**Normal Numbers – and random number generators**

Numberphile have a nice new video where Matt Parker discusses all different types of numbers – including “normal numbers”. Normal numbers are defined as irrational numbers for which the probability of choosing any given 1 digit number is the same, the probability of choosing any given 2 digit number is the same etc. For example in the normal number 0.12345678910111213141516… , if I choose any digit in the entire number at random P(1) = P(2) = P(3) = … P(9) = 1/10. Equally if I choose any 2 digit number at random I have P(10) = P(11) = P(12) = P(99) = 1/100.

It is incredibly hard to find normal numbers, but there is a formula to find some of them.

In base 10, we are restricted to choosing a value of c such that 10 and c are relatively prime (i.e share no common factors apart from 1). So if we choose c = 3 this gives:

We can now put this into Wolfram Alpha and see what number this gives us:

So we can put the first few digits into an online calculator to find the distributions

*0.000333333444444444444448148148148148148148148148148148148148148149382716049382716049382716049382716049382716049382716049382716049382716049382716049382716049382716049382716049382716049382716049827160493827160493827160479423863312 7572016460905349794238683127572016460905349794238683127572016460 9053497942386831275720164609053497942386831275720164609053497942*

4: 61

1: 41

8: 40

3: 38

0: 36

2: 33

7: 33

9: 33

6: 32

5: 10

We can see that we are already seeing a reasonably similar distribution of single digits, though with 4 and 5 outliers. As the number progressed we would expect these distributions to even up (otherwise it would not be a normal number).

One of the potential uses of normal numbers is in random number generators – if you can use a normal number and specify a digit (or number of digits) at random then this should give an equal chance of returning each number.

To finish off this, let’s prove that the infinite series:

does indeed converge to a number (if it diverged then it could not be used to represent a real number). To do that we can use the ratio test (only worry about this bit if you have already studied the Calculus Option for HL!):

We can see that in the last limit 3 to the power n+1 will grow faster than 3 to the power n, therefore as n increases the limit will approach 0. Therefore by the ratio test the series converges to a real number.

**Is pi normal?**

Interestingly we don’t know if numbers like e, pi and ln(2) are normal or not. We can analyse large numbers of digits of pi – and it looks like it will be normal, but as yet there is no proof. Here are the distribution of the first 100,000 digits of pi:

1: 10137

6: 10028

3: 10026

5: 10026

7: 10025

0: 9999

8: 9978

4: 9971

2: 9908

9: 9902

Which we can see are all very close to the expected value of 10,000 (+/- around 1%).

So, next I copied the first 1 million digits of pi into a character frequency counter which gives the following:

5: 100359

3: 100230

4: 100230

9: 100106

2: 100026

8: 99985

0: 99959

7: 99800

1: 99758

6: 99548

This is even closer to the expected values of 100,000 with most with +/- 0.25 %.

Proving that pi is normal would be an important result in number theory – perhaps you could be the one to do it!

Essential resources for IB students:

Revision Village has been put together to help IB students with topic revision both for during the course and for the end of Year 12 school exams and Year 13 final exams. I would strongly recommend students use this as a resource during the course (not just for final revision in Y13!) There are specific resources for HL and SL students for both Analysis and Applications.

There is a comprehensive Questionbank takes you to a breakdown of each main subject area (e.g. Algebra, Calculus etc) and then provides a large bank of graded questions. What I like about this is that you are given a difficulty rating, as well as a mark scheme and also a worked video tutorial. Really useful!

The Practice Exams section takes you to a large number of ready made quizzes, exams and predicted papers. These all have worked solutions and allow you to focus on specific topics or start general revision. This also has some excellent challenging questions for those students aiming for 6s and 7s.

Each course also has a dedicated video tutorial section which provides 5-15 minute tutorial videos on every single syllabus part – handily sorted into topic categories.

2) Exploration Guides and Paper 3 Resources

I’ve put together four comprehensive pdf guides to help students prepare for their exploration coursework and Paper 3 investigations. The exploration guides talk through the marking criteria, common student mistakes, excellent ideas for explorations, technology advice, modeling methods and a variety of statistical techniques with detailed explanations. I’ve also made 17 full investigation questions which are also excellent starting points for explorations. The Exploration Guides can be downloaded here and the Paper 3 Questions can be downloaded here.

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