If you’ve ever read a scientific paper, you’ve probably seen a statement like, “There was a significant difference between the groups (P = 0.002)” or “there was not a significant correlation between the variables (P = 0.1138),” but you may not have known exactly what those numbers actually mean. Despite their prevalence and wide-spread use, many people don’t understand what P values actually are or how to deal with the information that they give you, and understanding P values is vital if you want to be able to comprehend scientific results. Therefore, I am going to give a simple explanation of how P values work and how you should read them. I will try to avoid any complex math so that the basic concepts are easy for everyone to understand even if math isn’t your forte.
Note: for the sake of this post, I am not going to enter into the debate of frequentist (a.k.a. classical) vs. bayesian statistics. Although bayesian methods are becoming increasingly common, classical methods are still extremely prevalent in the literature and it is, therefore, important to understand them.
Hypothesis testing and the types of means
Before I can explain P values, I need to explain hypothesis testing and the difference between a sample mean and a population mean. To do this, I’ll use the example of turtles living in two ponds. Let’s say that I am interested in knowing whether or not the average (mean) size of the turtles in each pond is the same. So, I go to each pond and capture/measure several individuals. Obviously, however, I cannot capture all of the individuals at each pond (assume that there are hundreds in each pond), so instead I just collect 10 individuals from each pond. The mean carapace lengths (carapace = the top part of a turtle’s shell) from my samples are: pond 1 = 20.1 cm, pond 2 = 18.4 cm.
Now, the all important question is: can I conclude that on average, the turtles in pond 1 are larger than the turtles in pond 2? Maybe. The problem is that we don’t know whether or not my sample was actually representative of all of the turtles in the ponds. In other words, those numbers (20.1 cm and 18.4 cm) are sample means. They are the average values of my samples, and the sample means are clearly different from each other, but that doesn’t actually tell us much. It is entirely possible that, just by chance, I happened to capture more large turtles in pond 1 than pond 2.
To put this another way, we are actually interested in the population means. In other words, we want to know whether or not the mean for all of the turtles in pond 1 is the same as the mean of all of the turtles in pond 2, but since we can’t actually measure all of the turtles in each pond, we have to use our sample means to make an inference about the population means. This is where statistical testing and P values come in. The tests are designed to take our sample size, sample means, and sample variances (i.e., how much variation is in each set of samples) and use those numbers to tell us whether or not we should conclude that the difference between our sample means represents an actual difference between the population means.
For these statistical tests, we usually have two hypotheses: a null hypothesis and an alternative hypothesis. The null hypothesis states that there is no difference/relationship. So in our example, the null hypothesis says that the population means are not different from each other. Similarly, if we were looking for correlations between two variables, the null hypothesis would state that the variables are not correlated. In contrast, the alternative hypothesis says that the population means are different, the variables are correlated, etc.
The P value
Now that you understand the difference between the types of means and types of hypotheses, we can talk about the P value itself. For my fictional turtle example, the appropriate statistical test is the Student’s T-test (note: I got the values of 20.1 cm and 18.4 cm by using a statistical program to simulate two identical populations and randomly select 10 individuals from each population). When I ran a T-test on those data, I got P = 0.0597, but what does that mean? This is where things get a bit tricky. Despite what some people will erroneously tell you, the P value is not, “the probability that you are correct” or “the probability that the difference is real.” Rather, the P value is the probability of getting a result of your observed difference/correlation strength or greater if the null hypothesis is actually true. So, in our example, the difference between our sample means was 1.7 cm (20.1-18.4 cm) and the null hypothesis was that the population means are identical. So, a P value of 0.0597 means that if the populations means are identical, we will get a difference of 1.7 cm or great 5.97% of the time (to turn a decimal probability into a percent just multiply by 100). Similarly, for a correlation test, the P value tells you the probability of getting a correlation as strong or stronger than the one that you got if the variables actually aren’t correlated.
To demonstrate that this works, I took the same simulated ponds that I sampled the first time, and I made 10,000 random samples of 10 individuals from each population. For each sample, I calculated the difference between the sample means for pond 1 and pond 2, resulting in Figure 1. Out of 10,000 samples, 525 had a difference of 1.7 or greater. To put that another way, the two population means were identical and just by chance I got our observed difference or greater 5.25% of the time, which is extremely close to the calculated value of 5.97% (because the sampling was random, you wouldn’t expect the numbers to match perfectly).
When you look at Figure 1, you may notice something peculiar: I included both differences of =/> 1.7 and =/< -1.7 in my 525 samples. Why did I include the negatives? The answer is that our initial question was simply “are the turtles in these ponds different?” In other words, our null hypothesis was “there is no difference in the population means” and our alternative was “there is a difference in the population means.” We never specified the direction of the difference (i.e., our question was not, “are turtles in pond 1 larger than turtles in pond 2?”). A non-directional question like that results in a two-tailed test. In other words, because we did not specify the direction of the difference, we were testing a difference of the size 1.7 rather than testing the notion that pond 1 is 1.7 larger than pond 2.
You can do a one-tailed test in which you are only interested in differences in one direction, but there are two important things to note about that. First, your hypotheses are different. If, for example, you want to test the idea that turtles in pond 1 are larger than turtles in pond 2, then your null is, “the population mean of turtles in pond 1 is not larger than the population mean of turtles in pond 2” and your alternative is, “the population mean of turtles in pond 1 is larger than the population mean of turtles in pond 2.” Notice, this does not say anything about the reverse direction. In other words, your null is not that the means are equal, so a result that pond 2 is greater than pond 1 would still be within the null hypothesis and would not be considered statistically significant.
Second, if you are going to do a one-tailed test, you have to decide that you are going to do that before you collect the data. It is completely inappropriate to decide to do a one-tailed test after you have collected your data because it artificially lowers your P value by ignoring one half of the probability distribution. Look at the bell curve in Figure 1 again. You can see that just by chance you expect to get a result of +/-1.7 or greater 5.25% of the time, but if you ignore the differences on the negative side of the distribution (Figure 2), then suddenly you are looking at a probability of 3.95% because if the null hypothesis is true, getting a difference of =/> 1.7 is less likely than getting a difference that is either =/> 1.7 or =/< -1.7 (typically, one-tailed values are half of the two-tailed value, but because of chance variation, this sample came out with a slight negative bias). If you had a good biological reason for thinking that pond 1 would be greater than pond 2 before you started then you could and should use the one-tailed test because it is more powerful, but you can’t decide to use it after looking at your data because that makes your result look more certain that it actually is (this is something to watch out for in pseudoscientific papers).
What does statistical significance mean
At this point, I have explained what P values mean in technical terms, but the question remains, what do they mean in practical terms? In our example, we got a P value of 0.0597, but what does that actually mean? In short, we use various cut off points (known as alpha [α]) to determine whether or not the P value is “statistically significant.” What α you use depends on your field and question, but it always has to be defined before the start of your experiment. In biology, α = 0.05 is standard, but other fields use 0.1 or 0.01. If your P value is less than your α, you reject the null hypothesis, and if your P value is equal to or greater than your α, you fail to reject the null. In other words, if your α = 0.05 and your P value is less than that, your conclusion would be that the observed difference between your sample means probably represents a true difference between your population means rather than just chance variation in your sampling. Conversely, if your P value was 0.05 or greater, you would conclude that there was insufficient evidence to support the conclusion that the differences between the sample means represented a real difference between the population means. This is not the same thing as concluding that there is no difference between the population means (more on that later).
It should be clear by this point that we are dealing with probabilities, not proofs. In other words, we are reaching conclusions about what is most likely true, not what is definitely true. The astute reader will realize that if a P value of 0.049 means that you will get that result by chance 4.9% of the time if the null hypothesis is actually true, then for every 20 tests with a P value of 0.049, one of them will be a false result (on average). This is what we refer to as a type I error. It occurs when you reject the null, but should have actually failed to the reject the null, and it is the reason that we like to have small α values: the larger the α, the higher the type I error rate (I explained type I errors in far more detail here). This also explains why some published results are wrong, even if the authors did everything correctly, and it once again demonstrates the importance of looking at a body of literature rather than an individual study.
Now, you may be thinking that we should try to make the α values very tiny, that way we rarely get false positives, but that creates the opposite problem. If the α is tiny, then there will be many meaningful differences which get ignored (this is known as a type II error). Thus, the standard α of 0.05 is a balance between type I and type II error rates.
Statistical significance and biological significance are not the same thing
This is an extremely important point that is true regardless of whether or not you got a statistically significant result. For example, let’s say that chemical X is dangerous at a dose of 0.5 mg/kg, and you do a study comparing people who take pharmaceutical Y to people who don’t, and you find that people who take Y have an average of 0.2 mg/kg and people who don’t take Y have an average of 0.1 mg/kg and the difference is statistically significant. That doesn’t in anyway shape or form show that Y is dangerous because the levels of X are still lower than 0.5 mg/kg. In other words, the fact that you got a significant difference does not automatically mean that you found something that is biologically relevant. The different levels of X may actually have no impacts whatsoever on the patients.
Conversely, if you did not get a significant difference, that would not automatically mean that there isn’t a meaningful difference. Statistical power (i.e., the ability to detect significant differences/relationships) is very strongly dependent on sample size. The larger the sample size, the greater the power. Consequently, if the population means are very different from each other, then you can get a significant result with a small sample size, but if the population means are very similar, then you are going to need a very large sample size. This is part of why you fail to reject the null, rather than accepting the null. There may be an actual difference between your population means, but you just didn’t have a large enough sample size to detect it. For example, let’s say that a drug causes a serious reaction in 5 out of every 1000 people and that “reaction” already occurs in 1 out of ever 1000 people (those are the population ratios), but when you test the drug, you only use sample sizes of 1000 people in the control group and 1000 people in the experimental group, resulting in sample ratios of 6/1000 and 1/1000. When you run that through a statistical test (in this case a Fisher’s exact test) you get P = 0.1243. So, you would fail to reject the null hypothesis even though the drug actually does cause the reaction that you were testing. In other words, the drug does cause adverse reactions, but your sample size was too small to detect it. If, however, your sample sizes had included 2000 people in each group, and you had gotten the same ratios, you would have had a significant difference (P = 0.0128) because those extra samples increased the power of your test. This is why scientists place so much weight on large sample sizes and so little weight on small sample sizes. Research that uses tiny sample sizes is extremely unreliable and should always be viewed with caution.
Finally, it is worth noting that the population means of any two groups will nearly always be different, but that difference may not be meaningful. Going back to my turtle example, for any two ponds, if I measured all of the turtles in each pond, it is extremely unlikely that the two means would be identical. There is almost always going to be some natural variation that makes them slightly different, but, with a large enough sample size, you can detect even a very tiny difference. So, for example, if I had two turtle ponds whose population means were 18.01 cm and 18.02 cm, and I had several million turtles from each pond, I could actually find that there is a statistically significant difference between those ponds, even though that actual difference is extremely tiny and is not a meaningful difference between the two ponds. My point is simply that the fact that a study found a statistically significant result does not automatically mean that they found a meaningful result, so you should take a good hard look at their data before drawing any conclusions.
What do you do with a non-significant result?
The question of what to do with non-significant results is a complicated one, and it is probably the area where most people mess up. For various reasons (some of which I discussed above) you’re never supposed to accept the null hypothesis, rather, you fail to reject it. In other words, you simply say that you did not detect a difference rather than saying that there is no difference (in reality there is nearly always a difference, it just might not be a meaningful one). In practice, however, there are many situations in which you have to act as if you are accepting the null hypothesis. For example, let’s say that you are comparing two methods, one of which is well established but expensive, while the other is untried and cheap. You do a large study and you don’t find any significant differences between those two methods. As a result, scientists will begin using the cheap method and they will cite your paper as evidence that it is just as good as the expensive method.
Drug trials present a similar dilemma. Let’s say that we are trying a new drug and we find that there are no significant differences in side effect X between people who take it and don’t take it. The FDA, doctors, and general public will treat that result as, “the drug does not cause X” which is essentially accepting the null.
So how do we solve this problem? Do all drug trials violate a basic rule of statistics? No, the key here is sample size and statistical power. Remember from the section above that nearly all real population means will be different, but the difference may be very slight and not meaningful. So, when we accept that a novel method is as effective as the old, for example, we aren’t actually saying that there are no differences between the two. Rather, we are saying that there are probably no differences at the effect size that we were testing. To put this another way, we would say that the current evidence does not support the conclusion that they are different.
This may seem confusing, but you can think of it like a jury decision. We don’t declare someone “guilty” or “innocent.” Rather, we declare them “guilt” or “not guilty.” The “not guilty” verdict is essentially the same thing as failing to reject the null. It doesn’t mean that the person definitely didn’t commit the crime. Rather, it simply means that we do not have the evidence to conclude that they did commit the crime; therefore, we are going to treat them as if they didn’t.
Jumping back to science, ideally you should do something called a power analysis. This shows you what size difference you would be able to detect given your sample sizes, variance, and the level of power that you are interested in. So, let’s say that during the methods comparison test, anything less than a 0.01 difference between the two methods would be good enough to consider the new one reliable, and you had the statistical power to detect a difference of 0.001. That would mean that although there may be some very tiny difference between the methods, that difference is less than a difference that you would care about, and you had the power to detect meaningful differences. Similarly, if you are doing a drug trial and you have the power to detect a side effect rate of in 1 out of every 10,000 people, then you cannot conclude that the drug doesn’t cause that side effect, but you can say that if it does cause that side effect, it probably does so at a rate of less than 1 in 10,000 (note [added1-1-16]: as a general rule, power analysis should be done before conducting the study in order to determine what sample size will be necessary to detect a desired effect size).
All of this connects back to the importance of sample size. If you have a small sample size, then you won’t be able to detect small differences. Let’s say, for example, that a drug trial found improvements in 40 out of 100 people in the control group and 50 out of 100 people in the experimental group. That would result in a P value of 0.2008, which is not statistically significant, but that test would not have much power. As a consequence, that result is not very helpful. It could be that the drug simply doesn’t work, but it could also be that it does have an important effect, and this study just didn’t have a big enough sample size to detect it. Therefore, I am personally very hesitant to use results like this as evidence one way or other, and I think that when you have results like this, it is best to wait for more evidence before you try to say that there are no meaningful differences.
Some people, however, fall for the opposite pitfall. On several occasions, I have encountered people who look at studies with small samples sizes like this and say, “there isn’t enough power to actually test for differences, therefore we should just go with the raw numbers and assume that there is a difference.” This is completely, 100% invalid. Think back to the very start of this post, the whole reason that we do statistics is because without them we can’t tell if a result is real or just the result of chance variation. So you absolutely cannot blindly assume that the difference is real. If you don’t have enough power to do a meaningful test, then you simply cannot draw any conclusions.
This post ended up being quite a bit longer than planned, so let me try to sum things up with a bullet list of key points (note: for simplicity, I will talk about means, but the same is true for proportions, correlations, etc. so you can replace “no difference between he means” with “no relationship between the variables,” “no difference between the proportions,” etc.).
- In science, you nearly always sample subsets of the total groups that you are interested in (these are sample means)
- The means of those subsets will nearly always be different, but you actually want to know whether or not the means of the entire groups are different (these are the population means)
- You have two hypotheses. The null says that there is no difference between the population means, and the alternative says that there is a difference between the population means
- The P value is the probability of getting a result where the difference between the sample means is equal to or greater than the difference that you observed, if the null hypothesis is actually true
- The larger your sample size, the greater your ability to detect differences
- If you get a statistically significant result, you reject the null hypothesis; whereas, if you don’t get a significant result, you fail to reject the null hypothesis
- Statistical significance is not the same as biological significance
- Nearly all population means will be different from each other, but that difference may not be meaningful. Therefore, you cannot conclude that no difference exists, but you can provide evidence that if a difference exists, it is very small (or at least smaller than the effect size that you were testing)
Other posts on statistics:
- Basic Statistics Part 1: The Law of Large Numbers
- Basic Statistics Part 2: Correlation vs. Causation
- Basic Statistics Part 3: The Dangers of Large Data Sets: A Tale of P values, Error Rates, and Bonferroni Corrections
Very nice description. Should be worth 3 CE units at least. You may want to capitalize the word “student’s” in “student’s T Test” lest someone think you’re talking about an education-level tool, instead of the inventor’s (pseudo) name.
I am slightly unclear if you are advocating a post-hoc power analysis. It should be stressed that power analysis should not be undertaken post-hoc of a test which failed to reject the null. This is pernicious, incorrect but strangely popular belief amongst biologists and psychologists. The reason it does not work is that any frequentist test that fails to reject the null will always have low power. Power should only be calculated a priori to further analysis perhaps after a pilot study or in planning a study. It does not work as a post-hoc. Confidence intervals are a post-hoc way of looking at practical significance (perhaps adjusted for multiple comparisons if necessary).
As a general rule, I agree with you and should have made that more clear. I have added a line to attempt to clarify.


Submitted by Sarah Reinecker
The York Suburban Education Foundation (YSEF) has received a $5,000 donation from GIANT Food Stores. This contribution is through the Pennsylvania Educational Improvement Tax Credit (EITC) program. One of the many programs funded by YSEF with EITC contributions is the summer camp enrichment program Fizz, Boom, Read. In 2015 and 2016, this grant aimed to reinforce reading, writing, math and science skills for rising 1st-5th grade students through teacher led activities. STEAM (Science, Technology, Engineering, the Arts, and Math) concepts will be incorporated to guide student inquiry, dialogue, and critical thinking. Pictured are students from Indian Rock who represent a sampling of the age range of students who benefitted from Fizz, Boom, Read summer camp. YSEF thanks GIANT Food Stores for helping York Suburban teachers do even more for YS students!


[Extracts from The Pythagorean Perspective]
Literature, music, mathematics, art, are constituents of culture and each of them has its separate history. But each of them can also be seen as a manifestation of a human biological drive, a drive towards exploration, experimentation, the analysis of human perception.
Is mathematics an art? Can it reasonably be treated along with music, poetry and the visual arts? Can anything be said about its evolution? ... To answer these questions we perhaps need to have some clearer idea of what mathematics currently is, as a body of knowledge and as an activity. Mathematics can be defined, its content described and the way mathematicians approach their work examined. The objects with which mathematics deals can be stated, the emphasis on proof as characteristic of the mathematical process considered and then larger questions dealt with about the foundations and evolution of mathematics.
A contemporary definition is that mathematics is the science of pattern and deductive structure (replacing an older definition of mathematics as the science of quantity and space). The extent of mathematics is indicated by Davis and Hersh`s estimates that mathematics is contained in 100,000 volumes; that a skilled mathematician might know 10%, that there are 3000 categories of mathematical writing and that some 200,000 theorems are published every year. Mathematics is generally thought to place major emphasis on proof and to be concerned with a variety of mathematical objects such as circles, polyhedrons etc. Some professional mathematicians think the emphasis on strict proof is a mystification. G.H. Hardy, one of the most eminent pure mathematicians, commented: "There is strictly no such thing as mathematical proof; proofs are what Littlewood and I call gas, rhetorical flourishes, devices to stimulate the imaginations of pupils". How mathematicians see their activity is often equally surprising. Quoting G.H. Hardy again: "I have never done anything useful. The highest aspiration in mathematics is to achieve a lasting work of art". Poincaré also emphasised the aesthetic rather than the logical aspect of mathematics. Davis and Hersh (from whom the Hardy quotations are drawn 1983: 29, 85, 173). suggest that mathematics is a great art form distinguished from other humanities only by its science-like quality, its conclusions not subject to permanent disagreement like ideas of literary criticism. The meaning of mathematics is to be found in the shared understanding of human beings, not in any external nonhuman reality. The object of mathematics is to extract structure and invariance from apparent disorder. A mathematician, like a painter or a poet, is "a master of pattern" (Hardy). In their creative work, mathematicians rely, not on language or logic but on a semi-conscious stream of analogic thought, visual, or sometimes even musical (Hadamard). Views about the nature and foundations of mathematics are in process of change. Since the Greeks, mathematics has been seen as the `Queen of the Sciences', the only sure source of truth. Recent books such as those of Davis and Hersh, Lakatos,and Ormell contend that formalistic accounts of the nature of mathematics have had their day. Instead, mathematics is to be thought of as human-made, connected with the rest of knowledge, and just as much a cultural product as literature and music. With this change it starts to make sense to consider the evolution of mathematics in parallel with the evolution of other aspects of culture such as music, the visual arts and poetry.
There have long been speculations, both in philosophy and psychology, about the origin of mathematical knowledge and numerical concepts. The evolutionary view is that mathematics began with the study of real things, and was a product of the evolution of the brain, along with language and other aspects of culture. Mathematics is a form of perception which draws its unique power from the narrowness of the range of phenomena with which it is concerned, number, shape, pattern. At first sight, one might ask, with Davis and Hersh: How can man impose his mathematical will on the great cosmic processes? This is parallel to the problem considered by Lorenz how Kant`s categories could relate to the real world, to which he answered that our cognitive apparatus evolved over time precisely in such a way as to make its operations valid, effective in relation to the real world. Mathematics has evolved as a symbolic counterpart of the universe. Certain mathematical concepts and procedures have survived and others have been abandoned, survival of the fittest symbols, models, processes, constructs.
Mathematics began when the perception of three apples was freed from apples and became the integer three. There seems something specially natural about arithmetic. There are no preliminary axioms; arithmetic furnishes us with experimentally verifiable facts about the world. Quite young children can learn or discover basic arithmetic and even animals have some awareness, as experiments have shown with rats, canaries, parrots, raccoons, pigeons, and chimpanzees. Infants as young as 2.5 months old showed elementary addition abilities (1 + 1 = 2) as did rats and chimpanzees. Infants were able to recognise a group of three objects as such without counting. Wynn (1992), from whom the material in this paragraph is drawn, concludes that such abilities in a wide range of species and at a very early age in human infancy suggest that the knowledge is innate. This fits rather neatly with the long-disputed ideas of Brouwer on intuition as the foundation of mathematics. Brouwer's position was that the natural numbers are given to us by a fundamental intuition which is the starting point for all mathematics. Wynn comments moderately that it seems reasonable that our initial numerical knowledge somehow serves as a basis for the development of mathematics. For Kant, the truths of geometry and arithmetic are forced on us by the way our minds work but, following Lorenz, the way our minds work is the product of evolution of our cognitive structures to match the real structures of the environment in which our ancestors had to survive. Mathematical perception is a specialised segment of perception, which, along with other forms of perception, evolved in the service of survival. Davis and Hersh quote (1993: 319) Rene Thom: "I don't see why we should have any less confidence in this kind of perception, i.e. in mathematical intuition, than in sense perception"; both may represent an aspect of objective reality.
In summary, one can understand The Unreasonable Effectiveness of Mathematics in the Natural Sciences (title of an article by Wigner) if one accepts that there must be a structure in the universe and this structure is what is studied by every science, including mathematics, and determines the shape of mathematics. (Ormell) And one can add that mathematics as a specialised form of perception can legitimately be considered alongside music, the visual arts and poetry which are forms of representation of the contents of perception of rather more complicated, many-factored, aspects of the world and of the individual human being`s existence. Mathematics is another form, besides these other arts, which exploration of human presence in the universe can take. Like other arts, the prerequisites for mathematical exploration are to be found in the structure of the brain.
All the arts can be seen as a manifestation of one and the same impulse. ... The basic biological drive, impelling composers to compose, artists to paint, poets to write and even scientists to do their science is towards exploration of the perceived world (the outer world and the inner world) and the attempt to replicate it in some durable form. In a sense artists and musicians are engaged in exploration of the properties of the eye and exploration of the properties of the ear. ... We are perceiving creatures set in a multi-sensory world who have acquired an awareness of our perception and even perception of our awareness of our perception. We have a drive to the externalisation of our perception, of our awareness. Externalisation is transduction - transfer from one (neurological, physiological, cerebral) system to another - conversion of simultaneous patterning into time-patterning or extended experience into immediate unextended structure. The ability to transduce in this way must depend on cross-modal connectivity in the brain. On this view the production of art is a process of cross-modal transfer, for example, from neural emotional patterning to music, painting, the words of a poem.
Mathematics is no exception. It is not exceptional in the reliability of the information it provides about the world - visual and other forms of perception also produce reliable forms of information (verified by action). Mathematics is exceptional only in the narrowness of the category of information with which it deals. The mathematician's art materials are number and shape and the interrelation of number and shape. In the case of other forms of art, other forms of perception, the information is many-faceted, dependent upon the point-of-view, not susceptible to instrumental measurement. Between artistic and scientific perception there is no essential difference. As Hayek (1952: 165) suggests: "The apparatus by means of which we learn about the external world is itself the product of a kind of experience. We cannot regard the phenomenal world in any sense as `more real' than the constructions of science" - or, I would add, any less real.


The message reads (The tweet has been elaborated here as there is no 140 character constraint)
Take any 3 digit number, difference between first digit and last digit should be greater than 1. Reverse the digits. Find the difference. Reverse the digits in this number (difference). Add them to get 1089 always.For example, let the number as 109, reversing the digits gives 901, difference between them 901 - 109 = 792, reversing the digits in the difference gives 297, adding them gives 792 + 297 = 1089.
This is not something that I discovered, I just know about this property through some source long time back. But I will attempt to show why this happens.
Let the random 3 digit number be abc. Its value is 100a + 10b + c
Reversing the digits gives cba. Its value 100c + 10b + a.
Difference between them assuming that a > c (for sake of simplicity of analysis)
|abc - cba| = abc - cba = 100a + 10b + c - (100c +10b + a) = 99 x (a-c)
Let, |abc - cba| = xyz (another 3 digit number)
xyz, then is a multiple of 99 i.e. of both 11 and 9.
Properties of multiples
- In multiples of 9, the sum of digits is divisible by 9
- In multiples of 11, the difference between sum of all odd and even digits is divisible by 11
In this case,
Since xyz is divisible by 9, x + y + z is divisible by 9 or x + y + z = 9 or 18. It can't be 27 or more.
Since xyz is divisible by 11, x + z - y = 0 or 11. It can't be 22 or more.
If x + z - y = 0, then x + z = y, then x + y + z = 2y
2y cannot be 9, it can be 18, so 2y = 18 or y = 9
If x + z - y = 0, then x + z = y or x + z = 9, this is possible
If x + z - y = 11, then x + z = 11 + y or x + z = 20, this is not possible
So, we can conclude that, y = 9 and x + z = 9.
Reversing the digits in xyz, we get zyx.
Adding the 2 numbers, xyz + zyx = 100x + 10y + z + 100z + 10y + x
Total = 101(x + z) + 20y
Substituting for x + z = 9 and y = 9, we get,
Total = 101 x 9 + 20 x 9 = 909 + 180 = 1089!
By the way, 33 square is 1089. 99 x 11 is 1089 and 121 x 9 is 1089.


Fractions + Pizza =
We love Math
The third grade students have been studying fractions. They had a hands-on activity to show their understanding of how equal pieces add up to a whole and as a culminating activity to wrap up the unit, they made their own individual pizzas.
Ms. Franklin & Mr. Lepage
We pretended we were at a bug party and used cereal to help us understand parts and wholes in fractions! We also learned about numerators and denominators this way!
Ms. Georgiev & Ms. Mello


This course explores the practical usages of the Geogebra application in teaching. Skills for creating visually interactive user-friendly applets will be developed. Students learn about underlying programming structures/commands and methods of construction -then work to create an online portfolio of demonstrations, games and interactive challenge sets-which will help them to expand the curriculum.
I am a high school math teacher of almost 20 years experience who was introduced to the Geogebra Application about ten years ago. I had always wanted to make my curriculum more dynamic but up to that point never knew how. To that point in my career, all of my curriculum involved static images. So when I learned to use the Geogebra application, Curriculum Development became exciting. I thought with excitement --Is it possible to do this?! --I wonder if you could do that?! I was (and am) convinced that a dynamic creation which allows for manipulation of geometric figures and provides automatic feedback is superior to one that doesn't. As a long time Special Needs teacher I started off by creating Dynamic Demonstrations and then realised that Games would be a great way for students to practice skills. Over the past few years, (in my work with Regular Ed. & Honours level students) I have seen how Geogebra can easily challenge students with their logic and reasoning, with their ability to create/construct and with their ability to problem solve. Besides improving the curriculum, the program has made me a better teacher as it has helped me to see the content in new ways. The questions I ask are now more complex than they ever were before. I am convinced that jumping in to learn the Geogebra Application was the smartest move I ever made as a teacher and I am excited by the opportunity to help other teachers learn the program.
Frequently Asked Questions
Are you looking for new ways to captivate your students' attention?
When you are able to use Geogebra, the sky is the limit as to how dynamic and visually interactive your curriculum will be!


Enriched ∞-categories and Extended TFTs
While (weak) n-categories and (∞,n)-categories are of central importance in the study of extended topological field theories, they are notoriously hard to even define, let only work with. However, their description via iterated weak enrichment can be used to prove a surprising amount of statements from general principles. Currently, I am using the machinery of enriched ∞-categories to construct and characterize lax idempotents, also known as condensations, in them. This and related constructions should prove useful to construct and classify (derived) Extended TFTs.


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History of Mathematics - The Indian Contribution
The Contribution to Mathematics by
1. Zero and the place-value notation for numbers
2. Vedic Mathematics and arithmetical operations
3. Geometry of the Sulba Sutras
4. Jaina contribution to Fundamentals of numbers
5. The anonymous Bakshali manuscript
7. Classical contribution to Indeterminate Equations and Algebra
8. Indian Trigonometry
9. Kerala contribution to Infinite Series and Calculus.
10. Modern Contribution: Srinivasa Ramanujan onwards
Section 1: Zero and the Place-Value Notation
The number zero is the subtle gift of the Hindus of antiquity
to mankind. The concept itself was one of the most significant inventions in
the ascent of Man for the growth of
culture and civilization. To it must be credited the enormous usefulness of its counterpart,
the place value system of expressing all numbers with just ten symbols. And to
these two concepts we owe all the arithmetic and mathematics based upon them,
the great ease which it has lent to all computations for two millenia and the
binary system which now lies at the
foundation of communicating with computers. Already in the first three
centuries A.D.. the Hindu ancients were using a decimal positional system, that
is, a system in which numerals in different positions represent different
numbers and in which one of the ten symbols used was a fully functional zero. They called it 'Sunya'. The word and its meaning ‘void’ were obviously
borrowed from its use in philosophical literature. Though the Babylonians used
a special symbol for zero as early as the 3rd century B.C. , they
used it only as a place holder; they did not have the concept of zero as an
actual value. It appears the Maya civilisation of
1. A notation for powers of 10 upto the power 17 was already in use even from vedic times. Single words have been used to denote the powers of the number 10. The numbers one, ten, hundred, thousand, ten thousand, … are given by the sequence of words in the list: eka, dasa, Sata, sahasra, ayuta, laksha, prayuta, koTi, arbuda, abja, kharva, nikharva, mahA-padma, Sankha, jaladhi, antya, mahASankha, parArdha. Thus the decimal system was in the culture even in the early part of the first millenium B.C. . The Yajurveda, in its description of rituals and the mantras employed therein, the Mahabharata and the Ramayanaa in their descriptions of statistics and measurements, used all these words, with total abandon.
2. Counting boards with columns representing units and tens were in use from very ancient times. The numberless content of an empty column in course of time was symbolized to be ‘nothing’.
3. The thriving activity in astrology, astronomy, navigation and business during the first few centuries A.D. naturally looked forward to a superior numerical system that lent itself to complicated calculations.
4. Distinct symbols for the numbers 1 to 9 already existed and the counting system used the base 10 in all its secular, religious and ritual activities. Compare this with the Babylonian numeration which had only three figures, one for 1, one for 10, and one for 100, so that a number, say, 999, would require 27 symbols, namely, nine of each of the symbols.
Of these, the first and fourth factors are probably unique to Hindu culture and
contributed most to the thought process that led to the decimal place value notation as well as zero having a value. When exactly the invention of this most modest of all numerals
took place, we do not know. The first time it reached Europe was during the
Moorish invasion of
How the Sunya
of the Hindus became the Zero of the modern world is interesting. The 'Sunya'
of Sanskrit became the Arabic ‘sifr’ which means empty space. In
medieval Latin it manifested as ‘ciphra’ , then in
middle English as ‘siphre’, in English as ‘cypher’ and in American as ‘cipher’. In the middle ages, the word ‘ciphra’ evolved to stand for the whole
system. In the wake of this general
meaning, the Latin ‘zephirum’ came to be used to denote the Sunya.
And that entered English finally as ‘zero’.
Section 2. Vedic Mathematics and arithmetical operations
Vedic Mathematics provides an original and refreshing approach to subjects which are usually dismissed as mechanical and tedious. Bharati Krishna Tirtha who published his reconstruction of Vedic Mathematics in 1965, maintains that there are 16 aphorisms and 13 secondary aphorisms which forms his base of the so-called Vedic Mathematics. Though the origins of Vedic Mathematics have not yet been historically established, if nothing else, it provides tremendous insights into the place-value system of numbers without which it would not work. It is amazing that Vedic Mathematics does not require of cramming of multiplication tables beyond 5 x 5. One can improvise all the necessary multiplication tables for oneself and with the aid of the relevant Vedic formulae get the required products very easily, speedily, and correctly, almost immediately. The formulae can be used to evaluate determinants, solve simultaneous linear equations, evaluate logarithms and exponentials. Vedic Mathematics recognises that any algebraic polynomial may be expressed in terms of a positional notation without specifying the base. The same algorithmic scheme as applied to arithmetical operations will easily apply to algebraic problems. And this brings it to the Modern Algebra of Polynomials. It is difficult, in a historical introduction like this to get into the details of Vedic Mathematics. Suffice it to say that with today's over-dependence on calculators for even simple arithmetical computations, the Vedic methods have great pedagogical value and, through their revival, the skills of mental arithmetic may not be lost for posterity.
Section 3. Geometry of the Sulba Sutras
Hailing from the times of the Vedas, the ritual literature which gave directions for constructing sacrificial fires at different times of the year dealt with the their measurement and construction in a systematic and logical way, thus giving rise to the Sulba Sutras. The construction of altars (vedi) and the location of sacrificial fires had to conform to clearly laid down instructions about their shapes and areas in order that they may be effective instruments of sacrifice. The Sulba Sutras provide such instructions for two types of ritual - one for worship at home and one for communal worship. The instructions were mainly for the benefit of craftsmen laying out and building the altars. Bodhayana, Apastamba and Katyayana who have recorded these Sulbasutras were not only priests in the conventional sense but must have been craftsmen themselves. The earliest of them, The Bodhayana Sutras , in three chapters, (800 - 600 B.C.) contains a general statement of the Pythagorean theorem, an approximation procedure for obtaining the square root of two correct to five decimal places and a number of geometric constructions. These latter include an approximate squaring the circle, and construction of rectilinear shapes whose area is equal to the sum or difference of areas of other shapes. The Bodhayana version of the Pythagorean theorem sates as follows:
The rope which is stretched across the diagonal of a square
produces an area double the size of the original square.
It is therefore in the fitness of things that the Pythagorean theorem of Mathematics may be renamed as the Bodhayana theorem.! The other sutras are two centuries later but all of them are prior to Panini of the fourth century B.C. The geometry arising from these sutras give several geometric constructions. Some of these are:
1. To merge two equal or unequal squares to obtain a third square.
2. To transform a rectangle into a square of equal area
3. Squaring a circle and circling a square (approximately)
A remarkable achievement was the discovery of a procedure for evaluating square roots to a high degree of approximation. The square root of two is obtained as
the true value being 1.414213… . The fact that such procedures were used successfully by the Sulbasutra geometers to operations with other irrational numbers, is clear proof for negating the western-held opinion that the Sulba sutra geometers borrowed their methods from the Babylonians. The latter's calculation of the square root of two is an isolated instance and further they used the sexagesimal notation for numbers. The achievement of geometrical constructs in Indian mathematics reached its peak later when they arrived at the construction of Sriyantra, which is a complicated diagram, consisting of nine interwoven isosceles triangles, four pointing upwards and four pointing downwards. The triangles are arranged in such a way that they produce 43 subsidiary triangles, at the centre of the smallest of which there is a big dot called the bindu. The difficult problem is to construct the diagram in such a way that all the intersections are correct and the vertices of the largest triangles fall on the circumference of the enclosing circle. In all cases the base angles of the largest triangles is about 51.5 degrees. This has connections with the two most famous irrational numbers of Mathematics, namely p and f. The quantity f, called the golden ratio, has remarkable mathematical properties and is almost a semi-mystical number.
Section 4. Jaina contribution to Fundamentals of Numbers
By the time of the Jains, the role of rituals in the development of mathematics declined and mathematics began to be pursued also for its own sake. The Jains had a fascination for large numbers. Their definitions of the various types of infinities they comprehended are sophisticated, though lacking in mathematical precision. But it must be said to their credit that they were the first, in the chronology of scientific thinking, to have recognised that all infinities were not the same or equal. In fact this idea was established in the mathematical world only in the latter half of the nineteenth century when Cantor initiated his theory of sets.
The Jains were also aware of the theory of indices, though they did not have the modern notation or any convenient notation for the same. Calling the successive squares and square roots as the first, the second, etc. they make the following statement: The first square root multiplied by the second square root is the cube of the second square root. In modern notation this is nothing but the identity in the theory of indices:
a1/2 x a1/4 = (a1/4 )3
They have several such rules for working with powers of a number. They also seem to have had an idea of the logarithm of a number though they don't seem to have put them to practical use in calculation. Another favourite topic with them was the study of permutations and combinations. They had also a great interest in sequences and progressions developed out of their philosophical theory of cosmological structures. A Jain canonical text entitled Triloka prajnApati has a very detailed treatment of arithmetic progressions.
Section 5: The Anonymous Bakshali Manuscript
This manuscript was
discovered in 1881 A.D. near a village called Bakshali. It is written in an old
form of Sanskrit on seventy leaves of birch bark. It is probably a copy of a manuscript
composed in the early centuries A.D. It
is a handbook of rules and illustrative examples together with solutions, all
mainly on arithmetic and algebra. Fractions,
Square roots, Profit and Loss, Interest, Rule of Three, Approximation to surds,
Simple equations as well as Simultaneous equations, Quadratic equations,
Arithmetic and Geometric Progressions --
all these are covered. Very
unusually in the entire history
of Ancient Indian mathematics, the subject matter is organised in a
sequence: first, a rule or a sutra; then a relevant example in word form; the
same in notational form; then the solution and finally the demonstration or the
proof. Here for the first time in the
history of world mathematics, the Rule of Three is stated in its abstract form.
It was from here that the rule was taken to
If p yields f what will i yield?
Here p stands for pramAna, f for phala and i for icchA.
Here p and i are of the same denomination and
f is of a different denomination.
Write p, f, i in that order. Multiply the middle quantity by the last quantity and divide by the first. The result is fi / p.
The first appearance of
indeterminate equations is in the
Bakshali mss. This marks the beginning of the continued work on indeterminate
Section 6. Astronomy
The contribution to Astronomy by ancient Indians is so great that it does not befit it to include it as one of the contributions of Indian Mathematics to the rest of the world. It needs a separate forum all for itself. We shall leave it right there except to add a note on the ancient contribution to the problem of telling time at night by a look at the stars on the meridian. This part is usually not emphasized.
The ancients of
krittikA simhe kAyA
says that if you see the asterism krittikA (Pleides, in modern terminology) on the meridian, that is the time the Leo (= simha) constellation (of the zodiac) has risen above the horizon by an amount indicated by the word: kAyA. This latter word interpreted in katapayA sankhyA, which is the notation used by astronomers, astrologers and mathematicians to represent numbers, means in this context that the amount of Leo above the (Eastern) horizon is 27 minutes of time. From this and the known position of the Sun on the date in question, one mentally calculates the time of night. On November 7 for example, the Sun is in the middle of Scorpio. So if you see krittikA on the meridian it means Leo has risen 27 minutes before and this means the Sun is behind by
93m (remaining portion of simha)
+ 2h (full portion of kanyA)
+ 2h (full portion of
+ 60m (half portion of vrischika)
that is 6 hours 33 minutes. In other words it is 6h 33m before sunrise. So it is 11-27 P.M.
Suffice it to say these beautiful formulae constitute an intellectual marvel put to the most mundane use. Never perhaps was so much achieved with so little so early in the Ascent of Man. For details on this, see reference no. .
Section 7. Classical contribution to Indeterminate Equations and Algebra
The apex of Mathematical achievement of ancient
Aryabhata wrote the famous Aryabhatiyam which is an
exhaustive exposition of Astronomy. In
addition he gave a unique method of representing large numbers by word forms .
He systematized all the knowledge of astronomy and mathematics prior to him.
The first one in Indian mathematics to give the formula for the area of a
triangle was Aryabhata. Several results on Triangles and circles and on Progressions,
algorithm for finding cube roots, approximation of p, all these give him a unique position in the
development of mathematics. Aryabhata ushered in a Renaissance in Indian
Mathematics and Astronomy, that resulted in a remarkable flourishing of science
and technology in
Bhaskara I takes a large share of the credit of explaining the too brief and aphoristic statements of Aryabhata. On the important topic of indeterminate equations the Kuttaka method was introduced by Aryabhata and elucidated by Bhaskara I.
Brahmagupta is generally known as the Indian mathematician par excellence. His monumental work Brahma SiddhAnta has 24 chapters of which the latter 14 contain original results on arithmetic algebra and on astronomical instruments. The 12th chapter is on mensuration. The 18th chapter is on Kuttaka. Among his famous results are those on rational right-angled triangles, and cyclic quadrilaterals. He is the earliest one, in the history of world mathematics, to have discussed cyclic quadrilaterals. There is every reason for us to name cyclic quadrilaterals as Brahmagupta Quadrilaterals. It was partly through a translation of Brahma-siddhAnta that the Arabs became aware of Indian astronomy and mathematics.
Bhaskara II's famous work SiddhAnta Siromani has four parts of which the first two are Mathematics and the latter two are astronomy. The first part, LilAvati is an extremely popular text dealing with arithmetic, algebra, geometry and mensuration. The second part, BIjaganitam is a treatise on Advanced Algebra. It contains problems on determining unknown quantities, evaluating surds and solving simple and quadratic equations.
The sheer ingenuity and versatility of Brahmagupta's approach to indeterminate equations of the second degree of the form
N x2 + 1 = y2
is the climax of Indian work in this area. Bhaskara II's cakravAla method to solve such equations is world-famous. By using this powerful method he solved, as one example, the above equation with N = 61 and gave the least integral solution as
x = 226153980 and y = 1766319049.
The famous French mathematician, Fermat, in 1657 A.D. proposed this equation with N = 61 for solution as a challenge to his contemporaries. None of them succeeded in solving the equation in integers. It was not until 1767 A.D. that the western world through Euler, by Lagrange's method of continued fractions, had a complete solution to such types of equations, wrongly called Pell's equation by Euler. But the very same equation, though coincidentally, was completely solved by Bhaskara II five hundred years earlier.
The problem of determining integer solutions of such equations is called Diophantine Analysis after the Greek Mathematician Diophantus (3rd cen. A.D.). As soon as one finds a non-trivial solution (that is, other than the obvious solution x = 0, y = 1) an infinite number of new solutions can be found by repeated application of the Principle of Compositions, known as Brahmagupta's Bhavana Principle. It is Bhaskara's cakravAla method that makes the decisive step in determining a non-trivial solution. Under these circumstances it is appropriate to designate these equations as the Brahmagupta-Bhaskara equations.
Before we leave this topic it is important to mention Srinivasa Ramanujan, the 20th century genius, who revelled in such problems - namely, to determine the possible cases in which a number can be broken up into two or more equal sums of like or unlike powers or more generally to solve intermediate problems in rational numbers.
Bhaskara II introduces also the notion of instantaneous motion of planets. He clearly distinguishes between sthUla gati (average velocity) and sUkshma gati (accurate velocity) in terms of differentials. He also gave formulae for the surface area of a sphere and its volume, and volume of the frustum of a pyramid. Suffice it to say that his work on fundamental operations, his rules of three, five, seven, nine and eleven, his work on permutations and combinations and his handling of zero all speak of a maturity, a culmination of five hundred years of mathematical progress.
Section 8. Indian Trigonometry
Though Trigonometry goes
back to the Greek period, the character of the subject started to resemble
modern form only after the time of Aryabhata. From here it went to
Section 9: Kerala contribution to Infinite Series and Calculus.
produced rules for second order interpolation to calculate intermediate sine
values. The Kerala mathematician Madhava may have discovered the sine and
cosine series about three hundred years before
Section 10. Modern Contribution:
Srinivasa Ramanujan onwards
The second decade
of the 20th century compulsorily turned the attention of the
mathematical world to
Copyright © V. Krishnamurthy January 2007
1. Balachandra Rao, S. Indian Mathematics and Astronomy. Jnana Deep Publications. 1994
2. Barathi Krishna Tirtha. Vedic Mathematics. Motilal Banrsidas. 1992
3. Bhanu Murthy. T.S. A Modern Introduction to Ancient Indian Mathematics. Wiley Eastern. 1992
4. Durant, Will. The Story of Civilization. Part I: Our
Oriental Heritage. Simon and
5. Dutta and Singh . History
of Hindu Mathematics . 2 Volumes.
6. Hooda, D.S. & Kapur, J.N. Aryabhata - Life and Contributions. New Age International. 1996
7. Joseph, G.G. The Crest of the Peacock - Non-European Roots of Mathematics. East-West Press. 1995
8. Jones, Catherine C. Islamic
9. Krishnamurthy, V. The
Culture, Excitement and Relevance of Mathematics,. Wiley Eastern, New
10. Krishnamurthy, V. The Clock
of the Night Sky.
11. Mahalingam, N. (ed.) Ancient
12. Smith. D.E., History
of Mathematics 2 Volumes.
13. Struik, D.J. A
Concise History of Mathematics
14. Srinivasa Rao, K. Srinivasa Ramanujan, a Mathematical Genius. East-West Books, 1998
Copyright © V. Krishnamurthy January, 2007


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3 Sep 20 9 What is the cost of constructing a fence 6 feet, 6 inches, high around a lot measuring 90 feet by 75 feet, if the cost of erecting the fence is $ .25 per linear foot and the cost of materials is $0.825 per square foot of fence?


Attention Math Teachers: Engage in a half-day of problem-solving and mathematics direction setting for our region. Join others in building a strong mathematics infrastructure!
Location: Auburn Career Center, Presentation Center, 8140 Auburn Road, Painesville, OH 44077
March 8, 2016 : 8:30 - 11:00 am (K-2)
12:30 - 3:00 pm (3-5)
March 9, 2016 : 8:30 - 11:00 am (9-12)
12:30 - 3:00 pm (6-8)
Linda Gojak received the Presidential Award for Excellence in Mathematics and Science. She was also awarded the Christofferson-Fawcett Award for Leadership in Mathematics Education.
Register by contacting Jovette Hiltunen at firstname.lastname@example.org or Brenda Harriss at email@example.com.


An extremely large number. The term was coined by Milton Sirotta, a young nephew of American mathematician Edward Kasner (1878-1955), who said a googol was 1 followed by 100 zeros (10100). Since a googol is so large that it encompasses everything in the universe and then some, it just means a "gargantuan amount." An even larger amount is a googolplex. See Googleplex and Google.


Phone number: (406)-532-4590. Ext. 4599
Google Class codes
8th grade - ctu2dq
7th grade - rveronj
Welcome to Mr. Parrish's Classroom for the 2022/2023 school year. Mr. Parrish teaches mathematics and basic math skills. We are excited to begin the new year and look forward to all the learning opportunities we will have in mathematics. Please make sure that you always come prepared for class every day. Pencils, notebooks, textbooks, calculators, planners with recorded assignments, and a good attitude are required everyday! Let's have a great, successful year. Mr. Parrish.


Katherine Porter, Ph.D.
|Office: OM 409 Phone: (360)438-4312 Email: email@example.com|
All courses will be maintained
B.S. (Pure Mathematics) Montana College of Mineral Science and Technology 1983;
M.S. (Statistics) University of Delaware 1987;
M.S. (Applied Mathematics) University of Alabama at Huntsville 1998;
Ph.D. (Applied Mathematics with a minor in Industrial Systems Engineering)
University of Alabama in Huntsville 1999


Lewis is a typeface designed for mathematical typesetting, specifically for the TeX typesetting system. It consists of 3 text styles (Roman, Bold, Italic) and 3 math styles (Math Italic, Greek, Blackboard) for use as variables. The text Italic relates to the Roman while the Math Italic stand out with its cursive construction. Likewise, the Greek differentiate easily from latin characters. The Blackboard inlines are adapted for text sizes with their wide and open cut. Lewis features many size variants and extending shapes, ideal in displayed equations.


Repetition: Mantra are meant to be
repeated - classic mantra yoga consists of sequences of 108 repetitions.
Why 108? The
number 108 has mystical significance to both Hindus and Buddhists, and
even to mathematicians etc. etc.. However, in all my research, no one
has stated a satisfying answer to what is the actual meaning, what is
really significant about the number 108. A few examples of its use:
There are 108 beads on a mala (Buddhist and Hindu rosary).
Hindu deities have 108 names.
The Lankavatara Sutra has a section where the Bodhisattva
Mahamati asks Buddha 108 questions.
In Japan, at the end of the year, a bell is chimed 108 times.
The karate Goju-ryu kata "Suparinpei,"
translates as "108"
According to the Ayurveda, there are 108 pressure points in the body.
There are 108 moves of the Yang T'ai Ch'i long form.
In Homer's "Odyssey", there are 108 suitors coveting
Penelope, wife of Odysseus.
An official Major League Baseball ball has 108 stitches.
108 = (2 x 2) x (3 x 3 x 3), i.e. 2-squared times 3-cubed
The distance of the Sun from the Earth divided by the diameter of the
Sun and the distance of the Moon from the Earth divided by the diameter
of the Moon is approximately equal to 108.
In Euclidean space, the interior angles of a regular pentagon measure
108 degrees each.
The founders of Moosewood Restaurant in Ithaca NY (and the
artist) lived at 108 Schuyler Place during its inception.
In India,108 (1-0-8) is the toll-free emergency telephone number (of course).
If you research this number 108, you will be amazed at what you find.


Information sheet ECTS Syllabus
Course syllabus D1-SO - Image processing (FCE - 2019/2020 - post-graduate studies)
|University:||Slovak University of Technology in Bratislava|
|Faculty:||Faculty of Civil Engineering|
|Course unit code:||D1-SO|
|Course unit title:||Image processing|
|Mode of delivery, planned learning activities and teaching methods:|
|Recommended semester/trimester:||Applied Mathematics - doctoral (semi-compulsory), 1. year|
Applied Mathematics (in english language) - doctoral (semi-compulsory), 1. year
|Level of study:||3.|
|Prerequisites for registration:||none|
|Correctly elaborated all the assignments, exam.|
|Learning outcomes of the course unit:|
|Student will acquire knowledge in modern methods of 2D and 3D image analyses based on solving geometry driven nonlinear partial differential equations.|
|- Nonlinear diffusion filtering in image processing.
- Level-set methods in image filtering and segmentation.
- Lagrangian methods in image segmentation.
- Numerical methods in image processing.
|Recommended or required reading:|
|Language of instruction:||slovak and english or english|
|Assessed students in total: 3|
|Name of lecturer(s):||doc. RNDr. Zuzana Krivá, PhD. (examiner, instructor, lecturer) - slovak |
prof. RNDr. Karol Mikula, DrSc. (examiner, instructor, lecturer, person responsible for course, tutor) - slovak, english
|Last modification:||31. 10. 2017|
|Supervisor:||prof. RNDr. Karol Mikula, DrSc. and programme supervisor|
Last modification made by Ing. Peter Korčák on 10/31/2017.


Book Description Springer, More information about this seller Contact this seller.
Condition: New. Seller Inventory Book Description Condition: New.
Seller Inventory n. Book Description Springer , Brand new book, sourced directly from publisher.
Dispatch time is working days from our warehouse. Book will be sent in robust, secure packaging to ensure it reaches you securely. Condition: Neu. Neuware - Stochastic geometry deals with models for random geometric structures. Its early beginnings are found in playful geometric probability questions, and it has vigorously developed during recent decades, when an increasing number of real-world applications in various sciences required solid mathematical foundations.
Stochastic and Integral Geometry provides the mathematically oriented reader with a rigorous and detailed introduction to the basic stationary models used in stochastic geometry - random sets, point processes, random mosaics - and to the integral geometry that is needed for their investigation. Never used! This item is printed on demand. Rolf Schneider ; Wolfgang Weil. Publisher: Springer , This specific ISBN edition is currently not available.
View all copies of this ISBN edition:. Synopsis About this title Stochastic geometry deals with models for random geometric structures. Karlsruhe "About this title" may belong to another edition of this title.
Buy New Learn more about this copy. Other Popular Editions of the Same Title. Search for all books with this author and title.
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Customers who bought this item also bought. Stock Image. Published by Springer New Hardcover Quantity Available: 1. Seller Rating:. New Hardcover Quantity Available: Handschug , Chord length distribution density and small-angle scattering correlation function of the right circular cone , Math. Modelling 30 , no.develop.quickli.com/citas-online-matchcom.php
Theory 58 , no. Theory 59 , no. Theory 51 , no. Theory 53 , no.
Stochastic and Integral Geometry (Probability and Its Applications)
Mazzolo, B. Roesslinger, and C. Diop, On the properties of the chord length distribution, from integral geometry to reactor physics , Ann. Energy, 30 , Roberts and S.
Stochastic Geometry, Spatial Statistics and Random Fields | veymefdumaco.tk
Torquato , Chord-distribution functions of three-dimensional random media: approximate first-passage times of Gaussian processes , Phys. With a foreword by David Kendall. Sinica 34 , no. Dedicata 39 , no. Gardner, Geometric tomography , 2nd ed.
Related Stochastic and Integral Geometry (Probability and Its Applications)
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I was going to post a new thread but can I bump my old one with relevent updates?
I rode blank grips for a while but had a strange sudden inspiration to do this on my deck. It’s a Fibonacci approximation of the golden ratio - starting with the smallest square in the middle, if you add the sides of the next sized square next to it, it gives the width of the next square. The ratio of the widths will approach the golden ratio if you keep going in the pattern and you can also join them up to form a golden spiral. It’s pretty mathematical shit but can still be appreciated without even knowing about it.
Next up is my current grip. It’s a chibi version (super deformed) of the anime character Haruhi Suzumiya, probably one of the most well known modern animes in Japan and recently had a movie premier in Scotland.
After lots of white marker/correction fluid drawings, I decided to go with a different approach and try out griptape cut outs. For my first one, I think I’ve done a fairly decent job. The hardest part was actually finding the dimensions of the original NES controller so I could create this to scale.


This weekend something amazing happened…
Something that has never happened without a fight, without crying and stropping, something that we normally end up giving up on and not doing.
He did some homework!
That’s right, he did some homework!
He had to practice dividing by 2 for a maths test on Wednesday, and I know that he probably needs to do it more than once in the week.
Once, is one more time than he has ever previously done.
On Sunday, after we had attempted to do it without any success, and I could see it spiralling into a refusal which included lots of tears and the possible throwing of homework books.
We got some lego out.
I counted out 24 small square pieces, gave him 6 and asked him how many groups of 2 he could make, and he did it, he made 3 little towers of 2, counted them out loud and then wrote the answer in his homework book.
We did this for all of the questions until he had finished them all.
Not only did he do it, but he did it with a smile on his face, something that I never thought that I would see him do.


It’s often said that baking is a science – and that is certainly true – but the equally important skill when baking is math. (Have I mentioned that I was a math major?) Lucky for most of us, the math needed for baking is much simpler than algebraic equations and linear algebra, but basic math (addition, subtraction, multiplication, division) and geometry come in quite handy when baking. I always keep a calculator in the kitchen (the calculator on your phone works too), and a ruler nearby.
Today’s lesson is about the size of a baking dish:
A 10-inch pie plate is considerably larger than a 9-inch plate. That inch might not seem to make much difference, but you need considerably more pie dough and filling to make a 10-inch pie. Check out the math:
Area equals pi (3.14) times the radius squared.
A 9-inch pie plate has an area of 64 inches (4.5 x 4.5 x 3.14) whereas a 10-inch pie plate has an area of 79 inches (5 x 5 x 3.14). That’s a difference of 23%.
Whereas some recipes can be adapted for plate size (for example, a brownie recipe designed for a 9-inch-square pan can be made in an 8-inch-square pan by increasing the baking time, don’t try to use a 10-inch pie plate with a recipe that was designed for a 9-inch pie. You won’t have enough crust and the filling will fall flat.
If you must use a 10-inch pie plate with a recipe designed for a 9-inch pie plate you need to increase the ingredients by roughly one quarter (or 5 parts for every 4 that the recipe calls for).


Washington students take state and federal tests regularly to assess their learning growth as well as the progress of the educational system as a whole. State tests include the following:
Smarter Balanced Assessments (SBA): English language arts (ELA) and math tests for students in grades 3–8 and 10 for federal and state accountability. The high school ELA and math assessments can be used to meet a student's graduation pathway requirement.
Washington Comprehensive Assessment of Science (WCAS): Science test for grades 5, 8, and 11 required for federal and state accountability.
Washington - Access to Instruction and Measurement (WA-AIM): ELA, math, and science alternate assessments for students in grades 3–11 with significant cognitive disabilities documented in their Individualized Education Program (IEP) for federal and state accountability. The high school ELA and math assessments can be used to meet a student's graduation pathway requirement.
Washington Kindergarten Inventory of Developing Skills (WaKIDS): This program helps bring families, teachers, and early learning providers together to support each kindergarten student's learning and transition into public schools.
WIDA Screener: Used to determine initial eligibility for ELD services for English learners in grades K–12.
WIDA ACCESS Assessments: For all students in grades K–12 who qualify for English language development (ELD) services to determine which students remain eligible to continue receiving ELD services.
WIDA Alternate ACCESS: For students in grades K–12 with significant cognitive challenges who qualify for English language development (ELD) services to determine which students remain eligible to continue receiving ELD services.
National Assessment of Educational Progress (NAEP): NAEP is a national test that measures what students across the country know and can do in 10 subject areas, including mathematics, reading, writing, and science. Federal law requires every state to give the NAEP in reading and math at grades 4 and 8 every two years. The state selects a number of schools to participate each year.
- What is the Smarter Balanced Assessment?
- Who takes the test and when?
- What does the SBA look like?
- What is the science assessment students have to take in school?
- How is the SBA information used by schools and teachers?
- Are SBA scores used in determining students’ class grade or will it appear on the report card and transcript?
- Where can I find the SBA score report for my child?
- How can I use the SBA score report to support the academic growth of my child?
- What if my child does not meet Standard (Level 1 or 2)?
- What is the SBA graduation assessment requirement in high school?
- Should my child study to prepare for the test?
- How can I help my child better prepare for the test?
- Smarter Balanced Assessments – Sample Items


Santa Maria Sun / School Scene
The following articles were printed from Santa Maria Sun [santamariasun.com] - Volume 14, Issue 21
Algebra Academy boosts student math skills
By CAMILLIA LANHAM
A partnership between Cal Poly and Santa Maria High School boosted the math skills of 20 students from the Santa Maria Joint Union High School District during a weeklong algebra camp that ended on July 26.
Students from Santa Maria and Pioneer Valley high schools participated. They were given mind-teasing math problems and guided to the solutions with one-on-one instruction at Santa Maria High. During the next few months, students will get the opportunity to travel to Cal Poly and learn about university requirements and tour the campus.
Santa Maria High student Selena Valle said in a press release that the academy was both challenging and fun.
“It teaches you to open your mind and understand math better,” she said. “It’s better than being at home.”
Students volunteered to attend the second annual summer algebra program. District superintendent Mark Richardson said in the release that teachers and students found a common bond.
“It’s great to see students that are dedicated to their education and just as impressive to see our teachers dedicating their vacation time to the betterment of our students,” he said.
Cal Poly Professor Elsa Medina said she would have loved to participate in a program like the Algebra Academy when she was in high school.
“They are working together and finding solutions for themselves,” she said in the release. “Some students don’t do well at math in high school, but later fall in love with it.”
Meaningful connections: Volunteers offer friendship to isolated seniors through Wilshire's Caring Callers Program Fresh air: Elephant seals and the volunteer docents who watch over them Los Osos to get water conservation rebates, but who will fund it? Paso's two fire chiefs leave the city Revolution: SLO progressives look to shake up the Democratic establishment Accusations fly in supes spat over Nipomo substation Peschong elected chairman of SLO's bitterly divided board of supervisors


Review the attached Payment Time Case Study and Data Set.
Develop a 600-word report including the following calculations and using the information to determine whether the new billing system has reduced the mean bill payment time: Assuming the standard deviation of the payment times for all payments is 4.2 days, construct a 95% confidence interval estimate to determine whether the new billing system was effective. State the interpretation of 95% confidence interval and state whether or not the billing system was effective. Using the 95% confidence interval, can we be 95% confident that µ ≤ 19.5 days? Using the 99% confidence interval, can we be 99% confident that µ ≤ 19.5 days? If the population mean payment time is 19.5 days, what is the probability of observing a sample mean payment time of 65 invoices less than or equal to 18.1077 days?


InstructionsFizzBuzz is played by counting but saying "Buzz" if a number is divisible by three or contains the digit three and saying "Fizz" if a number is divisiable by seven or contains the digit seven.
The sequence starts: 1, 2, Buzz, 4, 5, Buzz, Fizz, 8, Buzz, 10, 11, Buzz, Buzz, Fizz, Buzz, 16, Fizz, Buzz, 19, 20, FizzBuzz, 22, ...
FizzBuzz can be played competitivly in a group by having each person say the next number. If you get it wrong you are out and the game starts over. There is also a version for the TI 81, TI 82, or TI 83 calculator.
This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version.
This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.
Copyright Stephen Ostermiller 2004-2021


Estimating beta with a financial calculator web extension 2b to illustrate how betas are calculated with a financial calculator, consider table 2b-1. Enter a company's stock-ticker symbol and get the company's wacc that's wacc is the best research and educational tool for weighted average cost of capital anywhere that's wacc automatically calculates a company's cost of debt, equity, and tax rate simply by entering their stock ticker symbol. The figure shown below provides a graphical illustration of the relationship between α, β & power.
To calculate the valuation of a stock based off its dividends, the most commonly used equation is the gordon growth model, which looks like this: in the equation, here's what the variables mean. Beta and r² are already normalized so no need to annualize them under the same assumptions, you are trying to explain one series of returns with the other using a linear model whatever the relation of monthly returns, you will have the same on annual returns. My co-authors and i have already provided standardized (stdyx) coefficients and 95% cis, but it seems like the editor wants something more his suggestion was to 1) run the model with and without each cross-lagged path, 2) calculate the change in r2 for the dv, and 3) convert to an f statistic. How to calculate beta five parts: beta calculator calculating beta using a simple equation using beta to determine a stock's rate of return using excel graphs to determine beta making sense of beta community q&a beta is the volatility or risk of a particular stock relative to the volatility of the entire stock market beta is an indicator of how risky a particular stock is, and it is used to.
The biochemistry questions site a free biochemistry question bank for premed, medical students and fmg number of rounds in the beta-oxidation necessary for converting the whole fatty acid to acetyl co a units: number of acetyl coa minus 1 [(n/2)-1] 27 thoughts on “ how to calculate the energetic balance of the total. Beta can be a useful tool when at least some of a fund's performance history can be explained by the market as a whole beta is particularly appropriate when used to measure the risk of a combined. Levered/unlevered beta of target corporation ( tgt | usa) beta is a statistical measure that compares the volatility of a stock against the volatility of the broader market, which is typically measured by a reference market index. Compuware cope virtualizes ims systems, enabling new development images of ims to be available practically on demand, significantly reducing the time, costs and technical difficulties associated with creating completely new ims subsystems.
A beta of 10 means that the investor’s stock or investment moves in direct coordination with the rest of the stock market an 8 percent return of the total stock market should result in the same corresponding return of a stock with a beta of 10. The beta is fairly simple to calculate the data required are the prices of the security being values, and an index of the market it is being valued against it is usual to measure market returns using broad index such as the ftse 350 (or all-share) or the s & p 500 it is easy to calculate daily returns by dividing each days index level by that of the previous day. Compuware is changing the way developers develop our products fit into a unified devops toolchain enabling cross-platform teams to manage mainframe applications, data and operations with one process, one culture and with leading tools of choice.
In the capm model, beta coefficient is a systemic risk indicator which is used to measure the volatility of an asset, a security or a portfolio relative to the overall market. Beta is a measure of co-movement, not risk instead, it is a relative risk measurement tool the higher the correlation between the security and the market (as measured by r2), the more meaningful is beta. Functions to calculate systematic or beta co-moments of return series description calculate higher co-moment betas, or 'systematic' variance, skewness, and kurtosis.
Calculating weighted average cost of capital-wacc relevering beta dec 30, 2010 by jawwad farid in beta also see calculating beta with respect to market indices and calculating alpha and beta related posts finance training course: wacc, beta, cost of capital posts review. Lowenstein sandler is a national law firm with approximately 275 lawyers based in new york, palo alto, new jersey, utah, and washington, dc the firm represents leaders in virtually every sector of the global economy, with particular emphasis on investment funds, life sciences, and technology. Unlevered beta (asset beta) is the volatility of returns for a business, without considering its financial leverage it only takes into account its assets it compares the risk of an unlevered company to the risk of the market. A stock's beta coefficient is a measure of its volatility over time compared to a market benchmark a beta of 1 means that a stock's volatility matches up exactly with the markets a higher beta.
Calculating beta for compuware corporation financial websites tend to calculate beta on securities differently for publicly traded companies beta is merely an estimable measure of an assets’ risk in relation to the market. Calculating correlation factors the most familiar measure of dependence between two quantities is the pearson product moment correlation co-efficient, or “pearson’s correlation” it is obtained by dividing the covariance of the two variables by the product of their standard deviations. Solution calculate competitor's market value of equity and market value of debt elfu co mve = $1•20 x 400m shares = $480m elfu co mvd = $96m use competitor’s information to estimate the project’s asset beta (includes business risk of the competitor only.
Another quantity used to describe particle energy loss is specific ionization the specific ionization beta particle range the maximum range, rmax, sufficient to stop all the beta particles emitted by the co-60 what is the x-ray flux as a function of distance. Calculating beta for compuware co calculating beta for compuware corporation financial websites tend to calculate beta on securities differently for publicly traded companies beta is merely an estimable measure of an assets’ risk in relation to the market. Covariance calculator - step by step calculation to measure the statistical relationship (linear dependence) between the two sets of population data, provided along with formula & solved example problems in probability & statistics theory for data analysis, the calculation of covariance is important in many experiments to find the statistical. The instructions for this vary a lot depending on your brand of calculator the best way to find out is to dig out the instruction book (or google it - most companies put thei r instruction books online now.


The Calcutta Statistical Association and the Department of Statistics, Calcutta University will jointly host the Eighth International Triennial Calcutta Symposium on Probability and Statistics between December 27 - 30, 2012 at the Department of Statistics, Calcutta University, Kolkata, India.
The symposium will feature invited and contributory papers, and posters sessions on theoretical and applied statistics and probability. Refer to the “Symposium Brochure” for futher details.
Notification of closure of accepting abstracts
Since the deadline for fresh submissions is long past, no new submissions can now be considered for inclusion in the programme.
Convenor: Arindam Sengupta