## Table of Contents

- Introduction
- Benford’s Law
- Explanation
- How Do The Tables Work?
- Conclusion

## Introduction

The numbers have always been interesting since I was a little kid. Especially the numbers that I saw in everyday life, whether in newspapers, magazines, or advertisement boards which I see on the streets, showed me how much mathematics processed into our lives. Seeing the numbers and figures in my daily life led me to wonder if they had come up with any sort of frequency or pattern. When I learned that I needed to do a Math Internal Assignment for the IB diploma program, I thought it was a great opportunity to make some research, find out if there is any kind of o pattern in numbers which we see every day, and if there is, learn how it works. I took this opportunity and decided to do an experiment myself to see if I can find anything. I figured that ıf I take a newspaper and investigate the numbers on it. Maybe I could find a pattern. So I took an economic newspaper (so it has many numbers -data-that I can work on) to do my experiment.

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My main my question was this: How many of the numbers on newspaper start with number 1? I started to count the numbers that is on this page. There were 127 numbers on the newspaper that I had. Ifwe think about the problem with the basic probity answer is supposed to be 1/9 because there nine possible numbers that can be at the front {1, 2, 3, 4, 5, 6, 7, 8, 9}. Therefore, if there are 127 numbers on the page there should be around 13 number starting with 1. However, that was not the case according to my experiment. In fact, there were 42 numbers starting with 1. I kept counting and came across these interesting results. There were: 42 numbers starting with 1 21 numbers starting with 2 12 numbers starting with 3 7 numbers starting with 4 6 numbers starting with 5 8 numbers starting with 6 9 numbers starting with 7 11 numbers starting with 8 11 numbers starting with 9 So scale of the leading figures was like this: 1 2 3 4 5 6 7 8 9 With the experiment I did, I found out that there were around 30% numbers starting with number 1. After a detailed research on the topic, I came across with Benford’s Law, which clearly shows the pattern of numbers that we see in our daily life in an impressive and captivating way. I saw that, his law matches with the results that I got from my experiment.

## Benford’s Law

This law was first discovered in 1882 by Simon Newcomb but then late rediscovered over 50 years later by the physicist Frank Benford when he was working general electric. They both noticed this pattern it in the same way. They noticed that pages near the beginning of the logarithm tables were getting more worn and dirty than the pages near the end. They realized that they were looking at the statistics that start with 1, more often the ones that start with other higher numbers. Apparently, Frank Benford did the same experiment that I did with the newspaper. But he did it on a big scale with a large amount of data. He took different numbers from our life such as newspapers, the population of different countries, articles and even street addresses and investigated. He got the same result a I did: Numbers starting with a 1 turn up about 30%. Here I have another example to show how 1 turns up so often in our daily life. If we imagine that we put 1€ in a bank which pays 10% interest every day. So we start with 1€ the next day we will have 1.10€ and then 1.21, 1.33, 1.46……. 1 1.10 1.21 1.33 1.46 1.61 1..77 1.94 2.14 2.35 2.59 2.85 3.13 3.45 3.80 4.18 4.59 5.05 5.56 6.11 6.72 7.40 8.14 8.95 9.84 10.83 11.92 13.11 14.42 15.86 17.45 19.19 21.11 23.22 25.55 28.10 30.91 34.00 37.40 41.14 45.26 49.79 54.76 60.24 66.26 72.89 80.18 88.20 97.02 - - We will end up with a table like this. We can easily see that there are far more numbers starting with 1 then there are starting with other higher numbers. We can also see here that numbers starting with 1 is about 30%. Note: This is the general idea and this can even be used to detect fraud. When people are faking data or making up numbers they might tend to spread out the numbers evenly or maybe start picking the middle numbers. If these numbers do not follow the Benford’s law, they may be committing fraud. This used many times in history in many places to detect the fraud. For example, this was used by Texas University to detect if there is a fraud in the American elections.

## Explanation

As I said before this was discovered when Benford noticed that beginning pages of log tables were getting worn and dirty.by the way Log tables are the booklets of logarithmic values which are used to multiply large numbers ın the days before calculators.

## How Do The Tables Work?

For example If we have number n we can easily find 10n n: 0 1 2 3 ….. 10n 1 10 100 1000 ….. As powers go up, we multiply by 10. As powers go down, we do the opposite, and divide by 10. Moreover, a logarithm to the base 10 or log is the reverse of 10n. If we have a number, we can get the original number n: 1 10 100 1000 ...... Log(n) 0 1 2 3 …… Then we can connect these points and see the log of numbers in between. For example log(50) = 1.7 Log tables were used to multiply large numbers because: log(xy) = log(x) + log(y) With this easy way, multiplication becomes only addition of logarithms and we could reverse the logarithm process and get the answer. Logarithmic tables were only need to go between 1 and 9 because it was the only range needed to do all the multiplications For Example ıf we had a number larger than 9: log(273)= log(2.73) + log(100) We could find the log(2.73) from the table and we knew that log(100) is 2. Benford noticed that: Prob(a number starts with) =log(n+1) - log(n) However, he could not explain why this was happening. I will show why this works in the further pages but the main idea essentially is that if we collect a lot of data we want the amount of data between, say 1 and 2 to be the same as the amount of data between 10 and 20 and we want that to be the same amount as data between 100 and 200. The only way to do that is if the probity of we start with a number 1 is 30%. If Benford Law exists and it is universal it should be unaffected by which units we choose, or what we measure with so it should always work no matter ıf we use kilometers, meters, feet, miles or any other units. Benford’s law should always hold true. If we imagine that, we have data like the one below 25 40 1220 2000 What this means is, we have a lot of data between 25 and 40 but we have fewer data between 1250 and 2000. Let us say we want to convert this data to something else. So we multiply by, say 50. So when we multiply everything by 50, data changes and we get something like this: 1250 2000 62500 100000 Now we have many data between 1250 and 2000. If this was scaled invariant the blue gap would be the same as the red gap. So blue gap represents the data there is 50 times larger than the red gap. If we took the log of this data, instead we would get this: 1.40 1.60 3.10 3.30 Now blue gap represents data that is 1.7 higher than the red gap. If we want this to be scale invariant, we want the size of the blue gap to be same as the size of the red gap. Therefore, we want this to be unaffected on the shifs. In other words, we want the log of data to be uniform like this: 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 Now, all we need to do is to reverse this to find the original distribution of something that is scale invariant: 1 2 4 10 20 40 100 200 400 1000 From this table we can see that ıf we double the distribution, the gap between, for example, 1 and 2 is equal to the gap between 2 and 4 which is equal to the gap between 4 and 8. If we multiply by 10 the gap between 1 and 2 is same as the gap between 10 and 20 which is the same as the gap btween100 and 200. If we want to know, the probability that a number starts with a particular digit we only need to consider the numbers between 1 and 9 as I said in the earlier pages. 1 2 3 4 5 6 7 8 9 For example, if we imagine that we throw a dart on this scale, what is the probability that we hit a number starting with 4 is the length of the section between 4 and 5 divided by the total length. Length of the section between 4 and 5 is equal to log(5)- log(4) and the total length is the log of 10 which is 1 : Prob( a number starts with 4)= log(5) – log(4) Therefore in general : Prob(a number starts with n): log(n+1) – log(n) This also works for many digits as well For example: Prob(a number starts with 123) = log(124) log(123) = 0.00352

## Conclusion

I always knew there were many mysteries and secrets all over your mathematics. Experiencing it by experimenting on my own was something I wanted to do for a long time but I never had the opportunity. Studying the Bendford law for Mathematics Internal Assessment was a good choice. This law has shown me how fascinating math is in our everyday life. My experiments and investigations showed me that this law works from the galaxies to the population of different countries. I also found out that this law can and is being used in detecting the fraud. I hope these pieces of information will be beneficial also in the future days of my life. Seeing the magic of mathematics with such an example made me love math more and I am sure that I will benefit from this throughout my life.