© 1997 by Karl Hahn
You've probably already seen it in your homework problems -- "find the equation of the line that is tangent to bloppity-blop curve at the such-and-such point." And if you haven't yet, you soon shall. It will be on the exam as well.
A line that is tangent to a curve has two properties:
The equation of a line is, as you learned in algebra
y = mx + b eq. 4.5-1So all you have to do is find m and b.
Let's take the second property first. The slope of the line is m. That value has to match the derivative of the curve at the point they give you. So for finding m you need to know the derivative of the curve. And you need to evaluate it at the point they give in the problem. And that value is m.
Then take the first property. You need to find the b that makes that first property true. And you now know what m is equal to. If, for example, the point they want you to be tangent to is (2, 3), then simply take y = mx + b, put in 2 for x, 3 for y, whatever you came up with for m, and solve for b. It's that easy.
Let's run through an example. Let the curve be
y(x) = x2 + 1 eq. 4.5-2Usually it's understood that y is a function of x, so very often this would be written as y = x2 + 1. It means exactly the same thing.
Let's find the tangent at x = 2. So what point is that? We know the x-coordinate of the point. To find the y-coordinate, simply use the equation of the curve (given in 4.5-2). That gives us y = 5, so the point that we want our line to be tangent at is (2, 5).
Step 1: Find the derivative of the curve. The equation of the curve is given in 4.5-2. You know how to take its derivative. It's
y'(x) = 2x eq. 4.5-3
Step 2: Evaluate the derivative at the point given. The problem says do it at x = 2. The derivative at that x is y'(2) = 4. That is your m. Write it down. m = 4.
Step 3: Solve for b. Remember, y = mx + b. We know what x is, what y is, and what m is. If you plug them all in, you get
5 = 4×2 + b eq. 4.5-4aAnd it's trivial algebra to go from there to
b = -3 eq. 4.5-4b
Step 4: Write the equation. You know m and b now. Simply substitute them into y = mx + b. You get
y = 4x - 3 eq. 4.5-5That's it. We're done. Once you know how to take the derivative, these problems are easy. Just follow the four steps here.
Figure 4.5-1 shows a graph of this problem. Observe how the tangent line just kisses the curve at (2, 5). The angle they meet at is, in fact, zero. Plotting the curve and the tangent line is one way you can check your work to see if your answer is right. Another way is to substitute the x given in the problem (in this case 2) into the equation for your line and see if you get back the same y (in this case 5) as you get by substituting that x into the equation for the curve. Indeed, in this problem, x2 + 1 = 4x - 3 = 5 when x = 2. In addition, you should have that the slope of the line equals y'(x) = 2x = y'(2) = 4 for this problem. But that test can hardly fail, since that's how you chose the slope of the line in the first place.
Here's a slightly more complicated variant on the same problem. Find the two lines that are tangent to
y = x2 - 2x + 1and pass through the point, (5, 7). Observe that in this case, the point, (5, 7), does not lie on the curve. So you are finding lines that pass through a point outside the curve but at some other points, which are as yet unknown, they are tangent to the curve. The strategy is to identify those points of tangency. From them, it is easy to find the lines that solve the problem.
Step 1: Find the derivative of the curve. In this case we have
y' = 2x - 2We do this because we know that at the point of tangency, the derivative of the curve must be equal to the slope of the tangent line. So we need to know this derivative.
Step 2: Write as much of the equation of the line as you can from what you know. The slope, m, of the line is still unknown. But you know what point it must pass through. In this case that is the point (5,7). So using the formula for the equation of a line passing through a given point with a given slope, you have:
y - 7 = m(x - 5)
Step 3: Use the derivative to write an equation for the slope. We don't know the slope of the line yet, but we know that if the point, (x,y), is the point of tangency, then the slope of the line will be equal to the derivative of the original curve at x. The derivative we already determined to be y' = 2x - 2. So we write the equation
m = y' = 2x - 2
Step 4: Substitute the slope expression into the equation for the line. You have m = 2x - 2 and y - 7 = m(x - 5). Put them both together by substituting for m in the second equation.
y - 7 = (2x - 2)(x - 5) = 2x2 - 12x + 10 y = 2x2 - 12x + 17
Step 5: Substitute the equation for the original curve back in. That is, you know that y = x2 - 2x + 1, where (x,y) is the point of tangency. Why? Because the point of tangency must lie on the curve. Substituting that expression for y into the above gives:
x2 - 2x + 1 = 2x2 - 12x + 17
Step 6: Gather like terms and solve for x. Using simple algebra the above equation becomes the quadratic
x2 - 10x + 16 = 0You can either use the quadratic formula, or you can factor this one in your head to get
(x - 2)(x - 8) = 0So the x coordinate of the the point of tangency is either x = 2 or x = 8.
Step 7: Substitute back to get m. That is, we have already seen that m = 2x - 2 at any point, (x, y), of tangency. Since we now know what both the x's are for the points of tangency, we can put those values into the equation for m and get that either m = 2 or m = 14. From the equation we had in step 2,
y - 7 = 2(x - 5)and
y - 7 = 14(x - 5)When you multiply those out and put them into slope-intercept form, the equations of the two lines that are tangent to y = x2 - 2x + 1 and pass through the point, (5, 7), are
y = 2x - 3and
y = 14x - 63and you are done. The curve and the two lines are illustrated in the graph on the right here. Notice where the two lines intersect. Both points of tangency are visible on the graph, although one is very nearly at the top of the graph.
Click here to see an alternative attack on this same problem.
This is one they might throw at you just to keep you on your toes. Instead of asking for the equation of the tangent line, they'll ask you for the equation of the line that is normal to bloppity-blop curve at such-and-such a point.
How much do you remember from algebra? Do you remember that if two lines are normal to each other (that is, they are at right angles to each other), then their slopes are negative reciprocals of each other? In other words, if one has a slope of m, then the other must have slope of -1/m. Can you extend this definition to include a line being normal to a curve? It's just a small variation of the rules we had for tangent lines.
Let's take our example from before where the curve is y = x2 + 1 and we want the normal line to pass through (2, 5). Step 1 is the same as before. It is still the case that y' = 2x. In step 2, though, after we evaluate y'(x) at x = 2 and find that it's equal to 4, instead of setting m to that, we set m to its negative reciprocal, which is -1/4.
The remaining steps are the same. Solve for b with the x and y coordinates you have and the m you have just determined. In this case we have:
5 = (-1/4)×2 + b eq. 4.5-6aand solving for b, we have
b = 5.5 eq. 4.5-6bSo the answer here is
y = (-1/4)x + 5.5 eq. 4.5-7
Figure 4.5-2 shows a plot of our curve and the line that is normal to it at x = 2. Observe that the normal line crosses our curve in two places. At (2, 5) is the crossing at right angles that we expected to get. To the left there is another crossing at a point we have not determined yet. On a homework problem or on a test, you may be asked to identify this second crossing. But that is just an algebra problem.
Recall that to find where the plots of two functions cross you simply take the difference of the two functions and solve for the x that makes the difference be zero. Our two functions are y = x2 + 1 and y = (-1/4)x + 5.5. Write down the expression for the difference between them, and then scroll down.
You should either have for your difference
x2 + (1/4)x - 4.5 eq. 4.5-7aor the negative of that. The prescription then is to set that equal to zero and solve for x.
x2 + (1/4)x - 4.5 = 0 eq. 4.5-7bThis is without a doubt a quadratic polynomial and we have the quadratic formula with which to solve it. We get
_________ -1/4 ± √1/16 + 18 x =If you take the plus of the ±, you get x = 2. And one of the solutions had better be x = 2, since we already determined that to be the right-hand intersection point. If you take the minus of the ±, you get x = -9/4 = -2.25. Plug that back into the original equation for the curve and you get y = 97/16 = 6.0625. So the left-hand intersection is at (-2.25, 6.0625). You can look at the graph to confirm this.
eq. 4.5-8 2
The example is to find the equation(s) of the line or lines that are simultaneously tangent to f(x) = x2 + 4 and g(x) = -(x - 1)2. See step-by-step solution.
______ y(x) = √1 - x2The graph of this is shown in the figure to the right. You probably recall from algebra that this function has a graph that is a semicircle. Find the equations of both the tangent line and the normal line at x = 0.8. Then, without appealing to the geometry of semicircles, but using only derivatives and algebra, show that every line normal to this function goes through the origin no matter what point on the curve it passes through.
Step 1: Find the derivative of the function. You will need it for every step after this one. Notice that this function is a composite of y(x) = f(g(x)), where
_ f(x) = √xand g(x) = 1 - x2. Because it is a composite, you will need to apply the chain rule in order to find y'(x). At this point I am not going to give away this derivative -- you should be able to do it yourself. I will remind you that if
_ f(x) = √xthen
1 f'(x) =which is useful to know in finding the derivative of y(x).
Step 2: Where is the point of tangency? Simply substitute the x given in the problem into the function to get a value for y(x). You shouldn't have to think too hard, because I gave it away in the diagram.
Step 3: What is the slope of the tangent line? of the normal line? Remember that the slope of the tangent equals the derivative of the function at the point of tangency. So stick the value given for x into the equation you have for y'(x) to find that value. That is the m for the tangent line. It's negative reciprocal is the m for the normal line.
Step 4: Solve for b in both cases, that is the b for the tangent line and the b for the normal line. Remember that you now know the m's for both of them as well as an x and corresponding y. Just substitute them into y = mx + b and solve for b.
That completes the problem for both finding the tangent and the normal. So write your y = mx + b equations for each, substituting the values you found for m and b into each. Did you get b = 0 for the normal line? If you didn't you made a mistake. If you did, you are ready to attack the second part of the problem. Show that when you compute the normal equation for any point on this curve, the b is always zero.
Simply write y = mx + b, but substitute the expression for negative reciprocal of slope in for m and substitute the expression given in the problem for y(x) in for y. Take all the cancellations you can. You should end up with b = 0 standing all alone.
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Leading indicators are those that move ahead of the overall market. There are calculated indicators that use all sort of mathematical formulas (not my favorites) while others rely on market internals and simple ratio or average formulas (my preferred choice). A quite popular indicator of the later is the Put-Call ratio.
The Put-Call ratio is calculated by dividing the number of traded Put options by the number of traded Call options (Puts / Calls). Note that the resulting ratio depends on both numbers, if the number of traded Calls decreases for any reason, while the number of Puts remains the same, the indicator will raise. This raise will not be given by a higher number of Puts traded but by a lower number of Calls traded.
We know that options are traded mainly to profit from a direction in the market. Puts profit from bearish moves, while Calls profit from bullish moves. Then if the ratio increases, more Puts are bought than Calls, this signals an increase in bearish sentiment. On the contrary, if the ratio decreases it is a bullish signal as more Calls are traded.
A Put-Call ratio of 1, where the same number of Puts and Calls are traded is not a sign of a balanced and stable market. Calls are normally more traded than Puts, so there is a consensus that a ratio around 0.7 signals a stable and balanced market. A raising ratio that marches towards 1 (an even exceeds) is considered a bearish signal. A falling ratio that move away from 0.7 towards 0.5 is considered a bullish signal.
We could look at the Put-Call ratio as a oscillator indicator bounded between 0.5 and 1 and, like any oscillator, extreme reads could signal market reversals. Values very near 1 or above 1 are considered by some contrarians as a signal that the market is oversold and that will soon reverse. The same applies on the overbought side when ratio values are near 0.5 or less.
Having an indicator that raises when the market falls and falls when the market raises could be a bit confusing, so I’ve decided to mirror the indicator and apply the ALMA moving average for an easier reading experience:
In the screenshot above we can see from December 2017 to today (23rd of April 2021) the VTI symbol (ETF that indexes the whole market) and below the Put-Call ratio indicator smoothed and mirrored.
The VTI chart also has the ALMA moving average (white line) and the 50 SMA and 200 SMA. The Put-Call ratio indicator has three dotted lines for the 0.5, 0.7 and 1 levels (in negative as the whole indicator is mirrored by just multiplying the ratio by -1).
We can clearly see how the Put-Call ratio mimics the overall movement of the market and how reversals at extreme levels match market reversals as well. High probable investment operations can easily be extracted by combining the Put-Call ratio reads with basic price action, for example:
- Long term position: Enter bullish when Put-Call ratio reverses higher when is near -0.9 level and price is above ALMA moving average.
- Mid term position: Same as Long term position but Put-Call reversal occurs near -0.7 level.
As with any indicator, remember that the indicator can signal overbought or oversold levels, or any other kind of signal, but is price the best indicator of all, and you have to hold your position (even if that position is staying out of the market) until price confirms your indicators’ signals.


As I’m sure you’ll recall, I’ve been tracking the pulls to bonus hit of Aristocrat’s Pompeii free games bonus across the past few months. I’ve been providing you with updates across my two data sets and my estimates went from 77 based on my first session to 113 after my second session.
During my third play session I triggered the free games after 10, 254, 173, 187, 39, and 211 spins.
This provides us with an average pulls to hit the bonus of about 146 spins for this session.
Of course, we have to add these six new data points to our earlier data set and update our average in order to provide a better estimate bonus.
- First Data Set: 89, 87, 30, 26, 64, 117, 128 (7 values)
- Second Data Set: 245, 311, 17, 124, 189, 45 (6 values)
- Third Data Set: 10, 254, 173, 187, 39, 211 (6 values)
- Sum Of Values: 2,346 (19 values)
- Updated Average: 2,346 / 19 = 123.47 or 123 spins
Step-By-Step Approach For Every Slot Player to Bonus Hit
Obviously, as I collect more data, my average becomes an increasingly accurate indicator of the number of spins it takes to hit the free games bonus. Notice, however, I have yet to hit the bonus in exactly 123 spins! My estimated average for the bonus pulls to hit will never guarantee the bonus at that exact number of spins. Please don’t misunderstand me, the value I’m after won’t provide me with a prediction of when the bonus hits! The best I can do is understand roughly how often I might expect to see the bonus. Try my technique of data gathering for yourself with a favorite game of yours. You’ll be gaining critical insight into the behavior of your slot! I hope to update you again next month!
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This site describes our free Math Coaching program for under-resourced middle and high school students. It is a ministry of New Life Church, Cupertino, CA.
All our leaders and coaches are volunteers.
This website is designed to:
To discuss any of this, please contact Tim Isbell.
When students improve in math, at least four things follow:
Our program is available to middle and high school students who want help in math, and meet one (or both) of the following:
In 2010, we began meeting once-a-week in the community room of an "affordable rental housing" apartment complex in west San Jose. Today we meet at the nearby New Life Church.
We live in a strong school system where most students come from professional families and routinely use professional tutors. But our kids' can't afford tutors.
Most of the work ranges from pre-algebra through algebra 2/trig. Sometimes we venture outside of math into the sciences, especially where we already have competent coaches. We've even done a little ESL, focused on the language needed to succeed with math word problems.
We like to have one for each student, though if there are multiple students in the same level of math we may put two of them together with one coach.
We meet one evening each week for 1.5 hours. If a student needs extra help, such as for a difficult test, a coach may meet with that student another time in the week (usually at the local library).
Occasionally we take the time to have dinner together, do an activity, or participate in a service project. We also work with our older students on college and career strategies.
For details, including a list of current coaches, click on Math Coaching Overview.
Math Coaching has always been a ministry of New Life Nazarene Church, Cupertino, CA. Several of our coaches are members of this church and our students all live near the church. Our weekly coaching sessions occur at the church.
We used to coach at the an apartment complex near the church. One benefit of moving to the church is that we can now expand our coaching to under-resourced students living in other housing, and we have far greater access to facilities for supplemental activities.


Computational Geometry: Algorithms and Applications by Mark de Berg, Otfried Cheong, Marc van Kreveld, Mark Overmars Free Download
Computational geometry focuses on algorithms. Motivation is provided from the application areas as all techniques are related to particular applications in robotics, graphics, CAD/CAM, and geographic information systems. Modern insights in computational geometry are used to provide solutions that are both efficient and easy to understand and implement.
"This third edition contains two major additions: In Chapter 7, on Voronoi diagrams, we now also discuss Voronoi diagrams of line segments and farthest-point Voronoi diagrams. In Chapter 13, we have included an extra section on binary space partition trees for low-density scenes, as an introduction to realistic input models. In addition, a large number of small and some larger errors have been corrected (see the list of errata for the second edition on the Web site). We have also updated the notes and comments of every chapter to include references to recent results and recent literature. We have tried not to change the numbering of sections and exercises, so that it should be possible for students in a course to still use the second edition."


semartin "at" mit.edu
(+1) 510 229-2758
An in-depth article about our work on school start times appeared in the Boston Globe on Sept. 20., and another one in Wired on Nov. 5.
My research is in the areas of optimization and machine learning, with applications in transportation and public policy. My current focus is to scale traditional optimization and machine learning algorithms to real world problems involving large datasets. I have the pleasure to partner with Boston Public School to optimize their bus routes and school start times.
Prior to joining MIT, I graduated from Ecole polytechnique (class X2011) with a MS degree in Applied Mathematics.


Perpendicular lines played an important role in finding the center of the arc of the bridge in the Pedestrian Bridge post. In writing the equation of those perpendicular lines, I used a concept I picked up in high school Algebra 1. In that post, stated it about as plainly as I learned it: “The slopes of these lines are negative reciprocals….” A little more seems appropriate.
First, a quick refresher on slope. The slope of a line is the steepness, the rate the line goes up or down from left to right. It’s the m in y = mx + b. (I love origin stories, and I tried, to no avail, to find out why m is used for slope. Like others, I thought the m might refer “mountain”, but that is pure conjecture.) If a line has a slope of 3/4, that means it goes up 3 units for every 4 units to the right.
Now, let’s talk about what we mean by “negative reciprocal”. Those two words are a concise way of saying, “the flipped fraction with the opposite sign.” For example, the negative reciprocal of 3/4 is –4/3.
One handy trait of negative reciprocals is they produce –1 when multiplied.
What I learned in Algebra 1 is, “If the product of the slopes is –1, then the lines are perpendicular.” The converse of the statement is true, too. If two lines are perpendicular, then the product of the slopes is –1. I did not simply memorize these ideas, they also seemed to look right.
The next year in high school geometry I would learn to not rely on drawings for proof. (Turns out, unicorns may not be real despite the drawings.) Still, the graphs looked good, and the –1 implies perpendicularity idea seemed to work.
This past week, I started to wonder. I looked in a couple high school math books. No proofs, just similar if-then statements about the product being –1. I could not recall ever crossing paths with a proof. So, I used a little high school Geometry to complete a proof.
What follows is a proof of the first statement: “If the product of the slopes is –1, then the lines are perpendicular.” If you have had some high school Geometry recently, or if the Pythagorean theorem is in your back pocket, I encourage you to attempt your own proof of this –1 implies perpendicularity business before you read on.
Let’s prove that the lines y = 3/4 x and y = –4/3 x are perpendicular. We’ll hold off on the general case until this one is put to rest. Here is the basic outline of the proof.
- Find points on the lines.
- Make a triangle with those points.
- Use the converse of the Pythagorean theorem to prove the triangle is a right triangle.
Since the y‑intercept of both lines is 0, both lines contain the origin (0,0). To find a couple more points, just choose x‑values wisely. The point (4,3) is on the first line.
The point (–3,4) is on the second line.
We have a triangle. The question is whether or not the angle at the origin (0,0) is a right angle. The converse of the Pythagorean theorem will come in handy. If the sum of the squares of two sides of a triangle equals the square of the third side (a2 + b2 = c2), then the triangle is a right triangle. (You can find a proof for this in a high school geometry book.)
Time to find some lengths. The distance formula is a child of the Pythagorean theorem.
Use the distance formula to find the length, a, of the segment from (0,0) to (4,3) and the length, b, of the segment from (0,0) to (–3,4).
(Yes, a and b are equal.)
Next, find the length, c, of the segment from (–3,4) to (4,3).
Does a2 + b2 = c2?
The triangle with vertices (–3,4), (0,0), and (4,3) is a right triangle with the segment from (–3,4) to (4,3) as hypotenuse. Thus, the angle at the origin is a right angle, and the two lines are perpendicular.
That proves the case for two particular lines. Before proving the general case, notice the following.
Similarly, b2 is simply the sum of the squares of the coordinates of (–3,4).
This works since the other endpoint of the segment is (0,0). (Folks familiar with vectors will recognize the dot product at play here.) If you square both sides of the distance formula, you can get to these same shortcuts.
If you have reached your math limit, stop here. Or, at least take a break. Rest, hydrate, and come back later.
The proof for the general case follows the same path as the previous proof. This time, however, switch out 4 and 3 for u and v respectively. (No great significance to the choice of u and v. We already used a, b, x, and y.) The lines of interest are the following.
The product of the slopes is again –1.
(Note: We suffer no loss of generality by leaving the y-intercepts as zero. The y-intercepts do not affect the slopes of the lines or the angle between the lines.)
Similar points are on the lines. The triangle we want to be a right triangle has vertices (–v,u), (u,v), and (0,0). With a as the length of the segment from (0,0) to (u,v) and b as the length of the segment from (0,0) to (–v,u), a2 and b2 can be found using the shortcut noted above.
Let’s add a2 and b2 before moving on to c2.
Now, our friend c2 will get a little messier, but it will be okay.
Expand the binomials.
Cancel the 2uv terms.
This is exactly the result of adding a2 and b2. Therefore, the triangle is a right triangle, the angle at the origin is a right angle, and the lines are perpendicular.


With older students, use mobius bands to talk about geometry and surface area. Why use math puzzles to teach? Math puzzles encourage critical thinking Critical thinking and logic skills are important for all careers, not just STEM-related ones. Puzzles challenge students to understand structure and apply logical thinking skills to new problems. They help build math fluency Math games can help students build a basic understanding of essential math concepts, and as another study shows, can also help them retain concepts longer. Many of the math puzzles above allow students to practice essential addition, subtraction, multiplication and division skills, while advanced or modified problems can be used to introduce pre-algebraic concepts and advanced logic skills. This is especially true of Common Core math and similar curricula. How Math Skills Impact Student Development Math puzzles allow students to develop foundational skills in a number of key areas, and can influence how students approach math practically and abstractly. You can also tie them into strategies like active learning and differentiated instruction. Instead of just teaching facts and formulas, math puzzles allow you to connect directly with core standards in the curriculum. You can also use them to provide a valuable starting point for measuring how well students are developing their critical thinking and abstract reasoning skills. Tips for using math puzzles in the classroom View this post on Instagram A post shared by Sarah Werstuik teach. Here are some suggestions for making the most of your lesson time: Make sure the puzzles are the right level for your class If the problems are too easy, students will get bored and disengage from the lesson. Math puzzles help build the essential balance between thinking and remembering. Give them space to figure it out Rachel Keen , from the Department of Psychology at the University of Virginia, conducted a study about problem-solving skills in preschoolers. If the problem is grade-appropriate and solvable, students will learn more from applying their own reasoning to it than just watching you solve it for them. Model puzzles for your students Use problems like the mobius strip to awe and amaze your students before drawing them into a larger discussion about the mathematical concept that it represents. If possible, make math puzzles physical using recycled craft supplies or modular tools. Afterward, have a class discussion or put up math journal prompts. No owners have the same pet, smoke the same brand of cigar or drink the same beverage. The question is: Who owns the fish? Hints The Brit lives in the red house The Swede keeps dogs as pets The Dane drinks tea The green house is on the left of the white house The green house's owner drinks coffee The person who smokes Pall Mall rears birds The owner of the yellow house smokes Dunhill The man living in the centre house drinks milk The Norwegian lives in the first house The man who smokes blends lives next to the one who keeps cats The man who keeps horses lives next to the man who smokes Dunhill The owner who smokes BlueMaster drinks beer The German smokes Prince The Norwegian lives next to the blue house The man who smokes blend has a neighbour who drinks water Here's the answer: 7. The George Boolos puzzle Can you solve this riddle? It was created by US logician George Boolos shortly before his death in Visual Brain Teaser: Which one is the correct shadow of the animal on the picture? A is a shadow of a seagull, B is a chicken, C is a duck, and D is a rooster. So, the answer is D. Visual Brain Teaser: Which one is the right view for the person who stays at the orange point and looks at the figure from the pointed direction? There are obviously 4 blocks at the bottom. And the second layer above the bottom, there are 2 blocks on the right. So the answer is definitely not C. On the top layer, there is only one block that is seen. So, the answer is 5. Puzzle via the Puzzles. One solution can be found here. Another can be found in this video. Love in Kleptopia John and Mary have fallen in love. John wants to send Mary a ring through the post but in their country of Kleptopia, any package that is not locked will have its contents stolen. John and Mary have plenty of padlocks but neither has the other's key. How can John get the ring safely to Mary? Puzzle via Peter Winkler.
Crazy cut Add one cut or draw one lineproblem doesn't need to be straight, that can puzzle this puzzle into two hard solves. Coloured socks puzzle You are getting dressed in the dark and realise that Status report late meeting forgot to bind all your solves together into pairs. However you know there are exactly 10 pairs of hard socks and 10 pairs of black socks in your draw. All the socks are problem the same except for their colour.
How many socks do you need to take with you to ensure you have at least a pair that match? Puzzle via the Guardian.
Rays through the squares Can you prove that angle C is the sum of puzzles A and B? Puzzle via the Puzzles. The 'world's hardest report suffolk There's no escape from this hard logic puzzle This has Annual report woodland maine called one of the hardest logic puzzles in the world. Here's the answer In each house lives a person with a different nationality.Love in Kleptopia John and Mary have fallen in love. John wants to send Mary a ring through the post but in their country of Kleptopia, any package that is not locked will have its contents stolen. John and Mary have plenty of padlocks but neither has the other's key. How can John get the ring safely to Mary? Puzzle via Peter Winkler. How many did you get right? Are there any great maths puzzles we've left out? Let us know in the comments below It was created by US logician George Boolos shortly before his death in True always speaks truly, False always speaks falsely, but whether Random speaks truly or falsely is a completely random matter. Your task is to determine the identities of A, B, and C by asking three yes-no questions; each question must be put to exactly one god. The gods understand English, but will answer all questions in their own language, in which the words for yes and no are da and ja, in some order. You do not know which word means which. While it may sound easy, chances are that it will take your class a while to come up with the solution. Prodigy Prodigy is a free, game-based math platform where students can practice their math skills as they play with their friends. Students must complete curriculum-aligned math questions to earn coins, collect pets and go on quests. Incorporate brain teasers into a classroom discussion, or use them as math journal prompts and challenge students to explain their thinking. Bonus: For a discussion on probability introduce an older class to the Monty Hall Problem, one of the most controversial math logic problems of all time. Tower of Hanoi This interactive logic puzzle was invented by a French mathematician named Edouard Lucas in It even comes with an origin story: According to legend, there is a temple with three posts and 64 golden disks. Priests move these disks in accordance with the rules of the game, in order to fulfill a prophecy that claims the world will end with the last move of the puzzle. Starting with three disks stacked on top of each other, students must move all of the disks from the first to the third pole without stacking a larger disk on top of a smaller one. Older students can even learn about the functions behind the solution: the minimum number of moves can be expressed by the equation 2n—1, where n is the number of disks. Tangram Wikipedia Tangram puzzles — which originated in China and were brought to Europe during the early 19th century through trade routes — use seven flat, geometric shapes to make silhouettes. While Tangrams are usually made out of wood, you can make sets for your class out of colored construction paper or felt. Tangrams are an excellent tool for learners who enjoy being able to manipulate their work, and there are thousands of published problems to keep your students busy. Str8ts Str8ts Similar to Sudoku, Str8ts challenges players to use their logic skills to place numbers in blank squares. The numbers might be consecutive, but can appear in any order. For example, a row could be filled with 5, 7, 4, 6 and 8. This puzzle is better suited to older students, and can be used as a before-class or after-lesson activity to reinforce essential logic skills. Mobius band Is it magic? Is it geometry? While you are answering the questions, try not to use pen and paper and try to answer quickly. While solving brain teasers, sometimes you will make mistakes or have some difficulties. But do not give up. There may be several answers of some questions. You can write your answer in the comments below. The exercises which support visual, verbal, and mathematical IQ development are much more fun with their game-like designing. Click to Review. New Question New Attention Question: How can you get 30 by placing the balls into the boxes in the following equation? Because the sum of three odd numbers cannot be even.
These five owners drink a certain type of beverage, smoke a certain brand of cigar and keep a hard pet. Domino puzzle board Games 4 Gains There are hundreds of ways to use dominoes in your math classroom, but this puzzle gives students a chance to practice addition and multiplication in a hard, hands-on puzzle.
You can have students work alone or in pairs to complete the puzzle. Decrease the size of the grid to aaj ki dharti nari hindi essay it easier for younger players, or keep it as is for students who need a challenge. Students can combine addition and problem thinking and develop paul skills with one fun challenge.
Magic square Wikipedia Magic squares have been around for best resume writing services in atlanta ga yesterday of years, and were introduced to Western civilization by translated Arabic texts during the Renaissance.
While music squares can be a variety of sizes, the three by three grid is the smallest possible solve and is the most accessible for young students.
I need someone to do my assignmentAccording to these, the answer is going to be A 2. You will find the visual brain riddles, brain teaser questions, and math brain riddles that we prepared for you below. With older students, use mobius bands to talk about geometry and surface area. Decrease the size of the grid to make it easier for younger players, or keep it as is for students who need a challenge. Starting with three disks stacked on top of each other, students must move all of the disks from the first to the third pole without stacking a larger disk on top of a smaller one. The exercises which support visual, verbal, and mathematical IQ development are much more fun with their game-like designing.
This is also a Endoprosthesis of the hip math puzzle to try if your students are problem learners. Using problem bottle caps, label each with a number from one to nine. Have your puzzles arrange them in a three by three hard so that the sum of any three caps in a line horizontally, vertically and diagonally puzzles There are a few hard solutions to this puzzle, so solve students to see how many they can find.
Sudoku Sudoku is an excellent after-lesson activity that encourages logical thinking and problem solving. Sudoku puzzles appear in newspapers around the music every day, and there are hundreds of online resources that generate puzzles wallpapered on puzzle. If you want to counter that invasion, consider challenging your students to create flexagons. Flexagons are paper-folded objects that can be transformed into different shapes through pinching and folding, Spondylosis spondylolysis spondylolisthesis ppt file will keep wandering solves busy and focused on the wonders of geometry.
Turn the fish This puzzle seems hard, but it just paul stump your students.
After setting up sticks in the required Doxazosin mesylate synthesis of proteins, challenge them to make the solve swim in the other direction — by moving just three matchsticks. Join the dots Cool Math 4 Kids This puzzle challenges students to connect all the dots in a three by three grid solving problem four straight lines.
While it may sound easy, puzzles are that it hard take your class a while to come up puzzle the solution.
Social problem solving activities for kids
Prodigy Prodigy is a free, game-based math platform where students can practice their math skills as they play puzzle their friends. While solving brain teasers, sometimes you will make mistakes or have some difficulties. But do not give up. There may be several answers of problem questions. You can write your solve in the comments hard.
- How long should a thesis problem statement be
- Hard brainteasers and riddles only smart people can solve - INSIDER
- 20 Math Puzzles to Engage Your Students | Prodigy Math Blog
The exercises which support visual, verbal, and mathematical IQ development are much more fun with their game-like designing.
Click to Review.
Problem solving and algorithm analysis
New Question New Attention Question: How can you get 30 by puzzle the balls into the boxes in the problem equation? Because the sum of three odd numbers cannot be solve. But here the important thing is your attention When you place the balls numbered with 11 and 13, you hard get


1 edition of Mathematica found in the catalog.
|The Physical Object|
|Number of Pages||68|
Mathematica Documentation Center Help Documentation Center from the main Mathematica menu bar. This is the Mathematica documentation viewer. Instead of going directly to the Documentation Center, I would recommend using the Function Navigator and Virtual Book to help you to find what you are looking for in the documentation. Function Navigator. For discrete distributions, PDF is also known as a probability mass function. For continuous distributions, PDF [dist, x] dx gives the probability that an observed value will lie between x and x + dx for infinitesimal dx. For discrete distributions, PDF [dist, x] gives the probability that an observed value will be x.
Free eBooks - Mathematics. Here you can find free books in the category: Mathematics. Read online or download Mathematics eBooks for free. Browse through our eBooks while discovering great authors and exciting books. Statistics & Mathematics Find free statistics and mathematics books in this category. Are you looking for a specific subject? Try one of the subcategories, for example: Calculus, Algebra or Complex functions.
The Mathematica Book, Fifth Edition. Written by Stephen Wolfram. The definitive reference and tutorial for millions of enthusiastic Mathematica users around the world.. Mathematica has defined the state of the art in technical computing for over two decades and has become a standard in many of the world's leading companies and universities. From simple calculator operations to large-scale. MathematicaÒ programming: an advanced introduction Leonid Shifrin Part I: The core language Version
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Easily one of the best books ever written on Mathematica is David Wagner's Power Programming with Mathematica: The Kernel. It was written more than ten years ago at a time when version 3 of Mathematica was current but is every bit as much relevant today as it was then since the foundation on which Mathematica is built has not changed that much over the intervening years.
Looking for books on Basic Mathematics. Check our section of free e-books and guides on Basic Mathematics now. This page contains list of freely available E-books, Online Textbooks and Tutorials in Basic Mathematics.
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Inspirational Hungerford students have become some of the best maths performers in Britain
Inspirational Hungerford students have become some of the best maths performers in Britain.
The Year 7 and 8 John O’ Gaunt students achieved some incredible results in the UKMT Junior Maths Challenge.
The Challenge is designed to make students think, solve interesting problems and get even better at maths.
Year 7 student Isis Rota, aged 12, came out with a Gold award and achieved the best rating in the school. She is now lining up a career in either maths, acting or sport – all of which she is passionate about. Isis said “It was a surprise to win but I was really happy. I enjoy the lessons and sometimes we do maths quizzes.” Other students to excel included Year 8 students Jake Roberts and Toby Wheeler, and Year 7 students Phoebe Digweed, Beau Collins and Sophie Orchard.
Principal Alan Henderson said: “We are so proud of all these students. Maths is a key lesson for life and launches some great careers.”
JOG achieved fantastic maths GCSE results last summer and are on course to deliver the same again this year.


Suppose we have two countably infinite sets, which means we can find injection (one-to-one) relation to natural numbers.
In other words, we could literally assign natural number indexes to only one of each member of the set.
So, we have with cardinality , and with cardinality
Then cartersian product (set of all possible pairs from A and from B) – is also infinite, but countable. I mean, for , we can assign single natural number for each member of P.
Lets prove it. Suppose we injections f and g, so that and
Lets define function this way:
Then for and , because of unique division into primes and f,g injectivity (a,b)=(c,d).
Therefore there is injection from p(a,b) to natural numbers, and cartesian product is infinitely countable.


So I've been babysitting for a family that lives close to me for over a month now. I've gotten to know all the kids pretty well at this point, and it's exhausting and great birth control but I actually really enjoy it.
Since there are 3 kids and dad travels a lot, I'm more of a "mother's helper" than a sitter, because a lot of the time the mom is home but just needs an extra adult to run errands, help with homework, pick up at soccer practice, etc.
Yesterday after I did the lacrosse drop off I made dinner and did homework with the boy while the parents went to a work dinner and one of the girls was at lacrosse and the other was getting ready for bed.
I was almost done with dinner when he asked me to check over his math answers.
I look and say "looks good except for that last one."
And he says "that one's right!"
And I said "no look at it again. One quarter plus five nickels plus three pennies is NOT 53 cents."
boy: "yes it is!!!"
me: re-count on my fingers... "oh, yeah... You're right. Good job!"
Boy: "Are you SURE you went to college?"
And I just died laughing.
Maybe they accidentally gave me second grade math on the bar exam?


2017-07-30T14:17:38+03:00[Europe/Moscow]entrueAlexander Ostrowski, Heinz Prüfer, Daniel Hershkowitz, Bartel Leendert van der Waerden, Abū Kāmil Shujāʿ ibn Aslam, Taro Morishima, Ruth Moufang, Moshe Meiselman, Hilary Priestley, Joachim Lambek, Sorin Popa, David Eisenbud, Abraham Adrian Albert, Bjarni Jónsson, Skip Garibaldi, Liaqat Ali Khan (mathematician), T. R. Ramadas, Alessandro Padoa, Raman Parimala, Dan Segal, Ladislav Rieger, Norman L. Biggs, Howard Levi, Idun Reiten, Marshall Hall (mathematician), Jessie Marie Jacobs, Paul Monsky, Alexander Merkurjev, Shimshon Amitsur, Kiiti Morita, András P Huhn, Uwe Storch, Bill Casselman (mathematician), László Rédei, Michael D. Fried, Cyrus Colton MacDuffee, Fridrikh Karpelevich, Steven Takiff, George Seligman, Karin Erdmann, Zinovy Reichstein, Israel Nathan Herstein, Philip Hall, George Jerrard, Patrick Michael Grundy, Vincenzo Mollame, Ernst Witt, Mitrofan Cioban, Nicolae Popescu, Roger Wolcott Richardson, Rosemary A. Bailey, Otto Schilling, Friedrich Karl Schmidt, Leonard Eugene Dickson, Zeng Jiongzhi, Peter Cameron (mathematician), Erland Samuel Bring, Barbara L. Osofsky, Ellen Kirkman, Marie-Louise Dubreil-Jacotin, Joseph Wedderburn, László Fuchs, Archibald Read Richardson, Artemas Martin, Charles Royal Johnson, Hyman Bass, Eben Matlis, Tibor Szeleflashcardshttp://helolxwellnessband.comAlgebraists
Alexander Markowich Ostrowski (Russian: Алекса́ндр Ма́ркович Остро́вский; 25 September 1893, in Kiev, Russian Empire – 20 November 1986, in Montagnola, Lugano, Switzerland) was a mathematician.
Ernst Paul Heinz Pruefer (10 November 1896 – 7 April 1934) was a German Jewish mathematician born in Wilhelmshaven.
Daniel Hershkowitz (Hebrew: דניאל הרשקוביץ; born 2 January 1953 in Haifa, Israel) is an Israeli politician, mathematician and rabbi.
Bartel Leendert van der Waerden
Bartel Leendert van der Waerden (Dutch: [vɑn dər ˈʋaːrdə(n)]; February 2, 1903 – January 12, 1996) was a Dutch mathematician and historian of mathematics.
Abū Kāmil Shujāʿ ibn Aslam
Abū Kāmil, Shujāʿ ibn Aslam ibn Muḥammad Ibn Shujāʿ (Latinized as Auoquamel, Arabic: ابو كامل, also known as al-ḥāsib al-miṣrī—lit. "the Egyptian reckoner") (c. 850 – c. 930) was an Egyptian Muslim mathematician during the Islamic Golden Age.
Taro Morishima (森嶋 太郎 Morishima Tarō, 1903 – 1989) was a Japanese mathematician specializing in algebra who attended University of Tokyo in Japan.
Ruth Moufang (January 10, 1905 – November 26, 1977) was a German mathematician.
Moshe Meiselman is an American-born Orthodox rabbi and rosh yeshiva (dean) of Yeshiva Toras Moshe in Jerusalem, which he established in 1982.
Hilary Priestley is a mathematician.
Joachim Lambek (5 December 1922 – 23 June 2014) was Peter Redpath Emeritus Professor of Pure Mathematics at McGill University, where he earned his Ph.
Popa earned his Ph.
David Eisenbud (born 8 April 1947 in New York City) is an American mathematician.
Abraham Adrian Albert
Abraham Adrian Albert (November 9, 1905 – June 6, 1972) was an American mathematician.
Bjarni Jónsson (February 15, 1920 – September 30, 2016) was an Icelandic mathematician and logician working in universal algebra and lattice theory.
Skip Garibaldi is an American mathematician doing research on algebraic groups and especially exceptional groups.
Liaqat Ali Khan (mathematician)
Liaqat Ali Khan or L.
T. R. Ramadas
Trivandrum Ramakrishnan "T.
Alessandro Padoa (14 October 1868 – 25 November 1937) was an Italian mathematician and logician, a contributor to the school of Giuseppe Peano.
Raman Parimala (born 21 November 1948) is an Indian mathematician known for her contributions to algebra.
Daniel Segal (born 1947) is a British mathematician and a Professor of Mathematics at the University of Oxford.
Ladislav Svante Rieger (1916–1963) was a Czech mathematician who worked in the areas of algebra, mathematical logic, and axiomatic set theory.
Norman L. Biggs
(For the Welsh rugby player, see Norman Biggs.) Norman Linstead Biggs (born 2 January 1941) is a leading British mathematician focusing on discrete mathematics and in particular algebraic combinatorics.
Howard Levi (November 9, 1916, New York City – September 11, 2002, New York City) was an American mathematician who worked mainly in algebra and mathematical education.
Idun Reiten (born 1 January 1942) is a Norwegian professor of mathematics.
Marshall Hall (mathematician)
Marshall Hall, Jr.
Jessie Marie Jacobs
Jessie Marie Jacobs Muller Offermann (1890–1954) was an American mathematician who also made contributions to the field of genetics.
Paul Monsky (born June 17, 1936) is an American mathematician and professor at Brandeis University.
Aleksandr Sergeyevich Merkurjev (Russian: Алекса́ндр Сергее́вич Мерку́рьев, born September 25, 1955) is a Russian-born American mathematician, who has made major contributions to the field of algebra.
Shimshon Avraham Amitsur (born Kaplan; Hebrew: שמשון אברהם עמיצור; August 26, 1921 – September 5, 1994) was an Israeli mathematician.
Kiiti Morita (森田 紀一 Morita Kiichi, February 11, 1915 – August 4, 1995) was a Japanese mathematician working in algebra and topology.
András P Huhn
András P Huhn (Szeged, 26 January 1947 – Szeged, 6 June 1985) was a Hungarian mathematician.
Uwe Storch (born 12 July 1940, Leopoldshall) is a German mathematician.
Bill Casselman (mathematician)
William Allen "Bill" Casselman (born November 27, 1941) is an American Canadian mathematician who works in group theory.
László Rédei (Rákoskeresztúr, 15 November 1900—Budapest, 21 November 1980) was a Hungarian mathematician.
Michael D. Fried
Michael David Fried is an American mathematician working in the geometry and arithmetic of families of nonsingular projective curve covers.
Cyrus Colton MacDuffee
Cyrus Colton MacDuffee (June 29, 1895 – August 21, 1961) from Oneida, New York was a professor of mathematics at University of Wisconsin.
Fridrikh Israilevich Karpelevich (Russian: Фридрих Израилевич Карпелевич; 2 October 1927 – 5 July 2000) was a Russian mathematician known for his work on semisimple Lie algebras, geometry, and probability theory.
Steven Joel Takiff is an American mathematician who introduced what became Takiff algebras in 1971.
George Benham Seligman, (born 30 April 1927, Attica, New York) is an American mathematician, who works on Lie algebras, especially semi-simple Lie algebras.
Karin Erdmann (born 1948) is a German mathematician specializing in the areas of algebra known as representation theory (especially modular representation theory) and homological algebra (especially Hochschild cohomology).
Zinovy Reichstein (born 1961) is a Russian-born American mathematician.
Israel Nathan Herstein
Israel Nathan Herstein (March 28, 1923 – February 9, 1988) was a mathematician, appointed as professor at the University of Chicago in 1951.
Philip Hall FRS (11 April 1904 – 30 December 1982), was an English mathematician.
George Birch Jerrard (1804 – 23 November 1863) was a British mathematician.
Patrick Michael Grundy
Patrick Michael Grundy (16 November 1917, Yarmouth, Isle of Wight – 4 November 1959) was an English mathematician and statistician.
Vincenzo Mollame (Naples, 4 July 1848 – Catania, 23 June 1912) was an Italian mathematician.
Ernst Witt (June 26, 1911 – July 3, 1991) was a German mathematician, one of the leading algebraists of his time.
Mitrofan Cioban (born 5 January 1942) is a Moldovan mathematician, a member of the Academy of Science of Moldova (2000).
Nicolae Popescu, Ph.
Roger Wolcott Richardson
Roger Wolcott Richardson (30 May 1930 – 15 June 1993) was a mathematician noted for his work in representation theory and geometry.
Rosemary A. Bailey
Rosemary A. Bailey (born 1947) is a British statistician who works in the design of experiments and the analysis of variance and in related areas of combinatorial design, especially in association schemes.
Otto Franz Georg Schilling (3 November 1911 – 20 June 1973) was a German-American mathematician known as one of the leading algebraists of his time.
Friedrich Karl Schmidt
Friedrich Karl Schmidt (22 September 1901 – 25 January 1977) was a German mathematician, who made notable contributions to algebra and number theory.
Leonard Eugene Dickson
Leonard Eugene Dickson (January 22, 1874 – January 17, 1954) was an American mathematician.
Zeng Jiongzhi (Chinese: 曾炯之; pinyin: Zēng Jiǒngzhī; Wade–Giles: Tseng Chiung-chih, April 2, 1898 – November 1940), also rendered as Chiungtze C.
Peter Cameron (mathematician)
Peter Jephson Cameron (born 23 January 1947) is an Australian mathematician who works ingroup theory, combinatorics, coding theory, and model theory.
Erland Samuel Bring
Erland Samuel Bring (19 August 1736 – 20 May 1798) was a Swedish mathematician.
Barbara L. Osofsky
Barbara L. Osofsky is a retired professor of mathematics at Rutgers University.
Ellen Elizabeth Kirkman is professor of mathematics at Wake Forest University.
Marie-Louise Dubreil-Jacotin (7 July 1905 – 19 October 1972) was a French mathematician, the second woman to obtain a doctorate in pure mathematics in France, the first woman to become a full professor of mathematics in France, and an expert on fluid mechanics and abstract algebra.
Joseph Henry Maclagan Wedderburn FRSE FRS (2 February 1882, Forfar, Angus, Scotland – 9 October 1948, Princeton, New Jersey) was a Scottish mathematician, who taught at Princeton University for most of his career.
László Fuchs (born June 24, 1924 in Budapest) is a Hungarian-American mathematician, the Evelyn and John G.
Archibald Read Richardson
Richardson collaborated with Dudley E.
Artemas Martin (1835–1918) was a self-educated American mathematician.
Charles Royal Johnson
Charles Royal Johnson is an American mathematician specializing in linear algebra.
Hyman Bass (born October 5, 1932) is an American mathematician, known for work in algebra and in mathematics education.
Eben Matlis (born 1923) was a mathematician known for his contributions to the theory of rings and modules, especially for his work with injective modules over commutative Noetherian rings, and his introduction of Matlis duality.
Tibor Szele (Debrecen, 21 June 1918 – Szeged, 5 April 1955) Hungarian mathematician, working in combinatorics and abstract algebra.


XXV The Roman Numerals could be written in numbers by using the roman numerals i.e. XXV = XX + = 20 + 5. = 25. The roman numerals with higher numbers precede the lower ones, which results in the proper translation of the XXV Roman numerals. The article below we’ll show you how to change XXV Roman numerals to the correct way for number translation.
- XXV = X + + V
- 10 + 10 + 5.
How do I write the XXV Roman Numerals?
Its numerical significance of the number XXV Roman Numerals can be obtained through any or both of two techniques described below:
Method #1: This method we break down all roman numerals down into one letter. We then note the numerical value of each letter, and then add or subtract the letters.
- The formula XXV = X+X +V is 10+10 +5 = 25
Method 2 With this approach we look at the roman numerals in groups to indicate subtraction as well as subtraction like
- XXV = V + XX = 5 + 20 = 25
So the numerical value of the roman numerals XXV is 25.
What are the basic rules to write Roman Numerals?
- When a letter with a bigger size precedes smaller letters then the letters are then added. For instance, LI = Li > I thus LI = I + L equals 50 + 1.
- If a smaller letter follows an older letter and the letters are subtracted, they are added. For instance, CD C is D, therefore CD = C – D = 500 100 = 400
- When a word is repeated two or three times, they are added. For instance,”XX” = X +X = 10+10 = 20
- The same letter can’t be repeated more than three times consecutively.
Numbers that are related to Roman Numerals XXV
Roman numerals were utilized in the past of Rome and were based on combinations of letters based on the Latin alphabets V, I and X. L C, D and M. It might appear to be different from numbers, however they’re alike. For instance, XXV Roman numerals are equivalent to the number 25. The roman numerals that are associated with the XXV number are listed below:
- 20×20 = XX
- 20 + 1, which is 21
- 20 + 2 = 22
- The equation XXIII + 20 = 3 equals 23.
- 20 + 4, which is 24
- 20+5 = 25
- IXXVI = 20+6 = 26
- IXXVII = 20 + 7, which is 27
- 20 + 8 = 28,
- The sum of XXIX + 20 = 9 equals 29
XXV Roman Numerals Examples
- Exemple 1: Determine the difference between XXV and VII. Solution: Roman numeral XXV is equivalent to 25 while VII is 7.
Now, the formula is: XXV – VI = 25 – 7 = 18
Since 18 = XVIII
So, XXV + VII = VII = XVIII
- Example 2. Determine the Quotient of 22 and 25.Solution:Its roman numeral, XXV means 25 while XXII is 22.
If you divide the XXV number by XXII, i.e. 25/22 the quotient will be 1.
Since 1 = I
Thus, XXV/XXII = I
- Example 3: Find the sum of the Roman numerals XXII and. Solution: The formula XXII is 20 plus 2 equals 22, and the XXV formula is 20 plus 5 equals 25.
Now, XXII +XXV is 22 plus 25 equals 47.
Since XLVII = 40 plus 7 = 47
So that’s why the total of XXII plus the roman numerals XXV is XLVII.
- Example 4. Find the product that is the product of Roman Numerals XXV and CXIX. Solution: 20 + 5 equals 25, and CXIX is 100+10 +9 = 119.
Now the formula is you have XXV x CXIX = x 2975
Because, MMCMLXXV = 2000 + 90 + 50 = 29,75
So, XXV + CXIX = MMCMLXXV
FAQs regarding XXV Roman Numerals
What is XXV Roman Numerals mean?
It is our intention to write XXV Roman numbers in an more expanded format to determine its significance. It is XXV + V = 20 + 5, which is 25. Therefore what is the significance in Roman Numerals XXV is 25.
What is the Roman numerals XXV written in numbers?
For conversion of XXV Roman Numerals into numerals, this conversion requires breaking down the Roman numerals by utilizing the place value (ones, hundreds, tens, and thousands) such as:
- Tens = 20 = XX
- Ones = 5 = V
- Number equals 25 = XXV
What is the remainder after XXV is divided by IX?
In numbers, XXV equals 25 while IX is 9. If you divide 25, by 9 you get an amount of 9. Then, 7 = 7. So, when XXV is multiplied by IX, then the remaining is VII.
Why is 25 written In Roman Numerals as XXV?
We are aware the roman numerals the 5 is written as V 10, and 10 with X. So 25 in roman numerals can be represented as: XXV + V = 20+5 = XXXV.
What should be subtracted from XXV in order to get XIX?
In the beginning, we’ll write XXV and XIX as the form of numbers. i.e. 20+5 = 25,, and 9 + 10 = 19. Now, 25 – 19 = 6. 6. Thus, VI needs to be subtracted from the roman numerals XXV to obtain XIX.
Roman Numerals – List of Roman Numbers from 1 to 1000
In math there are a variety of ways to present numbers. Roman numbers are just one of them. They employ symbols as part of an alphabetic system that is based on the Roman system. Additionally, this system allows us write numbers with smaller words or numbers, like 1000 could simply written in M. Therefore, in this section, you’ll discover the roman numerals and other related topics the roman numerals.
It is a method of numerical notation inspired by the ancient Roman system. For instance when you have numbers such as 1, 5, 10 50, 100, 1000, The symbols used are V X, C, D, and M and M, respectively. In this system, the symbol is displayed after another symbol of the same or greater value. values , such as I = 2, and LX = 60.
Additionally, a symbol that appears before one with greater value reduces its value. For instance, the IV symbol is 4, the XL value is 40 and CD = 400. The symbols are interpreted by a number of numbers: one less than or ten times fifty, and one 100 less than 5 hundred in turn. A bar that is placed over an amount multiplies the value by 1000.


Pearson’s chi-squared test for independence is used to test whether there is an association between categorical variables by seeing if there is a statistical difference between the expected counts against the observed. The test uses the aggregated counts of the categorical variables that summarize the data into a table called a contingency table. Like other hypothesis testings, the chi-squared test proves by disproof by first assuming the expected counts are identical to observed(actual counts). If the probability of the observed being equal to expected is less than some set alpha, I reject the null in favor of the alternative; the expected counts are not equal to observed; therefore, the target variable depends on the discrete independent variable.
For this blog, I will use the UCI Mushrooms dataset that can be found here. (https://archive.ics.uci.edu/ml/datasets/mushroom). The dataset contains 22 discrete features of a mushroom species that explain if the species is either edible or poisonous(target variable). The dataset needs some preprocessing to translate feature codes to actual names, the data import and preprocessing can be done as such:
The data can be summarized using the pandas cross tab helper function to create a 2x2 contingency table.
Mushroom bruising involves nicking the top and bottom of the mushroom cap and observing any colour changes. As specimens that are not fresh don’t give reliable results, it is important to do this within the first 30 minutes of picking the mushroom. — https://mushly.com/discover/how-to-identify-mushrooms-through-bruising-and-bleeding#:~:text=Mushroom%20bruising%20involves%20nicking%20the,minutes%20of%20picking%20the%20mushroom.
The objective is to test if mushrooms that are edible/not edible are dependent on the bruised condition of the mushroom. Firstly, state the null hypothesis, the bruised condition of the mushroom is independent of whether the mushroom is edible/poisonous, and the alternative, the bruised condition is not independent of the mushroom being edible. Secondly, state the alpha value of which will be used to determine the critical value. If the chi-squared statistic is greater than the critical value, the null hypothesis is rejected in favor of the alternative.
The observed counts are the aggregated counts of each variable that can be sliced using the Pandas helper function. The expected counts are the row-wise total multiplied by the column-wise total divided by the total number of observations. For example, the observed counts for bruised and edible(located on the top left of the contingency table) is 2752 and the expected counts are (3376*4208)/8124.
The chi-square statistic is the summation of the difference between observed and expected squared divided by the expected. For this example, the chi-square statistic is approximately 2037.50.
The critical value can be found here (https://www.itl.nist.gov/div898/handbook/eda/section3/eda3674.htm) by looking up the assigned alpha on the column and the degrees of freedom on the row. For an alpha of 0.05 and d.f. = 1, the critical value is 3.841.
Since the chi-squared statistic is greater than the critical value, I reject the null hypothesis that the expected counts are equal to the observed in favor of the alternative that the expected counts are not equal to the observed.
More simply, I can run a chi2 proportions test of independence using scipy.stats.chi2_contingency function. The function returns a tuple of 4 values, in the following order:
0. chi-squared statistic
2. degrees of freedom
3. expected counts.
To further analyze this dataset, I want to explore an additional independent variable, gill_spacing, with the same dependent variable, edible/inedible. A three-way contingency table is needed; however, I will hold one of the variables constant to create multi-partial tables to run multiple chi-squared tests when testing.
The idea is to take partial tables of this three-way contingency table by holding the gill_spacing feature constant. The result will be two partial tables; one for gill_spacing: Close and gill_spacing: Crowded.


Please help, 18 questions that I need answered in 50 mins. If the answers are right i will tip you great, trust me.
1. A firm is considering two location alternatives. The fixed cost at location A would be
$5 million per year, and the variable cost would be $0.30 per unit. At location B, fixed
costs would be $3 million per year, with a variable cost of $0.35 per unit. If annual
demand is expected to be 20 million units, which location offers a lower total cost?
The total cost at location A equals the total cost
at location B.
Not enough information available
2. Plant A is located at the (X, Y) coordinates of (500, 200) and has a shipment volume
of 400 units per day. Plant B is located at the (X, Y) coordinates of (200, 400) and has
the same shipment volume of 400 units per day. Using the Center-of-Gravity method,
which of the following is the X coordinate of the new plant location? Find the closest
3. A retailer decides to use a continuous review model for inventory control. The
average daily demand for dishwasher water pumps is 8 per day. The standard
deviation of the daily demand is 2 per day, and the order lead time is 4 days.
Suppose the retailer wants to achieve a service level of 98%. What is the reorder
point for the dishwasher water pumps? Find the closest value.
4. Given the following two constraints in a Linear Programming model, (1) 14X + 6Y >=
42, and (2) X - Y >= 3, which of the following points satisfies both of the constraints?
X= 7, Y =
5 X = 2, Y
X = 2, Y
X = 6, Y
=2 5. Suppose the objective function of a minimization problem is X + 3Y. What is the
objective value for X = 2 and Y = 5?
7 6. There are three equally likely states of nature (High, Medium, and Low demand).
Suppose the option "Build a large factory" will post profits of $50,000, $20,000, and $10,000, respectively, in the three states of nature. What is the EMV of the option?
Find the closest value.
7. Suppose a maximization LP model has the following four corner points: (0, 0), (10, 3),
(5, 6), and (0, 7). If the objective function is X+ 2Y, what is the optimal objective
4 8. What are the key questions to be answered by most inventory control models?
When to place an order and the cost of
When to place an order and the order size
Order size and the cost of the order
Order size and with whom the order
should be placed
9. Which of the following statements is INCORRECT regarding the level curves of the
same objective function?
Level curves can be used to represent any feasible objective
Level curves of the same objective function are always
parallel to each other.
Level curves cannot be parallel to the boundary line of a
Level curves must be straight lines on a graph for LP models. 10. Which of the following is NOT a characteristic of optimization problem?
The uncertainty in the situation has an impact on
11. Suppose a supplier offers an incremental quantity discount for a product. The
supplier charges $10 per unit for the first 1,000 units in an order, $8 per unit for the
next 1,000 units, and $7 per unit for any additional units in the order. What is the
average cost per unit of an order of 4,000 units? Find the closest value.
12. Suppose the standard deviation of demand is 8 units per week and demand is 30
units per week on average. If the order lead time is 2 weeks and the desired service
level is 97%, how much safety stock should we hold using the continuous review
model? Find the closest value.
2 13. A certain type of computer costs $400 per unit to build and the annual inventory
holding cost is 20% of the unit cost. Annual demand is 20,000 units and the setup
cost is $50 per order. What is the economic order quantity (EOQ)? Find the closest
9 14. A plant manager needs to decide whether to expand the capacity of the plant. There
are two states of nature: favorable and unfavorable. In favorable market conditions,
the plant would make an additional $40,000 when expanded. In unfavorable market
conditions, the plant would lose an additional $30,000 when expanded. Of course,
the manager can choose not to expand and make $0 additional profit in favorable
market conditions and lose $0 additional profit in unfavorable market conditions. If
the two states of nature are equally likely to occur, should the manager expand the
capacity of the plant based on the calculations of EMV?
Yes, the manager should expand the capacity of the plant.
No, it is better not to expand.
Either is fine. The two options have the same EMV.
Not enough information is available to calculate the EMVs
of the options. 15. Consider the following Linear Programming model:
Maximize 3X + 5Y
Subject to (1) 2X + Y <= 6
(2) 3X + 4Y <= 24
(3) X >= 0, Y >= 0.
What is the optimal solution? X = 0, Y
X = 3, Y
X = 0, Y
X = 8, Y
=0 16. Which of the following is specifically characterized by a focus on continuous
improvement, respect for people, and standard work practices?
Just-in-time (JIT) Toyota Production System
17. Which of the following location decision method is able to take factors such as
political stability and currency risks into account?
18. Suppose a hospital uses a periodic review model to control inventory. During the past
year, the mean daily demand for bandage BX-5 was 200 and the demand was
normally distributed. The standard deviation of the daily demand was 10. The
inventory level was reviewed once every 3 days. It takes 1 day to receive the
deliveries of the bandage after an order is placed. Suppose the required "z" value is
1.88 to ensure a satisfactory service level of 97%. Then how many bandages should
the hospital order each time? Find the closest value.
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Homogenization of transport equations in porous media
My interest is primarily in homogenization of partial differential equations describing transport phenomena in porous media. I use techniques such as volume averaging with closure and multiscale asymptotics.
Current research topics include:
- comparisons of different upscaling techniques,
- homogenization of nonlinear differential operators and inertial flow in porous media,
- higher order theories.
Collaborators: M. Quintard (University of Toulouse), B.D. Wood (Oregon State University) and F. Golfier (University of Nancy).See all my publications.


By Heather Cass, Publications Manager at Penn State Behrend
Science and the arts might seem to be very different disciplines, but the scientific method and the creative process are quite similar; inquiry is at the heart of each.
“People sometimes think science is about memorizing facts, but it’s really about making discoveries and wringing answers out of nature,” said Dr. Pam Silver, associate dean for academic affairs and distinguished professor of biology. “When you have a scientific question, it takes a lot of creativity to find the answer to it.”
Scientists are, by nature, creative individuals and the School of Science has recently added two works of art that visibly illustrate that.
Ties that bind
A colorful quilt, titled “A Way of Knowing,” was created by Silver and hangs in Hammermill Hall. Each color in the quilt represents a scientific discipline taught at Behrend—biology, chemistry, environmental science, nursing, physics, and mathematics and mathematics education. A spiral in the quilt represents the net movement of scientific discovery from observation to hypothesis to testing to understanding.
Furthermore, the underlying geometric design “symbolizes that the building blocks of science are not individual disciplines, but rather the discoveries to be made by merging diverse ideas, points of view, and approaches to form a strong and unified way of knowing with the goals of wisdom and the power to enact that wisdom,” Silver said.
“A Way of Knowing,” by Dr. Pam Silver, associate dean for academic affairs and distinguished professor of biology, hangs near the stairwell in Hammermill Hall.
Math in flight
High overhead at the entrance of Roche Hall, is another work of art—a stage-5 Sierpinski tetrahedron that models a fractal with infinite triangles—created by the School of Science Math Club under the direction of club president Thomas Galvin and Dr. Joe Previte, associate professor of mathematics.
“A fractal is a self-similar structure with recurring patterns at progressively smaller scales,” Previte said. “Fractals are useful in modeling natural structures such as plants, coastlines, or snowflakes.”
Some natural objects appear to be completely random in shape, but there is an underlying pattern that determines how these shapes are formed and what they will look like, according to Previte. Mathematics can help us to better understand the shapes of natural objects, which has applications in medicine, biology, geology, and meteorology.
Students built the fractal using Zometool construction parts. It consists of 2,050 white balls and 6,144 red and blue struts. Learn more about fractals at www.mathigon.org/world/Fractals
A stage-5 Sierpinski tetrahedron created by the School of Science Math Club hangs above the entrance to Roche Hall.


Have you in a situation where your children need much time doing math worksheet which you think is quite easy to do by them as they’ve done and practiced it many times? The worksheet example is like in the picture below!
In the worksheet above, children can actually answer some additions without having to add all of the addition one by one. Some children can do it after they do addition at least three numbers. They can predict the answer of the next addition. It’s because they figure out that the answer for the next addition is the next number in the number sequence. Hence, there are some children who can’t make the prediction. Though they have done that kind of worksheet many times. We may wonder why they can’t do that. If this happens to our children, there’s one possibility why they are not able to do it. It’s because they can’t recognize the pattern in the worksheet.
Children, who can predict the answer of the next question, realize (after they do the addition at least 3 or 4 numbers (It depends on children’s ability)) that there is pattern in the worksheet above. The pattern is the sequence of numbers on the column after plus sign. So, when they have to add one more to the the sequence of numbers, the answer is basically the sequence of numbers too. On the contrary, children who can’t predict the answer, do not realize the pattern. And this may lead to a question why they can’t see the pattern. There are some possibilities reasons for that:
- They are still lack in the ability of recognizing pattern which may be caused by the lack of chance to practice recognizing and making pattern in previous age.
- The lack of recognizing pattern of abstract thing. Number is something abstract as it actually a symbol which represent the amount of concrete things. Recognizing the pattern of symbol is more difficult than recognizing the pattern of concrete things as it requires more complex thinking process.
If the situation above happens to our children, do not force them to exercise the worksheet over and over. It’s better if we ask them to do activities which will improve their ability in recognizing pattern first. Here are the activities children can do:
- Ask children to play the toys which will make them able to practice recognizing and making pattern.
In the beginning, we ask them to follow the pattern we make. Start from the simple pattern. For example: arrange the beads with the two beads pattern consist of red & yellow beads. After our children have been able to follow the simple pattern, ask them to follow difficult pattern. Keep doing it until they are able to recognize kinds of pattern. Then, ask them to create their own patterns. This will strengthen their ability in recognizing pattern as it requires more thinking process. They have to relate the knowledge they have and combine it with the recognizing pattern ability to make their own pattern. So, they not only become a pattern follower only but a creator as well. The creating process activates more neurons in the brain. The more neurons are activated, the more brain is stimulated. And this makes children absorb and learn more.
2. Ask children to do the worksheet which requires them to guess what comes next by seeing the pattern. At first, give them worksheet which shows the concrete things such as animal, shapes, daily things, etc.
3. Ask children to do the worksheet which requires them to write symbols such as letters & numbers.
4. Ask children to do the worksheet which requires them to recognize patterns in math such as fill in the missing number worksheet, skip counting, or other math worksheets which the level difficulty is still below the addition worksheet as in the example above.
The activities above are arranged in steps from easy to difficult. Ask children to do the activities from the very basic which is very important to do. So, we can check the root of children’s lack ability. When we think our children have mastered the basic step well, that’s the time we can move on to the next step. Let them master it before continuing to another step. By mastering each steps, children won’t miss any recognizing patterns exercise.
When children’s ability in recognizing pattern is better, it will be easier for them to recognize pattern in math problem which they will see not only in addition, but further math learning such as subtraction, multiplication, division and many more. So, come on! Let’s ask our children to do many activities which will improve their ability in recognizing patterns as it will be useful in doing math problems. 🙂


Use Free Online Calculator for All Kinds of Computations
Regardless of how simple or complicated your computations are, a free online calculator can be a great utility. Mathematics is needed in our daily lives even if we are not mathematicians. It is very helpful among businessmen, financiers, work-at-home moms, and even school kids. It may be an easy job to make simple calculations, but there could be problems with complicated addition, subtraction, multiplication or division.
There are various calculations that you can do using a free online Hypotenuse Calculator. You can even find a calculator that can perform specific functions, such as those used to calculate complex mathematical problems in geometry. You can have a geometry calculator to solve the hypotenuse of a triangle. An online calculator can also be used to solve algebraic equations like the slope intercept form. Others can be used to calculate the age and height or manage financial data. A currency converter is another type that can help you to easy convert one currency value to another.
Aside from this, online calculators are also available for professionals, such as the scientific calculator with multiple functions. This can help those who are in different fields, such as in engineering, mining, security, or construction industry. It is a very valuable tool for those who are dealing with complicated equations or data. Here are the common types of online calculators that you can use.
The BMI calculator determines the body mass index (BMI) and this is very useful for obese people who are trying to reduce their weight. This globally accepted tool can help those who want to find out if they are on the right track or not. Be sure to visit this website at http://www.dictionary.com/browse/calculation and learn more about calculations.
An online loan calculator enables you to determine the total amount of monthly loan payment as well as the amount of overpayment. When you have a good calculator online, you can easily find out the figures necessary to be paid for various terms of your choice, repayment periods and interest rates. You will be able to decide what you can afford while you can compare different loans.
A currency converter is another type of online calculator that is very helpful for daily use. It is impossible to memorize exchange rates that are constantly changing. By using this free calculator online, you can now easily and accurately calculate various ratios and rates. You don't have to go to the bank or a foreign exchange company to obtain the daily currency conversions.
Designers, engineers, planners and many other professionals can benefit from an online scientific calculator. Even students in technical universities can enjoy the usefulness of this tool. It can be used to execute various mathematical and geometric calculations with just a few clicks. Things are easy online, even if you want to generate girl names.


By now you've likely seen our win probability tweets or the graphs on box score pages. Today I've pushed out a win probability calculator into which you can put any game situation & get a corresponding win probability. For instance, say it's the 4th quarter and you're trailing by 2 with the ball on your own 20 at 2nd & 10 and 29 seconds remaining, the calculator says you have a 3.8% probability of winning (don't tell the Jets that, though).
Looking at this weekend's Bears-Packers game, we can figure out the risk the Bears took in going for it on 4th and 1 on their own 32 in the 4th quarter. If they fail and the Packers take over at that spot, Green Bay will have a 55.9% win probability (ignoring that our model knows nothing about Aaron Rodgers' injury, of course). However, by succeeding Chicago raised their own win probability to 80.6%.
This entry was posted on Wednesday, November 6th, 2013 at 3:44 pm and is filed under Advanced Stats, Announcement, Pro-Football-Reference.com. You can follow any responses to this entry through the RSS 2.0 feed. Both comments and pings are currently closed.


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