But what about a 3 + b 3 = c 3? Mathematician Pierre de Fermat asked the same question about cubes and, in 1637, he claimed to have worked out a mathematical proof that, via line after line of painstaking logic, showed beyond any doubt that no, the sum of two cubes can't be a cube. For example, can the sum of two cubes be a cube? Remember the Pythagorean triples from the previous page? These trios of three numbers, such as (3, 4, 5), solve the equation a 2 + b 2 = c 2. The first step in building a shiny, new, mathematical theory, however, is asking a theoretical question about number relationships. They uphold their theories with axioms (previously established statements presumed to be true) and theorems (statements based on other theorems or axioms). Mathematicians formulate theories about the relationships between numbers and number groups. So, the world of math offers up numerous number types, each with its own particular properties. But just what is a theory of numbers? What goes into formulating a proof, and why do some mathematical questions remain unanswered for centuries? Number theory involves analyzing such mathematical relationships, as well as asking new questions about them. An example of this is 3 2 + 4 2 = 5 2, or 3, 4, 5. In some cases, we can add square numbers together to produce other squared numbers in what is called a Pythagorean triple, as they fit the Pythagorean theorem (a 2 + b 2 = c 2). Now let's add another level of intrigue to this example. We also express these four examples as 2 2, 3 2, 1 2 and 99 2. For instance, 2 x 2 = 4 and 3 x 3 = 9, so 4 and 9 are both square numbers. There are square numbers, produced by multiplying another number by itself. There are, of course, odd numbers (1,3, 5 … ), which cannot be divided evenly, and even numbers (2, 4, 6 … ), which can. What kind of relationships? Well, we actually categorize integers into different number types based on their relationships. It's part theoretical and part experimental, as mathematicians seek to discover fascinating and even unexpected mathematical interactions. Number theorists scrutinize the properties of integers, the natural numbers you know as -1, -2, 0, 1, 2 and so forth. The study of those sometimes subtle and far-reaching relationships is number theory, sometimes referred to as higher arithmetic.
Independent of their usage, numbers become fascinating entities, and their mathematical relationships express the complexity of a vast system underpinning nature itself.
The likes of 6, 28 and 496 turn into something more sublime than simple carriers of information. They look at the world of numbers and, just as you'd never define your human beloved solely by his or her profession or hair color, the math lover sees beyond the mere function of numbers. In the eyes of some, there is no finer beauty than that found in mathematics. They trace the undercurrents of our personality, and, to the observant and loving eye, they illuminate true beauty. Such interrelated details come to define us.
The way he or she lets old paperbacks stack up on the bedside table. The peculiarities of the other person's morning coffee ritual. The silly in-jokes shared at the end of the day. Andrea Pistolesi/The Image Bank/ Getty ImagesĪnyone who has ever fallen in love will tell you it's the little things about the other person that matter.
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