly following the goose in
front of him while daydreaming about his mate and not paying enough attention
to the blades.) Perhaps I used to care about
impressing and amazing my friends by figuring out the approximate answer in
less than a minute, but I’m actually happier now not being so hooked on the
need to be superior.
I enjoy most of my new values better than the ones they’ve replaced. I’ll put up with the occasional confusion.


Mathematics has always been a central component of human thought, and the growth of technology has increased its importance. It provides an effective way of building mental discipline and encourages logical reasoning and mental rigour. Mathematics is applied in various fields and disciplines, i.e., mathematical concepts and procedures are used to solve problems in Science, Engineering, Economics etc.
The Department of Mathematics is offering complimentary courses to various departments in this college. The department is established in 1995 when the Department of Tourism started the UG course in Tourism. It is offering a complementary course to the Microbiology department since 1999. And, the department is offering complimentary courses to Commerce Department also.
|Certificate Course in Quantitative Aptitude||Self Financing||Part Time||30 Hours||30|


Analytic Expression Format
In many cases, the coordinates, the sizes, locations and other numeric values can be represented as real numbers. However, often an analytic expression which is evaluated at runtime is needed. Example use cases are parametric building blocks and compact models describing bulding block performance. Therefore, openEPDA defines a standardized analytic expression format, which can be used anywhere where a numeric value is expected.
The expression grammar was chosen to cover a minimum yet comprehensive set of expressions. It supports standard mathematical operations, basic mathematical functions and variables defined at runtime.
Expressions can include the following items:
basic binary arithmetic operations (addition +, subtraction -, multiplication *, division /, exponentiation ^, and modulus %);
unary negation -;
mathematical functions (1 argument): abs, acos, asin, ceil, cos, cosh, exp, fac, floor, ln, log10, sin, sinh, sqrt, tan, tanh;
mathematical functions (2 arguments): atan2, pow;
constants: e, pi;
numbers (as defined in RFC 7159 - The JSON Data Interchange Format, Section 6. Numbers);
arbitrary variable identifiers, starting with a letter, and containing small and capital letters [a-zA-Z], digits [0-9], and an underscore _;
combinations of the above enclosed in the parentheses ( and ).
To be added soon.
A result of expression evaluation should be a real numeric value. This means that the information required for expression evaluation (such as variable values) should be available prior to evaluation.
A valid expression can be evaluated using the standard python’s eval function. When using this function, care should be taken of the variable names identical to the python keywords. See the list of keywords here: here.
A pure-python parser (e.g. to check expression validity) and an evaluator are available in the openepda package.
Also, a valid openEPDA expression is possible to evaluate using tinyexpr C parser. Note that we use ln for the natural logarithm.


Thought for the Day:
When you face the sun the shadows always fall behind you.
May 19-23, 2008
We would like to close the school year out with cooperation and fun with your children. We are going to have Field Day on Tues and an International Foods Day next Thurs. We will be eating with our team on Thurs with a catered meal and a movie. We need your help by sending in $5.00 to help with the food cost. We realize that we have asked for book fines, field day T-shirt money, etc. but our pockets are empty too. Remember the International Foods Day will be for your child’s lunch so that expense would be incurred anyway. Please help us make this a fun, cooperative week for everyone. Discuss your child’s behavior with them and your expectations of them. We have also done that, but as they get older, more will be expected from them. With additional responsiblility comes more freedom- which they all want. We want everyone to have a Great Summer!
Monday:T.L.W. take final test in SPI booklet.
Tuesday:T.L.W. participate in Field Day and complete a teacher-made worksheet on graphing a dinasour.
Wednesday:T.L.W. go to the computer lab for Math Games at Cool Math.com.
Thursday:T.L.W. complete a Teacher-Made worksheet- a spreadsheet on logical reasoning, and participate in International Foods Day.
Friday:T.L.W. complete a Teacher-made worksheet on addition, subtraction, multiplication and division of decimals and math terms.
Monday: T.L.W. work on the countries of South Africa Chapt. 26. p. 540-547. Complete Section #1 and #2 Review (1-5)
Tuesday: T.L.W. work on South Africa Today p. 548-555. Complete Sections #3 and #4 Review (1-5).
Wednesday: T.L.W. complete Chapt. 26 Review p.556. Vocabulary, Critical Thinking, and Reviewing the main ideas. Draw the map p. 557 and label A-H. WRITE A BRIEF ESSAY IN WHICH YOU COMPARE BOTSWANA AND MOZAMBIQUE. DISCUSS PHYSICAL FEATURES, ECONOMIES, AND MAJOR INDUSTRIES OF EACH.
Thursday: T.L.W. enjoy a movie for International Foods Day.
Friday: T.L.W. complete map Activity on Chapt. 26.
HAVE A GREAT SUMMER!


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buyers and/or users of this website. Note: The number of vends per day will always vary depending on number of customers within each machine location and Individual results will always vary.
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Are you new to the hoops game? Or a coach looking to improve the conditioning of your players? If so, you may be wondering how many laps around the basketball court make a mile.
When it comes time to run laps, keep in mind that all basketball courts don’t have the same dimensions. Find out how many laps around different sized courts you need to run to make a mile in this post. We’ll be using the dimensions of an indoor basketball court at each level.
Dimensions of Different Basketball Courts
Did you know that not all basketball courts are the same size? The perimeter of each court varies depending on the age and skill level of the basketball players. Here is a closer look at the dimensions for each different court.
Junior high basketball courts are 74 feet long and 42 feet wide. A kid playing youth basketball is normally in the sixth, seventh, or eighth grade, so the smaller sized court makes more sense for them. You wouldn’t want to put young players on a standard high school court.
A high school court is 84 feet in length. The width of the basketball court in high school is 50 feet. Since high school basketball players are older and larger than junior high players, playing on a junior high court would be too easy for them.
Basketball courts in the NBA and college basketball are 94 feet long and 50 feet wide. Note that there is no change in width from the high school level to the college and professional level. The only change is in the length of the court.
How Many Laps Around Each Court Makes a Mile
This question comes down to simple math. We know that the number of feet per mile is 5,280. Now, we simply have to take the length of the court and multiply it by 2. We will do the same calculation for the width of the court.
Junior High School
After finding this number, we are ready to do our calculations. So, for a junior high basketball court, we take the length of 74 feet and multiply it by 2 for a total of 148 feet. We also take the length of 42 feet and multiply it by two for a total of 84 feet.
Now, we’ll add those two numbers together for a total of 232 feet. Next, we take the one-mile total of 5,280 feet and divide it by 232 feet. This means that it will take 22.75 laps on a junior high school basketball court to make a mile.
For a high school basketball court, we take the 84 feet in length and multiply it by 2 for a total of 168. We take the width of 50 feet and multiply it by 2 to get 100. That leaves us with a total of 268 feet for one lap.
Again, we divide 5,280 feet by 268 feet. That tells us that it takes 19.7 laps to make a mile on a high school basketball court.
College and NBA
Now we get to the big boys. Both college basketball courts and NBA courts have the same dimensions of 94 feet long by 50 feet wide. Let’s follow the same mathematical procedure.
We take 94 and multiply by 2 to get 188 feet. We take 50 feet and multiply it by 2 to get to 100 feet. This gives us a total of 288 feet.
After dividing 5,280 feet by 288 feet, we see that it takes 18.33 laps to make a mile on both college and National Basketball Association courts.
How Many Laps Around Each Court Makes a Half-Mile
Now that we have the formula down, finding out how many laps it takes to make a half-mile is pretty simple. Just take our other answers and divide them by 2.
- So, using that formula we see that it takes 11.375 laps on a junior high school basketball court to make a half-mile.
- A high school basketball player will have to run 9.85 laps to make a half-mile.
- It takes 9.165 laps on an NCAA basketball court to make it a half-mile. It’s the same on an NBA court
How Many Laps Around Each Court Makes a Quarter Mile
Again, since we know the formula, we can find these answers fairly quickly. All we have to do is take our original answers for one mile and divide them by 4.
We already know that it takes 22.75 laps to make a full mile on a junior high basketball court. So, that means that it will take a player 5.69 laps to make it a quarter-mile on a junior high basketball court. (22.75 / 4 = 5.6875)
Using that same formula, we also see that a player will only need to run 4.93 laps to make a quarter-mile on a high school basketball court. How’d we get to that number? Take the 19.7 laps that it takes to make a full mile on a high school court and divide by 4. (19.7 / 4 = 4.925)
To figure out what makes a quarter-mile on an NCAA basketball court or an NBA court, take the 18.33 laps that it takes to make one mile and divide by 4. That standard set of measurements tells us that it takes 4.58 laps to run a quarter-mile on a college or NBA basketball court. (18.33 / 4 = 4.5825)
What Are the Dimensions of Olympic Basketball Courts?
Many folks assume that Olympic basketball courts have the same dimensions as NBA courts. Surprisingly, that is not the case.
An International Basketball Federation (FIBA) court is a tiny bit smaller than an NBA court. An NBA court is 50 feet wide, while an Olympic court is only 49 feet wide. A FIBA court is also two feet shorter in length than an NBA court (92 ft vs 94 ft).
Another stark difference between the two courts is the dimensions of the three-point line. The three-point line in the NBA is 22 feet away from the basket in the corners. The FIBA three-point line is only 21.65 feet away from the basket in the corners.
Directly above the break, the NBA three-point line is 23.75 feet away while the FIBA three-point line is 22.15 feet away. Those minute differences may not sound like a lot, but professional basketball players are remarkably more accurate the closer they are to the basket.
Why Do Basketball Coaches Make Players Run Laps?
At all levels of basketball, coaches have players run laps around the gym. They do this for a couple of reasons, which we will discuss below.
This reason is probably the most common. Coaches are notorious for making their players run laps as punishment for poor performance on defense during a game. They also may make players run laps when they do certain basketball drills incorrectly during practice.
For example, my junior high basketball coach used to make each player run one lap for every free throw that he missed during the game. Other coaches may make their players run one full-court sprint for each time that they turned the ball over during the game.
Basketball is a tough game and it requires that each player be in good cardiovascular condition to play full-court ball. For that reason, coaches will order players to run laps before or after practice. If you hate running, you should have reservations about playing organized basketball.
Running laps helps build up a player’s stamina so that he won’t get tired in the latter stages of a game. No basketball coach wants to see any of his players gasping for air during crunch time.
A popular form of conditioning is called running “suicides”. To run a suicide, a player starts at the end line underneath the backboard. He then runs from the end line to the free-throw line and back to the end line.
Next, the player sprints from the end line to the half-court line and back. He then runs from the end line to the other free throw line and back. Finally, he runs from one end line to the other end line and back. The crazy thing is that all of that sprinting only counts as one suicide.


I wish I could say that homeschooling is always sunshine and daisies, but that’s not how it really goes. Some days they don’t want to work, or I am dragging and don’t want to work, and sometimes things just don’t go as planned.
Today Little Yoda was doing his math and he just wasn’t getting it. We have been working on adding 9’s to numbers. (ex. 26 +9) Yesterday he went through this skill like he had been doing it his whole life, but now he was looking at me like he had never seen a problem like this this ever! His frustration added to my frustration and we went in circles until we both blew up. After we took a little break from each other we came back and talked about what was happening.
Part of our problem was that he just wanted to be finished. He thought if he could just write down some numbers on the sheet we could move on with our day. I explained to him that our goal is to never to just check off a task, but rather to have real understanding and learning. I didn’t care how many problems he did or didn’t do, I just wanted him to truly understand how to add a 9 to a number. So we got out the abacus and practiced giving a one to the nine to make it a ten and adding up the numbers that way in our heads. I modeled my thinking aloud for him so he could see what my thought processes were as I moved through the addition. After a few minutes of talking through the steps I gave him a few problems to do on his own. He quickly and easily did all of them without a hesitation. Success!
So in the end it wasn’t very pretty, and I really hate it when I lose my patience with the boys, but he did learn how to add nines to a number. More importantly he learned that our goal is not to simply complete a task, but to truly learn and understand the skills we are covering.


The passage above comes from pages 84-87 of
emeritus of science at Westmont
College Peter Stoner has calculated the probability of one man
major prophecies made concerning the Messiah. The estimates were
out by 12 different classes of 600 college students.
carefully weighed all the factors,
discussed each prophecy at length, and examined the various
might indicate that men had conspired together to fulfill a particular
prophecy. They made their estimates conservative enough so that there
finally unanimous agreement even among the most skeptical students.
Professor Stoner took their estimates
and made them even more conservative. He also encouraged other
or scientists to make their own estimates to see if his conclusions
than fair. Finally, he submitted his figures for review to a committee
the American Scientific Association (ASA). Upon examination, they
verified that his calculations were dependable and accurate in regard
scientific material presented.
concerning Micah 5:2, where Micah
prophesied the Messiah would be born in Bethlehem,
Stoner and his students determined the average population of Bethlehem from
the time of Micah to the
present. Then they divided that figure by the population of the earth
during the same period. They concluded that the chance of one man
born in Bethlehem was one in 2.8 x 105 - or rounded, one in 300,000.
examining only eight different prophecies, they
conservatively estimated that the
chance of one man fulfilling all eight prophecies was one in 1017.
how large the number 1017
is (a figure with 17 zeros), Stoner gave this illustration: Imagine covering the entire state of Texas with
dollars to a level of two feet deep. The
total number of silver dollars needed to cover the whole state is 1017. Now choose just one silver dollar, mark it,
and drop it from an airplane flying over the state. Then
thoroughly stir all the silver dollars
all over the state.
When that has
been done, blindfold one person,
and tell him he can travel wherever he wishes in the state of Texas in an
attempt to reach down into all
those silver dollars and pull up the one that was marked.
The chance of his finding that one silver
dollar – in the entire state of Texas
– would be the same chance the Old Testament prophets had for eight of
prophecies coming true about any one man in the future.
terms, is there anyone who would
fail to invest in a financial venture if the chance of failure were
only one in
1017? This is the kind of
sure investment we are offered by God for belief in his Messiah.
course, Jesus fulfilled many more than
eight prophecies. In another
calculation, Stoner used 48 prophecies (even though he could have used
as 456) and arrived at the extremely conservative estimate that the
of 48 prophecies being fulfilled in one person is 10157.
How large is
the number one in 10157? Well, 10157
zeros. Let’s try to illustrate the size
of this number, this time using electrons.
very small objects. They are smaller than atoms. It would take 2.5 times 1015 of
them, laid side by side, just to add up to one inch. If
we counted four electrons every second and
counted day and night, it would take us 19 million years just to count
of electrons one inch long.
But how many
electrons would we have to count if
we were dealing with 10157 of them?
Imagine building a solid ball of electrons that would extend
directions from the earth a length of six billion light years. The distance in miles of just one
light year is 6.4 trillion
miles. That would be a big ball!
But not big enough to measure 10157
In order to
do that, you must take that big ball
of electrons measuring six billion light years long in all directions
multiply it by 6 x 1028! How big is that? It’s
the length of the space required to
store trillions and trillions and trillions of the same gigantic balls
more. In fact, the space required to
store all of these balls combined would just start to “scratch the
the number of electrons we would need.
have some idea of the number of
electrons we are talking about, imagine marking just one of those
electrons. Then stir them all up, and
appoint a person to travel in a rocket for as long as he wants,
wants to go in that mass of electrons.
Tell him to then find that one marked electron.
What do you think his chances of success
would be? One in 10157.
Remember, this number represents the chance of
only 48 prophecies coming true in one person.
This affirms that it is absolutely impossible for anyone to
fulfilled all the Messianic prophecies solely by chance.
In fact, a leading authority on probability
theory, Emile Borel, states in his book Probabilities
and Life that once we go past one chance in 1050 (from a
perspective), the probabilities are so small it’s impossible to think


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First, install R Statistical Software on your local systems. You can download the installation file from here. Next, download the Desharnais data set from the Promise Repository and read its descriptions available at the top of the file. Load the data set into R as a data frame [Verzani, pp. 32] (Hint: You only need the comma separated values to read the data into R. Also, use the header option to read in column names.) For an introduction on how to use R, download and read the tutorial by Verzani. Note that you may read only relevant sections from it that are suffice to undertake the following tasks on the Desharnais data set -
Summarise the above your results (in groups of two) and submit them, preferably as PDFs, to Rahul Premraj on 7th May latest by 4PM.
We more or less repeat last week's exercises, but with some extenions/modifications. Your list of tasks to accomplish this week are as follows:
- Calculate the mean, median and mode of Project Effort and Length [Verzani, Section 2].
- Calculate the Pearsons and Spearmans correlation between Project Effort and Length. Can you explain the difference [Verzani, pp. 26]?
- Build a linear regression model using Length as the independent variable and Project Effort as dependent [Verzani, pp. 28].
- Search the conditions that the data must meet to be used in a linear regression model. Summarise your findings in a bulleted list. Can you check if the conditions hold true for the regression model built in the previous step?
You are required to undertake the following tasks this week -
- Rebuild the linear regression model using Effort as the dependent variable and Length as indepdent. However, this time, convert both features into their natural logarithms (not log to the base 10) first. Compare the R-square values of both models. Comment on which model do you think is better.
- Divide your data into a training and testing (2:1 ratio) set by randomly allocating projects to the two groups. Remember to filter out the 4 incomplete projects first. Do this only once, in contrast to what we discussed in the Exercises class.
- Rebuild linear regression models using Effort as the dependent variable and Length as indepdent (with and without log transformations) using the training set. Using the predict function in R, predict the Effort value of projects in the test set. Compare the prediction accuracy of each model. Remember that for the log model, you will have to re-transform the prediction value of Effort into its anti-log form. You can do this by issuing the command e^(log-value) where e is a constant and its value is 2.718.
- Build a linear regression model on the training data using Effort as the dependent variable and all other variables (except Project ID) as independent. Take only the raw form of the data, i.e. no log transformations. Comment on the prediction accuracy of this model and compare it to the ones above. Any thoughts on the effect of having Language as an independent variable in this model?
- In your exercise sheets, include the indices of the projects that comprise your training and testing set in the appendix.
- Build a linear regression model by substituting the Language feature by dummy variables. Use the same training and testing data from Exercise 2.
- Build a backward elimination stepwise regression model, again using dummy variables for the Language feature.
- Compare the prediction accuracy of both models by predicting projects in your testing set using sum of residuals as an indicator.
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Cartographers have long struggled with map distortions that occur when the Earth’s curved, two-dimensional surface is flattened into a flat surface. For example, the Mercator projection makes landmasses and other features far from the equator appear larger than they actually are. Equal area projections, on the other hand, distort the shape of various features on the map.
A wrinkle pattern is formed by the length mismatch between the curved 2D surface and the plane. Nonlinear mechanics and tension field theory successfully explain limited surface wrinkling. However, that approach cannot explain why wrinkle patterns appear on surfaces with low tension.
Ian Tobasco (University of Illinois at Chicago), Joseph Paulsen (Syracuse University), Eleni Katifori (University of Pennsylvania) and their colleagues looked at the problem from a mathematical angle rather than a physical one. A simple set of geometric rules they have derived is a precise solution to the wrinkle patterns that occur when thin, curved sheets are flattened.
Researchers have found that the patterns fall into two categories. The first rule is that for negative curvature (think of a horse saddle, for example), wrinkles occur along lines (blue) perpendicular to the surface boundary, as in panels a and d of the figure below. predict it will form. The wrinkles intersect each other along the central axis, defined as the set of points where the edge can be reached fastest from two or more paths. In particular, these segments constitute equal legs of a special family of isosceles triangles (panels b and e).
The second rule predicts that for spheres and other positive curvature surfaces (panels c and f), wrinkles will form along opposite legs (green) of the same isosceles triangle. To the researchers’ surprise, the wrinkle patterns of the positively and negatively curved sheets are related: their interrelationship allows one pattern to be inferred from the other.
The area covered by the isosceles triangles has a regular wrinkle pattern that is preserved even on uneven curved surfaces. The remaining regions support chaotic patterns. Geometrically, regions correspond to points on the central axis that have 3 or more shortest paths to the boundary (green polygons in panels e and f).
To test the new shape-based prediction, Paulsen and some co-authors observed the wrinkling that occurs when polystyrene films of various shapes are formed on a curved glass surface and then placed on a flat liquid surface. Did. The results of these experiments, and the simulations of the phenomenon led by Catifoli, showed encouraging agreement with the geometric interpretation.
Researchers led by Tobasco found that at the limit of infinitely fine wrinkles, the curved sheet tried to cover as much of the liquid bath as possible, resulting in a wrinkled pattern. That coverage maximization is driven in part by the surface tension of the water pulling the edges of the sheet apart as much as possible. This is the dominant role of liquids in experiments. However, gravity is also acting on the system, and simulations driven entirely by gravity with zero surface tension produced the same coverage maximization and the same wrinkle pattern as the experiment. (I. Tobasco et al., nut. Physics. 181099, 2022.)


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To mark International Women’s Day, this post was contributed by Professor Sarah Hart, Head of the Department of Economics, Mathematics and Statistics at Birkbeck.
Here, she explains why there is more to do to ensure representation of women in mathematics – and how that can be achieved
Today is International Women’s Day – a day which as well as celebrating the achievements of women also serves as reminder we still have work to do to achieve gender parity. In my field of mathematics, the last century has seen a series of women pioneers who have blazed a trail for future generations. An example was highlighted in the recent film “Hidden Figures”, which tells the story of the female African-American mathematicians and engineers working at NASA in the 1950s and 1960s. Those women overcame many obstacles, including having to go to court to gain the right to attend graduate classes in engineering. We have come a long way in the last 50 years, but not as far as you may think. For example, in mathematics it is still the case that fewer than 10% of the professors are women.
Could it be the case that women just aren’t as good at maths on average? No. Girls do just as well in maths at school – what happens is that the proportion who choose to pursue maths as a career is lower. This question isn’t even asked in most other areas where women are under-represented – are women “not as good as men” at being MPs? Or judges? The issue is certainly wider than mathematics. As Head of the Department of Economics, Mathematics and Statistics at Birkbeck, I’m very aware that women are underrepresented in the UK in all three of these disciplines. This matters because these fields are missing out on the contributions of those women who aren’t there but could be.
So what can we do? Research has shown that role models are very important. With that in mind, this month the department is involved with organising two events around women in maths and economics. Today, Birkbeck’s student-run Economics and Finance Society is putting on a Women in Economics Event with two eminent economists – Vicky Pryce and Oriana Bandiera – discussing barriers and opportunities, leadership and recognition, quotas and pay gaps. All are welcome to attend.
At the end of March we will be holding an event in collaboration with global investment management firm Winton, aimed at encouraging girls and young women to consider pursuing a career in mathematics. The Winton Women Trailblazers in Mathematics conference will be a two-day event. The first day is for girls in years 11-13 at school, where they can meet women working in mathematics and statistics and get an idea of what it’s like to take the subjects further. The second day, which is also supported by the London Mathematical Society, is a Women in Mathematics day, bringing together postgraduate and postdoctoral students to meet fellow mathematicians and hear from established women mathematicians from academia and industry about their work and careers.
Why are these activities important? Because it is our collective responsibility as a society to do what we can to further gender parity. “But things are changing”, you say. “Professors are old and there are more women doing maths and economics now, they’ll become professors eventually”. The World Economic Forum has calculated that yes, things are indeed changing, but that gender parity will not be achieved until 2186. I’m not prepared to wait that long. Are you?


Our Vision for Maths
At All Saints CE Academy, our vision is to foster positive attitudes, curiosity and excitement for maths. Our aim is to encourage our children to have the confidence to tackle problems and to have the resilience to persevere. Children learn to see maths in the wider world and to make connections across the curriculum. We feel that it is important for children to achieve their full potential, regardless of ability, taking account of learning needs and preferred learning styles. We ensure that each child is a respected part of our school family and is loved and valued as one of God’s children.
Intent, Implementation and Impact
Our Maths curriculum is ambitious for all and prepares children for later life. It is planned coherently using White Rose. It supports the learning of all children.
Our experienced teachers present the subject matter clearly and with competence. They respond to the needs of all children and skillfully adapt their teaching to challenge or address misconceptions. Maths is related to real-life situations to give the children context.
Children have a detailed knowledge of Maths concepts, using acquired skills and knowledge and applying them to new learning. They speak positively about the subject and relish challenge.
What is Maths like at All Saints, Denstone?
At our school, the children are provided with a rich and varied Maths curriculum. They are encouraged to use a wide range of equipment throughout their time here to support and consolidate their understanding of key mathematical concepts.
We support a Mastery approach that is intended to ensure key mathematical concepts are secure before moving on to new concepts. All children are challenged using a comprehensive problem solving and reasoning approach to Maths.
Reasoning about their maths learning is key and the children are encouraged to explain how they have reached their answers to maths problems.
We use a wide range of resources, including White Rose, the government’s Maths initiative, which has been devised by a range of maths experts. We also use other resources, such as NRich, Oxford Owls, and Classroom Secrets to ensure that the children get access to the best resources available and that the most and least able are both supported and challenged.
All children have access to online practice and support via Maths Seeds and Times Tables Rockstars. Through diagnostic tools within the websites, children work at their own pace with the aim of addressing misconceptions, plugging gaps, and helping to move children forward.
We also believe that the children’s Maths should have its foundations in the real world. This is particularly relevant when using money. The children handle money weekly to pay for tickets to enter our very popular cake raffle! They also handle money during fundraising events and the school council are responsible for counting the money and suggesting ways of spending it!
We also use our mathematical skills in other subjects. For example, we create timelines in history, collect data about weather and take measurements in Science.
Maths approaches and resources
It is important that all children have the opportunity to work with concrete apparatus, abstractly and using pictorial representation.
These all help with understanding and can be developed as part of calculation strategies. We use a range of resources to help us picture our problems and work out the answers by physically handling apparatus.
The main resources we use for numbers include dice, cubes, compare bears, Numicon, number fans, straws, Base 10, and place value counters.
Further resources include money, 2D and 3D shapes, rulers, scales, clocks, etc.
When using number resources, we use a variety of models. The main models we use are part, part-whole, number lines, and ten frames.
Part, part-whole models are used during addition and subtraction activities. Children use two or three smaller circles when adding numbers together, with the bigger circle being used for the answer. When subtracting, children work backwards. They use the bigger circle for their starting amount, subtract the given amount away into one of the smaller circles and see what numbers are leftover. This can be extended to understand how to find out the inverse and is useful for problem-solving. This model helps children understand the links and relationships between a whole number and its component parts. It helps them to interpret, visualise and solve a range of problems.
Number lines are used as a visual aid during activities such as number recognition, number ordering, counting on or back, finding one more or one less, and adding and subtracting, for estimating and rounding. These can take many forms dependant on the needs and skill levels of the child.Ten frames are used during number recognition, number bonds, addition and subtraction, place value and counting activities. They help the children to know that if filled, each frame is worth ten
Ten frames are used during number recognition,number bonds, addition and subtraction, place value andcounting activities. They help the children to know that if filled, each frame is worth ten
Children have the opportunity to count with a range of things. These include: cubes, bears, pasta, dinosaurs, counters, etc. The children are used to using the equipment when they need concrete or real things to physically add or take away.
Place value counters have the numbers 1, 10 and 100 on them. Children can make a two or three digit number using the counters. They can also be used to support understanding of adding and subtracting when used with a place value chart.
Straws are used singly or in bunches of ten to help with addition and subtraction and for place value
Base 10 is apparatus used for a range of activities. It helps with addition, subtraction and place value in particular. It is especially useful when adding or subtracting bridging 10 and exchanging tens for ones is required.
Place value dice are another way of helping children to understand place value, addition, and subtraction. They either have a single-digit or a 2-digit multiple of 10. The children roll both dice and put the two together to make a 2-digit number.
Bar models are a visual representation of a problem or idea that can be used for any of the four operation: addition, subtraction, multiplication and division. In a word problem, bar models have the added benefit of helping children to decide which operations they can use or how to visualise a problem.
The range of apparatus, models and resources is used throughout the school at different levels.
Why do we use models and images?
Using models and images helps us to:
- Secure a child’s understanding of what, for example, ten looks like
- Understand how addition and subtraction are linked
- Support the teacher in recognising where and why mistakes are being made. Sometimes, the errors that children make can be addressed by using models. For example, making 13 and 31 would show if a child has understood the different values of the two numbers.
What can you do at home to support your child?
Often the simplest ideas can be the most effective. Here are just a few examples, but you may be able to think of more!
- Memory games such as pairs
- Cards (Snap)
- Board Games (using a dice)
- Shape hunt
- Cooking (weighing and measuring)
- Sing nursery rhymes


The Effectiveness of Using the Flipped Learning Strategy Using Pen Tablet in the Achievement and Motivation towards Mathematics among Tenth Grade Female Students in Wadi Al-Seer District
فاعلية استخدام استراتيجية التعلم المقلوب باستخدام Pen Tablet في التحصيل والدافعية نحو مادة الرياضيات لدى طالبات الصف العاشر الأساسي في لواء وادي السير
This study aimed to identify the effectiveness of flipped learning strategy using pen tablet in the achievement and motivation towards mathematics among tenth grade female students in Wadi Al-Seer district. The study used a Quasi-experimental design, Where the study was applied to a sample of (93) female students from the tenth grade, it was divided into (48) female students in the experimental group and (45) female students in the control group. The experimental group was taught using a flipped learning strategy by activating the pen tablet, while the control group studied in the traditional way. The Study tools were used achievement test and Motivation scale for learning mathematics.
The results of the study showed a statistically significant difference at the level of significance (α = 0.05) between the mean achievement of female students in mathematics in the two experimental and control groups on the post test for the benefit of the experimental group students who studied using the flipped learning strategy using the pen tablet, where the arithmetic mean reached (15.96) of (20), The results also showed a statistically significant difference at the level of significance (α = 0.05) in the scores of the motivation scale towards mathematics between the experimental and control groups on the dimensional scale in favor of the experimental group students who studied using the flipped learning strategy using the pen tablet, where the arithmetic mean reached (2.89) From (3.) the differences were in favor of the experimental group.
In light of the results of the study; it recommends the use of the flipped learning strategy by activating the pen tablet tool in teaching math for the tenth grade, and further research into the use of the pen tablet in teaching other educational materials and for other educational stages.


To change a quadratic equation from a standardized form to a vertex form is quite easy to derive. You can certainly do so if you understand what a perfect square is. In this article, we would show you all the ideal concepts regarding vertex form and vertex formula. Let us start with a technique called “Completing the Square”. Using this technique, we can change a quadratic equation into a perfect square and you would get to know that it can be easily factorized. So, let us check how it is done.
Theories regarding a perfect square
Let us start with some examples over the application of the technique called “Completing the Square”. The main goal of CTS or Completing the Square is to take any quadratic equation that is not a perfect square and change it into a squared one without even changing the value. We have a few theories regarding this which is listed below.
Theory #1: Squares can be easily factorized
Any quadratic equation that is a square can be easily factored. Let us take an example:
x2 – 16x + 64 is a square which can again be denoted as (x – 8)2.
Theory #2: Finding the pattern within squares
From the above quadratic equation, we can say that it has a pattern since the leading co-efficient is a perfect square. Thus we can say that squaring half of b is always equivalent to c. So, from the above example we can say that half of b is equal to –(16/2) = -8. Now if we square -8, we get (-8)2 = 64.
Theory #3: Retaining the Value
Let us take an example of an equation, y=5x – 9. We can add or subtract values in the equation without changing the original value of the equation. For instance, we can add 3 to each part of the equation.
Thus we can write, y + 3 = 5x – 9 +3. However, it is not an appropriate purpose to add in this equation, but mathematically, it does not change the value of the equation. Similarly, we can add and subtract the same values from one part of the equation simultaneously. Using the above example, we can say,
y = 5x – 9 + 3 – 3. Here we are adding and subtracting 3 within one part of the equation without changing the value of the equation.
Finding the vertex and the vertex form
Let us take another example of an equation for instance.
y = x2 + 8x – 2
This equation cannot be factorized and apparently it is not a perfect square either. As we have said before, to be a perfect square you should square the half of b to get c. But within this equation it is not the same. So, what would be the value of c to make this a perfect square?
c should have to be 16 to make the equation a perfect square. Let us add and subtract 16 within one part of the equation. So the equation looks like, y = x2 + 8x + 16 – 2 – 16. Thus, we get a perfect square, x2 + 8x + 16, with some extra values.
Let us factor the perfect square and combine the extra values which would lead to:
y = (x + 4)2 – 18.
This is actually the vertex form of the original equation, y = x2 + 8x – 2 and the vertex is (-4, -18).
Thus to summarize, for changing a quadratic equation to vertex form, we need to change it into a perfect square with few extra values. Eventually, we use the half of b and then square it. After that we add and subtracted the squared value within one part of the equation. Lastly, we factorize the perfect square and combine the extra values.
For further details regarding vertex formula, book a session with Cuemath for online math classes.


By Douglas E. Ensley Discrete Mathematics, Student Solutions Manual: Mathematical Reasoning and Proof with Puzzles, Patte (1st Edition) on Amazon. com. Set Components. Discrete Mathematics: Mathematical Reasoning and Proof with Puzzles, Patterns, and Games. Douglas E. Ensley, J. Winston Crawley. $ Discrete Mathematics, Student Solutions Manual: Mathematical Reasoning and Proof with Puzzles, Patterns, and Games. Douglas E. Ensley/J. Winston Crawley.
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Return to Book Page. Discrete Mathematics Student Solutions Manual: Did you know that games and puzzles have given birth to many oftoday’s deepest mathematical subjects? Now, with Douglas Ensley solutkon Crawley’s Introduction to Discrete Mathematics, you canexplore mathematical writing, abstract structures, counting, discrete probability, and graph theory, through games, puzzles, patterns, magic tricks, and real-world problems.
You will Did discrets know that games and puzzles have given birth to many oftoday’s deepest mathematical subjects? You will discoverhow new mathematical topics can be applied to everyday situations, learn how to work with proofs, and develop your problem-solvingskills along the way.
Discrete Mathematics, Student Solutions Manual by Ensley, Douglas E ; Crawley, J Winston
Online applications discrefe improve your mathematical reasoning. Highly intriguing, interactive Flash-based applications illustratekey mathematical concepts and help you develop your ability toreason mathematically, solve problems, and work with proofs. Explore More icons in the text direct you to online activities atwww. Improve your grade with the Student Solutions Manual.
A supplementary Student Solutions Manual contains more detailedsolutions to selected exercises in the text. Paperbackpages.
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Our new math curriculum is called Eureka! We look forward to our students becoming master mathematicians!! Look back often for tips and tricks!!!
How are number sentences related? Same addend? Doubles? Doubles plus 1? Same sum? Students investigated all of these ways and more!
Students in Mrs. Talbert’s class were practicing their doubles facts using the best resource ever….their hands!
During the math meeting held on September 7th, Mrs. Carlson talked about our new math series. During this meeting she introduced the helpful math homework sheets in Google Classroom. If you go to the homepage for IW Elementary and click on the parent info tab, the first choice is the Parent Math Resources!!!!
Another helpful website!
Check out this website for information about Eureka Math!
Click below to check out more helpful tips!


This is a lovely drawing I did of Minjong on the slowest day ever. I drew it after I wrote the following stuff you're about to read. The picture has nothing directly to to do with the stuff so don't think it's a visual aid or you will become very confused.
you may procede
I'm sorry. I really am. I promise I'm not bein smart, but I want to share with you, my spam bots, something I thought about that was kind of groovy. First I had this thought:
Ideally, for a caricature artist, the unseen and the always seen get the same treatment.
Then I thought, well that's pretty solid. Maybe I can graph that.
so here's our x axis then
and our y axis then
Now, as we go from "always seen" and move leftward toward "unseen," we're going to gradually increase because we're going toward more bizarre and unusual and interesting features. Away from things like, his head is connected to his neck. He has two eyes. When he smiles his teeth show. His eyebrows have a space between them.
But when we move in from the opposite side, from unseen, the things you don't draw because they aren't there--things like..he has hair on his tongue, his face is on fire--as soon as you enter from the left axis, into the the "almost hardly ever, once in a lifetime, see" zone--as soon as you move this direction the tiniest imaginable distance, you're immediately in the "should draw" zone as deep as you can possibly be. So the graph goes up like this.
I think this would have to be aaaaaaaaaalmost a right triangle, but we can't really express it visually, but if we use math, maybe it's like this, but i wish I was more math savy.
Here I'm in a situation where if I knew useless geometry more good I could solve this.
And I don't know if "point zero repeating, one" is an actual number, because I would have to use a Google search to find that out, and I was unable to figure out how to type a repeating symbol on a Mac. The above 9 was done with copy and paste.
One thing I did learn was that the idea of a repeating decimal is actually controversial. Some mathematicians say .9 is actually 1. If we at the blog realm discover that to be the case, you can be sure we'll change it to .99999999999999999999999999999999999999999999999999999999999999999999999999999999999. So dont' worry about that.
I use to love math back in 7th grade. And then 8th grade I lost my grip on all of it. It depends so much on the teacher and I did not like my 8th grade teacher's teaching. I need a teacher that says everything with a sly grin out the side of their mouth like they got the scoop. my 8th grade teacher taught like she was doing announcements at a company union meeting.
And I had a great chemistry teacher, but he said something that I still just don't get. He said "When you're driving, every time a mosquito hits your wind shield your car stops very briefly." Is that crazy?! Of course it is, but from knowing the guy, I know he knows what he's talking about, but it didn't make sense to me then, and it doesn't make sense to me now.


By Derek Schlennstedt
Students at Badger Creek Primary have received new maths equipment as part of a donation from the Healesville Rotary Club.
With the help of various maths apparatus; from calculators and abacuses to a measuring wheel, no math equation will be too hard for the students to tackle.
At the school assembly on Friday December 1, Principal Scott Crawford unveiled the table full of equipment and said that it was only a small taste of what else was to come, with more equipment to be donated from Healesville Rotary.
“We have so much measurement, shape and maths gear for you all to use and it’s all thanks to Rotary,” Mr Crawford said at the assembly.
“This is just a small taste of what we’ve got – two more packs still to come.”
The Mail spoke to Assistant Principal Sue Hartley who said the packs would help improve their applied maths program, which had recently undergone extensive changes and required better equipment.
“We had limited material for the kids to use, and so we’ve gotten two kits for each applied maths area … it’s really boosted our maths equipment,” Ms Hartley said.”
“It’ll make lessons more hands-on and it will bring higher engagement for the applied maths program that includes measurement, shape, and space.”
“We’ve been working hard on our maths program and have had great results in the students’ growth and learning … the maths equipment will enhance the program even more.”
Graeme Chester, vice-president of Healesville said they were glad the maths paraphernalia component of a $3000 grant from Rotary would be put to good use, and improve the students’ education.
“They’ve made major improvements and exceeded the averages in the state, but they still want to improve more and have reached a level where they needed some new numeracy resources to do that,” Mr Chester said.


I'm still new (only 5 batches so far), so please pardon me if my response is uninformed (stupid). Just a few weeks back I had a related discussion (argument) with the "help desk" at Mr.Beer. The CarbDrops package instructions specify 2 Drops/750 ml bottle. I'm using their 1/2 liter (500 ml) bottles, so I asked them how many drops I should use per bottle. They insisted that only 1 Drop should be used per 500 ml bottle. This of course resulted in more discussion (argument). With degrees in Chemistry and Chemical Engineering, it was obvious to me that if it takes 2 CarboDrops to carbonate 750 ml, 1 drop will only carbonate 375ml, not 500 ml. "Can you do the math?" I asked them. Unfortunately, I guess they didn't have a calculator handy to divide 750 by 2, so my explanation to them went nowhere. Their ultimate response was "Well 1 Drop per 500 ml bottle is what our Brewmasters recommend. If you don't agree, please take it up with them". So against my better judgement, I used only 1 drop per 500 ml bottle. As a result, right now I have 2 cases of finished product carbonating. Needless to say, I'm going to be really troubled (pissed - and I don't mean drunk) if these cases come out flat because of their mistake (or because of my own mistake for listening to them). I'll know either way in about another week. If anyone knows for sure that 1 Drop will be enough, please send me a reassuring pat on the head that everything will be OK. Thanks!


You have a 100 numbers from 1-100 in random order.
You have an array of length 99.
Taking 99 of the 100 numbers available into the array, how can you determine the number that is not in the array?
Hint #1: You can sort the array but that is overkill. Solve it without sorting.
The sum of numbers from 1…n = (n * (n+1)) / 2
Sum of 1…100 = (100 * 101) / 2 = 5050
202 – sum of array = Missing Number


1st grade math games for free. Addition, subtraction, place value, and logic games that boost first grade math skills.
1st Grade Math Games Free Math Games Turtle DiaryThey'll learn fractions, measurements, money counting, data analysis, and more through educational and interactive online activities. These exciting and engaging 1st grade math games will help kids reinforce and master important mathematical skills necessary for their academic future.
2 days ago · 4. College Math Games (Ages 18+). Yes, there are math games for college students and up! There isnt as much to sort through on College Math Games as the other sites, but you can still play free
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Do all the presents you give away over the holidays look like they were wrapped in the dark? Don’t worry: this video features a series of mathematical tricks to help you ensure your gifts always look neatly wrapped.
Covering the full gamut of geometrical shapes and squishy items, mathematician Katie Steckles shows us how to wrap all kinds of gifts in this super-useful four-minute video. Be warned, though: the best techniques of all does require an equation, as well as a little practice.
[Katie Steckles via Flowing Data]


Quiz for Mathematics at a Glance
If you’re stumped on how best to finish a question correctly, ask others to assist you through the discussion board. You will receive the exact answer. Finding the right answer on the very first clue fetches 5 points.
Understanding Quiz for Mathematics
Now that you’ve identified the issue, almost half your job is finished. Since education is just one of those places which receive minimum funding from the government, it is crucial for parents to http://sweet63.ru/ look for assorted options that may help give their child a better education. Money is always the problem so people have a tendency to budget each and every thing they can.
Then you’ll come back to the screen listing each of the quizzes. After you click the Quiz” link, you are going to be directed to a listing of all the quizzes that are available that you take. Pick the worksheet link to look at the alignment note in the top left corner of its page.
Life After Quiz for Mathematics
At JumpStart, you’ll find many grade-based math activities and worksheets for children. The first kind of math worksheet has an assortment of similar math issues or exercises. The worksheets are generated randomly, and that means you look at this now get a different one each moment.
If you’re not happy, please tell us. Please specify where you would like to go. Count to inform the variety of objects.
Even if you choose not to finish this quiz at all now and you would like to use this time for a tutorial only, a minimal score on that quiz isn’t going to matter. During the previous week of the quarter, you will need to take a MATLAB quiz. Please read all the instructions on this page before taking your very first quiz.
Finding out how to use mathematics in poker may be a tough process at the beginning, but we highly advise that you continue trying, since it will let you develop into a long-term winning poker player. The youngest children begin by simply counting. Furthermore, they have the chance to communicate using their own words.
The Dirty Facts on Quiz for Mathematics
Otherwise, state the reason. Often some kind of calculation prior to medication administration is necessary. Just take this test to see whether you have what it requires to pass the GED test.
You are going to receive credit only as long as the circles are filled in correctly. With masterpapers no stoppage a individual travels a particular distance at a mean speed of 80 kmph, and with stoppages he covers the exact distance with an average of 60 kmph. Pot odds take into consideration the amount your opponent has bet in regard to the pot, and the chance of completing your draw to inform you about whether you need to call or fold.
What You Don’t Know About Quiz for Mathematics
The test isn’t official, it’s merely a wonderful approach to learn how much you know, or don’t know, regarding the jQuery library. Here it is possible to find information on all CUNY’s placement tests. Give an proper reaction.
There’s a array of views among mathematicians and philosophers concerning the precise scope and definition of mathematics. There’s no limit to the range of worksheets you generate for the class or tutorials and they’re able to be used for distribution purposes. Add to discover the perimeters of polygons on such worksheets.
The Quiz for Mathematics Pitfall
Multiplication is among the critical elements of mathematics, although it can be challenging for some young learners as it requires memorization together with practice. Although mathematics will allow you to arrive at elegant and efficient techniques, the usage of a computer and programming skills will have to solve most problems. Just take this test to learn how well you comprehend some simple math concepts.


Qualities of the Best Learning Math Website
The word Math is the short form of the word Mathematics. In math which is a branch in science, one plays with numbers in order to determine the arrangement, shape, and quantity. In math, there are formulae which assist one in finding solutions to the given problems. The following are some branches of math; Probability, Numbers, Algebra, Vectors, Statistics and the Singapore Math. Math is very important and it is offered in the various levels of education. Nowadays, learning math has been simplified by the World Wide Web. There are websites which provide math learning resources, skills, classes, tips, and guidelines. A websites that assist in math learning is known as a leaning math website. The following are attributes of the competent math tutoring online sites.
A wide variety of math grades and branches is a quality of a competent learning math website. There are many branches in math. Different math learners require help in different branches of math, therefore, a good math tutoring site should cover all the branches so as to serve all the learners. In order to serve all the pupils and students, a good learning math site should provide lessons and tutorials on all the math learning grades. The Common Core has resources on all the math branches and grades.
Reliability is another quality of a good learning math website. In order for a learning math website to be considered reliable, it should only cover math and should never be unavailable. In order for it to be 24/7 available, it should be well created and hosted. A good math tutoring site should only have resources for math. By clicking here, you will get to know more about a reliable math tutoring site.
The best math worksheets websites provide resources free of charge. Worksheets are effective in learning math and the best math learning sites provide them free of charge. Some incompetent learning math websites will ask you to make a payment in order to access their resources but this is not advisable.
A good online math tutoring should have learned and skilled math tutors. No one would want his/her child to be misled by an unqualified tutor. A competent math tutor is the one who has undergone training in teaching math and has passion in this subject. With this kind of a math tutoring website, you will see improvements in your child’s math performance.
The competent learning math websites are responsive. Of late, various devices can access the World Wide Web. These devices have various screen properties, therefore, a good learning math site should be designed such that it will be viewed on all these devices. In order to achieve this, the site should be designed by a competent web designer.
The competent online math tutoring sites have the above qualities.


Pi Day is a celebration of the mathematical constant π, or Pi. It is observed today because when the date is written out in month/date format, it shows the first three significant digitals of Pi: 3.14. If you have a sweet-tooth, check out these four videos so that you can celebrate Pi Day with real pie!
“The sweet science of baking pie” – (above)
“Learn Pi’s digits with this irrationally long song”
“Understand Pi with Pie”
“A recipe for pie since isn’t a pi-party without pie”


Recently Chandoo.org posted a question about how to graph data when you have a lot of small values and a few larger values. It’s not the first time that I’ve come across this question, and I’ve seen a lot of answers, many of them really bad. While all solutions involve trade-offs for understanding and interpreting graphs, some solutions are better than others.
Data graphs tell stories by revealing patterns in complex data. Good data graphs let the data tell the story by revealing the patterns, rather than trying to impose patterns on the data.
As William Cleveland discusses in The Elements of Graphing Data and his 1993 paper A Model for Studying Display Methods of Statistical Graphics, there are two basic visual operations that people employ when looking at and interpreting graphs: pattern perception and table look-up. Pattern perception is where we see the geometric patterns in a graph: groupings; relative differences (larger/smaller); or trends (straight/curved or increasing/decreasing). Table look-up is where we explore details of values and names on a graph. These two operations are distinct and complimentary, and it is through these two operations that the data’s story is told.
So suppose that we have some data like that at right, where we are interested in the patterns of smaller, individual values, but there are also a few extremely large values, or outliers. We describe such data as being skewed. How do we plot this data? First, for such a small data set, a simple table is the best approach. People can see the numbers and interpret them, there aren’t too many numbers to make sense of and the table is very compact. For more complicated data sets, though, a graph is needed. There’s a few basic options:
- Graph as-is;
- Graph with a second axis;
- Graph the logarithm of the data;
- Use a scale break.
- Plot the data multiple times.
This is the simplest solution, and if you’re only interested in knowing about the outliers (Dec ’09 and Jan ’10) then it will do. However, it completely hides whatever is happening in the rest of the months. Pattern recognition tells us that two months near the end of the series have the big numbers. Table-lookup tells us the approximate values and that these months are around December ’09 and February ’10, but the way the labels string together and overlap the tick marks, it’s not clear exactly what the labels are, let alone which label applies to which bar (which months are those, precisely? Is that “09 Dec” and “09 Feb?” Do the numbers even go with the text, or are they separate labels?).
For all but the simplest of messages, this rendition defeats both pattern recognition and table look-up. We definitely need a better solution.
Use a Secondary Axis
Excel gives us an easy solution: break the data into two columns (“small” numbers in one and “large” numbers in the other) and plot them on separate axes. Now we can see all the data, including the patterns in all the months.
Unfortunately, pattern recognition tells us that the big-sales months are about the same as all the other months. It’s only the table look-up that tells us how big of a difference there is between the two blue columns and the rest of the data. This is why I’ve added data labels to the two columns: to aid table look-up.
Even if we tweaked around with the axes to set the outliers off from the rest of the data, we’d still have the same basic problem: pattern recognition would tell us that there is a much smaller difference than there actually is. By using a secondary axis, we’ve set up a basic conflict between pattern recognition and table look-up. Worse, it’s easy to confuse the axes; which bars go with which axis? Reproduction in black and white or grayscale would make it impossible to correctly connect bars to the correct axis. Some types of color blindness would similarly make it difficult to interpret the graph. Table look-up is easily defeated with secondary axes.
The secondary axis presents so many problems that I always advise against using it. Stephen Few, author of Show Me The Numbers and Information Dashboard Design, calls graphs with secondary axes “dual-scaled graphs.” In his 2008 article Dual-Scaled Axes in Graphs, he concludes that there is always a better way to display data than by using secondary axes. Excel makes it easy to create graphs like this, but it’s always a bad idea.
Take the Logarithm
In scientific applications, skewed data is common, and the usual solution is to plot the logarithm of the values.
With the logarithm, it is easy to plot, and see, all of the data. Trends in small values are not hidden. Pattern perception immediately tells us the overall story of the data. Table look-up is easier than with secondary axes, and immediately tells us the scale of the differences. Plotting the logarithm allows pattern perception and table look-up to compliment each other.
Below, I’ve created the same graph using a dot plot instead of a bar chart. Dot plots have many advantages over bar charts: most obviously, dot plots provide a better arrangement for category labels (e.g. the months); also, dot plots provide a clearer view of the data by plotting the data points rather than filling in the space between the axis and the data point. There are some nice introductions to dot plots, including William Cleveland’s works and a short introduction by Naomi Robbins. The message is clear: any data that you might present with a bar chart (or pie chart) will be better presented using dot plots.
Use a Scale Break
Another approach, which might be better for audiences unfamiliar with logarithmic scales, is to use a scale break, or broken axis. With some work, we can create a scale break in Excel or OpenOffice.org.
There are plenty of tutorials for how to accomplish this in Excel. For this example, I created the graph in OpenOffice.org Spreadsheet, using the same graph with the secondary axis, above. I adjusted the two scales, turned off the labels for both y-axes and turned off the tick marks for the secondary y-axis. Then I copied the graph over to the OpenOffice.org Draw application and added y-axis labels and the break marks as drawing objects.
That pretty much highlights the first problem with this approach: it takes a lot of work. The second problem is that those break marks are just too subtle; people will miss them.
The bigger problem is with interpretation. As with the secondary axis, this subtle scale break sets up a basic conflict between the two basic operations of graph interpretation. Pattern recognition tells us that the numbers are comparable; it’s only table look-up that tells us what a large difference there is.
Cleveland’s recommendation, when the logarithm won’t work, is to use a full-panel scale break. In this way, pattern recognition tells that there are two distinct groups of data, and table look-up tells us what they are.
The potential disadvantage of this approach is that pattern perception might be fooled. While the scale break visually groups the “large” values from the “small” ones, the scale also changes, so that the broader panel on the left actually represents a much narrower range of values (about 1100 dollars range) than the narrower panel on the right (about 17000 dollars range). Our audience might have difficulties interpreting this correctly.
Edward Tufte has popularized the idea of small multiples, the emphasis of differences by repeating a graph or image with small changes from one frame to the next. In this case, we could show the full data set, losing fidelity in the smaller values, and then repeat the graph while progressively zooming in on a narrower and narrower slice with each repetition.
This shares many similarities to Cleveland’s full scale break, but provides greater flexibility. With this data, there are two natural ranges: 0 – 100000 and 0 – 1200. If there were more data between 1200 and 85000, we might repeat the graph several times, zooming in more with each repetition to show lower levels of detail.
I think there are two potential pitfalls. As with the full scale break, the audience might fail to appreciate the effect of the changes to scale. Worse, the audience might be fooled into thinking that each graph represented a different set of data, rather than just a different slice of the same data. Some care in preparing such graphs will be needed for successful communication.
When presenting data that is, like the data above, arranged by category, use a dot plot instead of bar charts. When your data is heavily skewed, the best solution is to graph the logarithm of the data. However, if your audience will be unable to correctly interpret the logarithm, try a full scale break or small multiples.
8 thoughts on “Graphing Highly Skewed Data”
How would I attach a picture of how I think this chart should look?
I don’t think WordPress supports attachments in comments, so I would suggest posting it to a free image hosting site, like Imageshack.us, Flickr or Picassa Web Albums, and then providing the link here.
OK. I tried imageshack.us… here is the URL http://img839.imageshack.us/i/chartswithsmalllargeval.png/
I tried this link and it seems to work. Interesting.
dropforge, there are some nice features about your graph. I particularly like the presentation of both absolute values and percentages. It’s certainly an improvement on my “As-Is” example. Still, I see a few problems with this approach.
Adding tabular data only bolsters table lookup (i.e. reading values); pattern perception is not enhanced. It is because pattern perception tells us so little in this graph (nothing at all, for eleven of the thirteen months) that you’ve had to add so many labels to the graph. Put another way: if the small values are important enough for us to add their values to the chart, then they are important enough for us to graph them in a way that supports the user in both pattern perception and table lookup operations.
Also, this technique of overcoming the limitations of the graph by adding table data only works for a small data set. If there were a few dozen categories, or two or three values for each category, then such data labels would be cumbersome to read.
Finally, though I didn’t mention it above, a good data graph follows the maxim of Antoine de Saint Expury, “perfection is achieved, not when there is nothing more to add, but when there is nothing left to take away.” The “As-Is” graph with added tabular data plots the same data three times on one graph (bars + axis labels, absolute values and percentages), and this added clutter can actually make it harder to correctly interpret the story that the data tells. The best presentation will strip the redundancies and present the data once, cleanly.
Taking the logarithm or using a scale break, while they have their own difficulties, are preferable to adding labels on a graph design that does not meet the demands of the viewer.
>I agree with your observation that this technique only works with a small data set if by small you mean number of values per month… in this case we have only one value per month. On the other hand you could probably support another 100 or so months of data on a chart that is allowed an 11″ x 8 1/2″ space.
>Since this is a chart of sales (and that really makes all the difference in how the data is presented as it provides a clue on who the intended audience is), more months will either reinforce the extreme sales seasonality or show that we were seeing the beginning of a new product launch or a combination of both or a new market introduction or… whatever
>By adding the absolute labels I get rid of the need have the left side y-axis scale, this scale is the real lookup issue as it cannot possibly support the very small and very high value lookup adequately.
>I would welcome additional categories of data… since this is a sales chart I could imagine there might be different markets or even more fun would be expenses and/or profits on a monthly basis. Customer penetration by month would be great. I would probably set up a way to cycle through the categories but might also look for a relationship that might lend itself to being overlayed through a 2nd y-axis that is scaled to stay out of the way of the bars and values of the 1st y-axis… well maybe just a new chart with the same x-axis scaling.
Thanks for sharing. Do you think “[Ed: link deleted]” can be a potential solution to this issue?
I like that your patented algorithm for dual-scale axes attempts to create a visual impression of similar scale. However, as I write in my blog, I think that there are always better solutions than dual-scale axes. I encourage you to follow the links to understand why comparing different series of data using dual scales is not bad just because the scale ranges aren’t chosen intelligently enough. Regarding the specific example in this post, I don’t see how a dual scale—however intelligent the ranges are chosen—could reasonably work to compare data within the same series.


Spreadsheets formulas. Easy method to make each bowl game worth a different point value so the national championship game and semi finals can be worth more points or however you want to customize it. Formulas are great for. It also includes a complex formula example that relies on excels order of operations to calculate the correct answer.
Leverage the full power of excel formulas excel 2016 formulas is fully updated to cover all of the tips. Formula acceleration saves time and mitigates errors by surfacing relevant formulas as you type. These functions can be used to create formulas that manipulate data and calculate strings and numbers.
Features for this years bowl prediction pool over the previous college football bowl pool manager spreadsheets include the following. This tutorial covers in detail how to create and use formulas and includes a step by step example of a basic excel formula. Spreadsheets developed as computerized analogs of paper accounting worksheets.
Excel 2016 formulas mr. A spreadsheet is an interactive computer application for organization analysis and storage of data in tabular form. Power editor dssresourcescom spreadsheets have been used by accountants for hundreds of years.
Free shipping on qualifying offers. Free shipping on qualifying offers. Each cell may contain either numeric or text data or the results of formulas that automatically calculate and display a value based on the.
Formulas in excel are used to perform calculations or other actions on data entered into the formula andor stored in program files. The program operates on data entered in cells of a table. Computerized or electronic spreadsheets are of much more recent origin.
Excel may refer to. They can range from basic mathematical operations such as addition and subtraction to complex engineering and statistical calculations. Google sheets supports cell formulas typically found in most desktop spreadsheet packages.
Microsoft excel a spreadsheet program by microsoft corporation.


Two years ago, I was assigned to develop a platform where user can deposit money into e-wallet and make payment for game, telco, bill and many other services from that e-wallet. Instead of target to direct end users, the platform are specially catered for small business (Internet cafe, convenience store etc) owners, also called as a retailer, who charged cash from customer and perform the payment for them. The difference between the cash charged from customer and the money that is deducted from his e-wallet will be the profit or commission that a retailer can earn.
Fixed-point V.S. Floating-point
The very first function in this system is to track how much money user has deposited in his e-wallet, i.e, we need to keep a number for the balance of e-wallet. What's more, the number may not be integral, i.e, it may have fraction. Most of programmers are familiar with the floating-point data type that is the formulaic representation that approximates a real number so as to support a trade-off between range and precision. However, there is also a fixed-point number representation for numbers that have a fixed number of digits after (and sometimes also before) the radix point. Fixed-point number is usually the preferred choice when storing critical information like monetary data where exact precision must be preserved. Most of programming languages and DBMS offer support for both data types. In Python, the following statement defines a decimal number with a precision of 2 digits after decimal point.
>>> from decimal import Decimal >>> balance = Decimal('2.78')
One caveat here is if we omit the quota mark:
>>> balance = Decimal(2.78) >>> print balance Decimal('2.79000000000000003552713678800500929355621337890625')
The result may be different on different platforms and CPU architectures.
decimal to store exact numeric data types. Here we choose decimal to store the balance:
CREATE TABLE `retailer_tab` ( ... `balance` DECIMAL(20,2) NOT NULL ...
Django's ORM model translate the decimal type in Python to SQL with single quotation marks. However, in MySQL 5.x the following operation will trigger a warning, which caused exception in Django ORM:
mysql> UPDATE retailer SET balance = 10.00; mysql> UPDATE retailer SET balance = balance - '1.79'; Warning: Data truncated for column 'balance' at row 1
One workaround is using floating point in Django ORM so that the translated SQL becomes:
mysql> UPDATE retailer SET balance = balance - 1.79;
but again, this involves floating point conversion and may lose precision.