Influential Mathematicians: Gauss (2)

Read the first part of this series here.

Although Gauss made contributions in many fields of mathematics, number theory was his favourite. He said that

“mathematics is the queen of the sciences, and the theory of numbers is the queen of mathematics.”

A way in which Gauss revolutionised number theory was his work with complex numbers.

Gauss gave the first clear exposition of complex numbers and of the investigation of functions of complex variables. Although imaginary numbers had been used since the 16th century to solve equations that couldn’t be solved any other way, and although Euler made huge progress in this field in the 18th century, there was still no clear idea as to how imaginary numbers were connected with real numbers until early 19th century. Gauss was not the first to picture complex numbers graphically (Robert Argand produced the Argand diagram in 1806). However, Gauss was the one who popularised this idea and introduced the standard notation a + bi. Hence, the study of complex numbers received a great expansion allowing its full potential to be unleashed.

Furthermore, at the age of 22 he proved the Fundamental Theorem of Algebra which states:

Every non-constant single-variable polynomial over the complex numbers has at least one root.

This shows that the field of complex numbers is algebraically closed, unlike the real numbers.

Gauss also had a strong interest in astronomy, and was the Director of the astronomical observatory in Göttingen. When Ceres was in the process of being identifies in the late 17th century, Gauss made a prediction of its position. This prediction was very different from those of other astronomers, but when Ceres was discovered in 1801, it was almost exactly where Gauss had predicted. This was one of the first applications of the least squares approximation method, and Gauss claimed to have done the logarithmic calculations in his head.

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Source: The Story of Mathematics

Part 3 coming next week!

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Catalan’s Conjecture

Although popularly known as Catalan’s conjecture, this is in fact a theorem in number theory, proven by Preda Mihăilescu in 2002, 158 years after it was conjecture in 1844 by French and Belgian mathematician Eugène Charles Catalan.

The theorem states that the only solution in the natural numbers of

{\displaystyle x^{a}-y^{b}=1}

for a,b>1; x,y>0 is x = 3, a=2, y=2, b=3.

Catalan ‘Near Misses’ (i.e. xa – yb <= 10; 2 <= x,y,a,b <= 100; a,b prime)

  • 33 – 52 = 2
  • 27 – 53 = 3
  • 23 – 22 = 62 – 25 = 53 – 112 = 4
  • 25 – 33 = 5
  • 25 – 52 = 42 – 32 = 27 – 112 = 7
  • 42 – 23 = 8
  • 52 – 42 = 62 – 33 = 152 – 63 = 9
  • 133 – 37 = 10

Proving the Conjecture

Similarly to Fermat’s last theorem, the solution to this conjecture was assembled over a long period of time. In 1850, Victor Lebesgue established that b cannot equal 2. But then it was only after around 100 years, in 1960, that Chao Ko constructed a proof for the other quadratic case: a cannot equal 2 unless x = 3.

Hence, this left a,b odd primes. Expressing the equation as (x-1)(xa-1)/(x-1) = yb one can show that the greatest common divisor of the two left hand factors is either 1 or a. The case where the gcd = 1 was eliminated by J.W.S Cassels in 1960. Hence, only the case where gcd = a remained.

It was this “last, formidable, hurdle” that Mihailescu surmounted. In 2000, he showed that a and b would have to be a ‘Wieferich pair’, i.e. they would have to satisfy ab-1 ≡ 1 (mod b2) and ba-1 ≡ 1 (mod a2). In 2002,  he showed that such solutions were impossible!

Video

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MATHS BITE: Kaprekar Numbers

Consider an n-digit number k. Square it, and then add the right n digits to the left n or n-1 digits (by convention, the second part may start with the digit 0, but must be nonzero). If the result is k then it is called a Kaprekar number. They are named after D. R. Kaprekar, a recreational mathematician from India.

We can extend the definition to any base b:

Let  X  be a non-negative integer and  n a positive integer.  X  is an n-Kaprekar number for base  b  if there exist non-negative integer A, and positive integer B  satisfying:

X2 = Abn + B, where 0 < B < bn
X = A + B

-Wikipedia

Examples in Base 10

  • 297: 2972 = 88209, which can be split into 88 and 209, and 88 + 209 = 297.

  • 999: 9992 = 998001, which can be split into 998 and 001, and 998 + 001 = 999.
  • In particular, 9, 99, 999… are all Kaprekar numbers.
  • More generally, for any base b, there exist infinitely many Kaprekar numbers, including all numbers of the form bn − 1.
  • 100: 100 is NOT a Kaprekar number as, although 1002 = 10000 and 100 + 00 = 100, the second part here is zero.

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    (P.S. Another post on Kaprekar is coming soon!)

Friedman Numbers

A Friedman number is an integer that can be obtained combining all its digits with the 5 arithmetic operations (+, -, x, /, ^) and concatenation. Note that parentheses can be used in the expressions in order to “override the default operator precedence“. These numbers are named after mathematician Erich Friendman.

For example, 13125 is a Friedman number as it can be written as $21\cdot5^{3+1}$.

nice Friedman number is one where the digits in the expression can be arranged to be in the same order as the number itself. An example of this is 127 = -1 + 2^7.

Friedman numbers can be repdigits, such as

\[999999999= ((9 + 9 + 9)^{9 - 9}+9)^9 - 9/9\]

or pandigital numbers, like

\[9108432576 = 251^3\cdot4\cdot6\cdot(7 + 8 + 9 + 0).\]

This link will take you to a list of Friedman numbers and their decompositions up to 10^6.

Michael Brand showed that the density of Friedman numbers among the naturals is 1. This means that the probability of a number chosen randomly and uniformly between 1 and n to be a Friedman number tends to 1 as n tends to infinity.

Finding 2-digit Friedman Numbers

2-digit Friedman numbers are the easiest to find, although there are less of them than 3-digit Friedman numbers.

Working in base 10 Let’s represent a 2-digit number by 10m + n, where n is an integer from 0 to 9. Now, we only need to check each possible combination of m and n against the equalities:

10m + n = m^n and 10m + n = n^m 

We do not have to worry about n, m x n, m – n and m/n as these will always be smaller than 10m + n when n < 10.

To read more click here.

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Surreal Numbers

Surreal numbers were first invented by John Horton Conway in 1969, but was introduced to the public in 1974 by Donald Knuth through his book ‘Surreal Numners: How Two Ex-Students Turned on to Pure Mathematics and Found Total Happiness‘.

What are Surreal Numbers?

Surreal numbers are the ‘most natural’ collection of numbers that include both real numbers and the infinite ordinal numbers of Georg Cantor. The surreals have many of the same properties as the reals, including the usual arithmetic operations. Hence, they form an ordered field.

For a surreal number x we write x = {XL|XR} and call XL and XR the left and right set of x,respectively. These will be explained below.

Conway Construction

“Surreal numbers are constructed inductively as equivalence classes of pairs of sets of surreal numbers, restricted by the condition that each element of the first set is smaller than each element of the second set.”

– Wikipedia

Using the Conway construction, we construct the surreal numbers in stages along with an ordering ≤ such that for any two surreal numbers a and b either a ≤ b or b ≤ a.

Every number corresponds to two sets of previously created numbers, such that no member of the left set is greater than or equal to any member of the right set. Therefore, if x = {XL|XR} then for each xL ∈ XL and xR ∈ XR, xL is not greater than xR.

In the first stage of construction, there are no previously existing numbers so the only representation must use the empty set: { | } = 0.

Subsequent stages yield the following:

  • {0|} = 1, {1|} = 2, {2|} = 3, etc;
  • {|0} = -1, {|1} = -2, {|2} = -3, etc.

For more, click here.

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Lychrel numbers

Lychrel number is a natural number that cannot form a palindrome by the 196-algorithm: an iterative process of repeatedly reversing a numbers’ digits and adding the resulting numbers.

Whilst in other bases (powers of two) certain numbers can be proven to never form a palindrome, in base 10 (the base system we use in everyday life) no Lychrel numbers have been proven to exist. However many numbers, such as 196, are suspected to be a Lychrel number on “heuristic and statistical grounds“.

The name Lychrel was coined by Wade Van Landingham in 2002 as an anagram of Cheryl, his girlfriend’s name.

196-Algorithm

The reverse-and-add process is when you add a number to the number formed by reversing the order of its digits.

Examples of non-Lychrel numbers are (taken from Wikipedia):

  • 56 becomes palindromic after one iteration: 56+65 = 121.
  • 57 becomes palindromic after two iterations: 57+75 = 132, 132+231 = 363.
  • 59 becomes a palindrome after 3 iterations: 59+95 = 154, 154+451 = 605, 605+506 = 1111
  • 89 takes an unusually large 24 iterations to reach the palindrome 8,813,200,023,188.
  • 1,186,060,307,891,929,990 takes 261 iterations to reach the 119-digit palindrome 44562665878976437622437848976653870388884783662598425855963436955852489526638748888307835667984873422673467987856626544, which is the current world record for the Most Delayed Palindromic Number. It was solved by Jason Doucette‘s algorithm and program in 2005.

196

196 is the smallest number suspected to never reach a palindrome in base 10 and has thus received the most attention:

  • In 1985 a program by James Killman ran unsuccessfully for over 28 days, cycling through 12,954 passes and reaching a 5366-digit number.
  • John Walker began his 196 Palindrome Quest in 1987. His program ran for almost three years, then terminated (as instructed) in 1990 with the message:

    Stop point reached on pass 2,415,836.
    Number contains 1,000,000 digits.
  • In 1995, Tim Irvin and Larry Simkins reached the two million digit mark in only three months without finding a palindrome.
  • Jason Doucette then reached 12.5 million digits in May 2000.
  • Wade Van Landingham used Jason Doucette’s program to reach a 13 million digit. By May 2006, Van Landingham had reached the 300 million digit mark.
  • In 2011 Romain Dolbeau completed a billion iterations to produce a number with 413,930,770 digits, and in February 2015 his calculations reached a number with billion digits.

 A palindrome has yet to be found.

For more on Lychrel numbers, click here or here.

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New Books in Maths: March 2017

Today I decided to do another instalment of my series ‘New Books in Math’, where I talk about books which have been recently released or will be released soon.

Bad Choices: How Algorithms Can Help You Think Smarter and Live Happier – Ali Almossawi

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Release Date: April 2017

Ali Almossawi, the author of the popular book Bad Arguments, has returned with a funny and smart introduction to algorithms.

This book aims to answer questions such as “why is Facebook so good at predicting music?” and “how do you discover new music?

To demystify a topic of ever increasing importance to our lives, Almossawi presents us with alternative methods for tackling twelve different scenarios, guiding us to better and more efficient choices “that borrow from same systems that underline a computer word processor, a Google search engine, or a Facebook ad”.

“Bad Choices will open the world of algorithms to all readers making this a perennial go-to for fans of quirky, accessible science books.”

Are Numbers Real?: The Uncanny Relationship of Mathematics and the Physical World – Brian Clegg

Release Date: March 2017

28220651.jpgWhat was life like before numbers existed? Numbers began as simple representations of everyday things, but mathematics rapidly took a life of its own and grew into what it is today.

In Are Numbers Real?, Brian Clegg explores the way that maths has become more and more detached from reality, but yet remains the driving force of the development of modern physics.

“From devising a new counting system based on goats, through the weird and wonderful mathematics of imaginary numbers and infinity to the debate over whether mathematics has too much influence on the direction of science, this fascinating and accessible book opens the reader’s eyes to the hidden reality of the strange yet familiar world of numbers.”

The Mathematics Lover’s Companion: Masterpieces for Everyone – Edward R. Scheinerman

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Release Date: March 2017

How can a shape have more than one dimension but fewer than two? What is the best way to elect public officials when more than two candidates are vying for the office? Is it possible for a highly accurate medical test to give mostly incorrect results? Can you tile your floor with regular pentagons? How can you use only the first digit of sales numbers to determine if your accountant is lying? Can mathematics give insights into free will?

Edward Scheinerman answers these questions and more in The Mathematics Lover’s Companion, in bite-size chapters that require only secondary school mathematics. He invites readers to get involved in solving mathematical puzzles, and provides an engaging tour of numbers, shapes and uncertainty.

The result is an unforgettable introduction to the fundamentals and pleasures of thinking mathematically.

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