Types of Numbers - Taxicab numbers

There was a mathematician named G. H. Hardy, who discovered and then later worked closely with a brilliant young mathematician named Srinivasa Ramanujan. One day in 1918 or 1919, Hardy went to visit Ramanujan in his private nursing home. Hardy said about Ramanujan:

"I remember once going to see him when he was lying ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me a rather dull one, and that I hoped it was not an unfavourable omen. 'No,' he replied, 'it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways'."

 This can be generalised to the following definition:

Ta(n) is the smallest number that can be expressed as the sum of 2 positive cubes in n distinct ways.

These numbers have been dubbed Taxicab numbers in honour of this anecdote. At the moment, only 6 taxicab numbers are known. The first few, with the cubes that make them up, are:

Ta(1) = 2 = 1^3 + 1^3

Ta(2) = 1,729 = 1^3 + 12^3
                      = 9^3 + 10^3

Ta(3) = 87,539,319 = 167^3 + 436^3
                                = 228^3 + 423^3
                                = 255^3 + 414^3

The remaining three known taxicab numbers are 6,963,472,309,248, 48,988,659,276,962,496 and 24,153,319,581,254,312,065,344.

I would love to know how they discover which number is the smallest one. Is is just a brute force algorithm that goes through every possible combination of cubes for each number? Or are there ways to narrow down the variables? At a guess, I would say it is mostly brute force, as there has been a lot of time between the discovery of each new number, the latest one only being discovered in 2003.

Although we only know the smallest number for six equal sums of two cubes, it has been proven by Hardy and Wright that the number of sums can be made arbitrarily large. In other words, the list of taxicab numbers goes on forever. There is not a biggest taxicab number.

Some people take taxicab numbers to be the list of numbers that are the sum of two cubes in two or more distinct ways. The first few are 1729, 4104, 13832, 20683, 32832, 39312, 40033, 46683, 64232, ...

I can see how you could find these numbers. You would have to limit the number you look up to, say to N. Then you take every cube number between 1 and N and add them to 1. Next add every cube number between 2^3 = 8 and N and add it to 8. Do the same up until N^3 + N^3, then sort all of the sums into order of the answers. Then you can go through the list and see which sums have the same answers. Obviously this would take very long to do by hand, but hey, that's what computers and algorithms are for!

Types of Numbers - Imaginary Numbers

Last week I told you about the building blocks of numbers, the number line. I also said that these numbers are not sufficient to solve every equation possible. Imagine you had the equation:

x^2 = -1

(You probably know x^2 means x squared, but I just thought I'd check!)

At this point you are probably thinking, "but you can't have a negative square number," which is absolutely right, if you are dealing with real numbers. So the answer to the question of how to solve the equation is to not look at real numbers. This is where imaginary numbers come in.

An imaginary number is represented by the letter i. i is equal to the square root of -1, so as you can probably guess, the answer to the above equation is in fact:

x = i

This is a very strange concept for some people, because imaginary numbers are not something that you can imagine in real life. Whole numbers, fractions and even irrational numbers are easily translated into real life problems, but imaginary numbers are an abstract concept.

There is another type of number that uses imaginary numbers. These are called complex numbers. These are simply numbers of the form a + bi, where a and b are real numbers. All previous numbers I have talked about are complex numbers too. If we take a or b to be equal to zero (in maths, or usually means one or both) then we can see how real numbers and imaginary numbers are both of the same form as complex numbers.

Complex numbers can create some very beautiful things. For example, take the Mandelbrot Set.

Mandelbrot Set

Zoomed in

This is an example of a fractal, which is created by feeding the result of z^2 + c (where z and c are both complex numbers) back into z.

Types of Numbers - The Number Line

There are so many different types of numbers with amazing properties that make them just beautiful, but I wouldn't be able to talk about these numbers without the most basic of numbers. So I am going to talk through the different numbers that make up the number line. (I said number a lot in that paragraph didn't I?)

First of all, we have the natural numbers or counting numbers. These are, as the name suggests, the numbers used to count things: 0, 1, 2, 3, 4, 5...

Next, we have the integers. These are the negative whole numbers along with the natural numbers: ...-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5...

Next are the rational numbers. These are numbers that are quotients of integers. In other words, numbers of the form a/b, where a and b are integers. There is an additional restriction where b cannot equal 0, since dividing by 0 causes all sorts of problems! Basically, rational numbers are just fractions: 1/2, 37/90, 3/4, -4/2 etc.

Next are the real numbers. These are the rational numbers along with numbers called the irrationals. These are numbers that cannot be written as a quotient of integers. They are numbers where the decimals don't terminate or repeat (in other words they go on forever without repeating). Any number that isn't rational is irrational. The most famous examples are: pi, e, the golden ratio and the square root of 2.

So these are the numbers that make up the number line. However, these numbers are not sufficient to be able to solve every equation. But more on that next time!