![]() ![]() ![]() But what would you call these lists: -1, -1, -1. The message to take away is that counting numbers, integers, rational numbers, and real numbers can all be represented simply as sequences of digits and a few more symbols. What was on the left of the decimal point is a single integer of any size, and all other digits are separated by commas: 1Īnd of course we can write negative numbers as well: -1 Instead of a decimal point separating sequence of digits, we can adopt a more flexible notation. Now we can, if we wish, write our numbers in a different manner. 1įor those who have forgotten their grade school math, you can prove the lower number is the same as the first: x For instance, the number "1" has two representations. Now note that there is not necessarily just one way of representing numbers. 1/ 2Īnd irrational numbers don't come to an end or repeat: π Rational numbers are represented like integers except that to the right of the decimal point, you have lists of digits that either come to an end - they terminate - or repeat endlessly. 1Īnd integers just have positive or negative signs added and include zero. For instance, counting or natural numbers are the numbers greater than zero that you can write using the digits followed by a decimal point and nothing but zeros. ![]() First of all, all of these numbers can be represented as combinations of the 10 digits (0, 1, 2, 3, 4, 5, 6, 7, 8,9) plus a few more signs (. So you might suspect you could represent the complex systems in terms of the simpler ones.Īnd you would be right. We see then that the simpler systems are subsets of the more complex systems. And if you lump the irrationals with the rationals, integers, and counting numbers, we have the real numbers. We say that counting numbers and integers are subsets of the rational numbers.īut then you have numbers like π (3.14159.), the square root of 2 ( √ 2 or 1.41421.), and the famous exponential number e (2.71828.). Since fractions are ratios of integers they are called rational numbers.īut notice that some rational numbers are counting numbers and integers: 1/ 1, 2/ 1, -8/ 4, and so on. These make up a set mathematicians represent - not as I, but Z.Īnd you certainly know you can take the integers and put one on top of another separated by a line, ½, ¼, ⅔, -⅔, -¼ and such. These are also called natural numbers and they make up a set designated as N.Īnd if you take the counting numbers and stick in the number zero ( 0) and allow the option of the minus sign, you have integers. Everyone knows what counting numbers are. But in any case, the new numbers Abraham wrote about are most unusual. Others, though, think numbers are a human and therefore artificial construct. Platonists believe numbers exist independently of the human mind. ![]() Of course, it's a moot point if Abraham invented the new numbers or if he discovered them. To show that Gottfried, Isaac, and Seki really did know what they were talking about, Abraham had to invent a new kind of number. By the way, if you didn't reach this page through the usual CooperToons Caricature introduction, you can read a bit about Abraham's interesting - and for a mathematician rather adventurous - life if you click here. He decided that if the current numbers wouldn't do the job, then he'd just invent numbers that would. He was, as you'll figure out if you read on, one of the more inventive mathematicians of the 20th century - literally. It's very simple really.Īnd, yes, it was Abraham Robinson who showed us the way. I thought you would as Captain Mephisto said to Sidney Brand. So came the 20th century and incomprehensibility.īut we, on the other hand, would like to see how to do calculus without using limits - provided, of course, it is with sufficient rigor. Sadly, the inventors of calculus who used simpler systems - Isaac Newton, Gottfried Leibniz, and Seki Takakazu - lacked the rigor demanded by later mathematicians. It was trying to prove the limit of a formula using the dreaded epsilon-delta notation that drove you nuts. That is, as long as you didn't have to prove how to calculate derivatives or figure out the integrals.Īctually, proving formulas for derivatives and even integrals wasn't the problem. If you've ever taken a modern calculus course, you probably did OK when calculating derivatives and figuring out the integrals. ![]()
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