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Peter Harremoes 3 2011 Is zero natural number
NumbersDate: 12/10/97 at 19:14:39From: Joanna GreenoSubject: NumbersHow do integers and whole numbers, rational numbers, transcendentalnumbers, and counting numbers relate to each other?Date: 12/10/97 at 21:41:50From: Doctor LuisSubject: Re: NumbersNATURAL NUMBERS (COUNTING NUMBERS)First of all, there are the natural numbers. These are the mostfundamental of all and they are sometimes referred as counting numbersNatural Numbers 1,2,3,4,5,....Here, the dots are taken to signify that the list goes on forever.They give us a basic and primitive notion of infinity. That is why theset of natural numbers is so essential to mathematics.Although you cannot literally "count" how many natural numbers thereare, mathematicians have given a name to this "infinitely" largenumber that represents how many natural numbers there are. It isusually called "aleph nought" (aleph is the Hebrew letter "A"). Now,this is a different kind of number from what we are used to, and youcan use it to define the cardinal numbers, which in a way let us speakabout various degrees of infinities. For example, there are more realnumbers than natural numbers, and more irrational numbers thanrational numbers.Anyway, we now have the natural numbers.WHOLE NUMBERSIf you add zero to your list then you have the whole numbers,Whole numbers 0,1,2,3,....The number zero has an interesting history. The ancient Mayans werethe first to introduce the concept of zero as a number, centuries ago.Why did we have to introduce a symbol to denote "nothing" or a "zeroquantity" of something? The reason for this is apparent when you startconsidering numbers that are too large for us to represent by a singlesymbol.What are some ways to represent numbers? Well, we can start namingthem; that is, representing each number by a symbol (they would allhave to be different). Eventually, we would run out of symbols, or thelist of symbols would be too large for us to remember (imagine tryingto multiply numbers - you'd have to remember ALL the symbols for everynumber and memorize an INFINITELY long multiplication table.)Is there any way around this problem? Fortunately, there is a veryclever way to represent numbers that solves most of our problems. Itis called place value notation.PLACE VALUE NOTATIONIn place value notation, you assign a value to each symbol accordingto where it is in the number representation (its place). (The valueyou assign is related to the base of your number system.)Example: 123 (base 10)In the example, the symbol 3 is worth 3 units BUT the 2 is worth 2*10units, and the 1 is worth 1*100 units.This allows us to represent more numbers with a shorter list ofsymbols. However, in order to represent the entire set of naturalnumbers in place value notation you NEED the number zero (that way youcan say that you DON'T have any units of 100, for example, in thenumber 1025).This completes our representation of the natural numbers very nicely,AND it gives us a cool new number to work with. Okay. We have thenatural numbers and the whole numbers.INTEGERSTo get the integers, just add all the negative numbers to your list!Integers ...,-3,-2,-1,0,1,2,3,....The dots before the -3 mean that the list continues FROM NEGATIVEinfinity, so that if you go back all the way, you'll see all of thenegative numbers.Note: for obvious reasons, the natural numbers are often calledpositive integers; and the whole numbers are also oftencalled nonnegative integers.So far, our use of number has been restricted to mean integers.However, you can define different kinds of numbers as wellRATIONAL NUMBERSNow, the rational numbers are simply defined as ratios of integers.1/2 is a rational number. 2/3 is also a rational number. Note that allof the integers are rational numbers, because you can think of them asthe ratio of themselves to 1, as in 2 = 2/1 which is certainly theratio of two integers, and so 2 is a rational number.The story of the rational numbers is also interesting. Ancient Greekmathematicians were very fond of rational numbers. In fact, when theydiscovered that there were other numbers which were not rational, theyswore that "terrible" discovery to secrecy.IRRATIONAL NUMBERSThe square root of 2 is a classic example of an irrational number: youcannot write it as the ratio of ANY two integers. Here's the squareroot of 2 to a few decimal places (it continues forever, and there isno pattern!):sqrt(2) = 1.414213562373095048801688724209698078570TRANSCENDENTAL AND ALGEBRAIC NUMBERSTo understand transcendental numbers, you also need to understandanother type of numbers called algebraic numbers.A number is called algebraic if it is the root of a polynomial (of anydegree) with rational coefficients. Any number that is not algebraicis called transcendental.2 and 2/3 are algebraic (they are also rational) because they are theroots of the rational polynomial 3x^2 - 8x + 4 (it is rational becausethe coefficients are rational numbers)3x^2 - 8x + 4 = 0or (3x - 2)(x - 2) = 0 so x = 2 or 2/3Note that the square root of 2 is also algebraic (it is alsoirrational) because it is a solution of the rational polynomialx^2 - 2 = 0A classic example of an transcendental number (that is, not algebraic)is the number pi (shown here to an accuracy of a thousand decimaldigits) It goes on forever, just as the square root of 2.3.141592653589793238462643383279502884197169399375 1058209749445923078164062862089986280348253421170679821480865132823066 4709384460955058223172535940812848111745028410270193852110555964462294 8954930381964428810975665933446128475648233786783165271201909145648566 9234603486104543266482133936072602491412737245870066063155881748815209 2096282925409171536436789259036001133053054882046652138414695194151160 9433057270365759591953092186117381932611793105118548074462379962749567 3518857527248912279381830119491298336733624406566430860213949463952247 3719070217986094370277053921717629317675238467481846766940513200056812 7145263560827785771342757789609173637178721468440901224953430146549585 3710507922796892589235420199561121290219608640344181598136297747713099 6051870721134999999837297804995105973173281609631859502445945534690830 2642522308253344685035261931188171010003137838752886587533208381420617 1776691473035982534904287554687311595628638823537875937519577818577805 32171226806613001927876611195909216420199....It turns out that pi is also irrational (you cannot write it as theratio of two integers).It is impossible to cover all possible aspects of these numbers in asingle page, so I have kept their description at a bare minimum. Ifyou are interested and want to find out more, you can find a lot ofinformation in any good history of mathematics book.If you have any questions, feel free to reply.-Doctor Luis, The Math ForumCheck out our web site! http://mathforum.org/dr.math/
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