Date: 12/10/97 at 19:14:39 From: Joanna Greeno Subject: Numbers How do integers and whole numbers, rational numbers, transcendental numbers, and counting numbers relate to each other?
Date: 12/10/97 at 21:41:50 From: Doctor Luis Subject: Re: Numbers NATURAL NUMBERS (COUNTING NUMBERS) First of all, there are the natural numbers. These are the most fundamental of all and they are sometimes referred as counting numbers Natural 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 the set of natural numbers is so essential to mathematics. Although you cannot literally "count" how many natural numbers there are, mathematicians have given a name to this "infinitely" large number that represents how many natural numbers there are. It is usually called "aleph nought" (aleph is the Hebrew letter "A"). Now, this is a different kind of number from what we are used to, and you can use it to define the cardinal numbers, which in a way let us speak about various degrees of infinities. For example, there are more real numbers than natural numbers, and more irrational numbers than rational numbers. Anyway, we now have the natural numbers. WHOLE NUMBERS If 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 were the 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 "zero quantity" of something? The reason for this is apparent when you start considering numbers that are too large for us to represent by a single symbol. What are some ways to represent numbers? Well, we can start naming them; that is, representing each number by a symbol (they would all have to be different). Eventually, we would run out of symbols, or the list of symbols would be too large for us to remember (imagine trying to multiply numbers - you'd have to remember ALL the symbols for every number and memorize an INFINITELY long multiplication table.) Is there any way around this problem? Fortunately, there is a very clever way to represent numbers that solves most of our problems. It is called place value notation. PLACE VALUE NOTATION In place value notation, you assign a value to each symbol according to where it is in the number representation (its place). (The value you 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*10 units, and the 1 is worth 1*100 units. This allows us to represent more numbers with a shorter list of symbols. However, in order to represent the entire set of natural numbers in place value notation you NEED the number zero (that way you can say that you DON'T have any units of 100, for example, in the number 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 the natural numbers and the whole numbers. INTEGERS To 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 NEGATIVE infinity, so that if you go back all the way, you'll see all of the negative numbers. Note: for obvious reasons, the natural numbers are often called positive integers; and the whole numbers are also often called nonnegative integers. So far, our use of number has been restricted to mean integers. However, you can define different kinds of numbers as well RATIONAL NUMBERS Now, 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 all of the integers are rational numbers, because you can think of them as the ratio of themselves to 1, as in 2 = 2/1 which is certainly the ratio of two integers, and so 2 is a rational number. The story of the rational numbers is also interesting. Ancient Greek mathematicians were very fond of rational numbers. In fact, when they discovered that there were other numbers which were not rational, they swore that "terrible" discovery to secrecy. IRRATIONAL NUMBERS The square root of 2 is a classic example of an irrational number: you cannot write it as the ratio of ANY two integers. Here's the square root of 2 to a few decimal places (it continues forever, and there is no pattern!): sqrt(2) = 1.414213562373095048801688724209698078570 TRANSCENDENTAL AND ALGEBRAIC NUMBERS To understand transcendental numbers, you also need to understand another type of numbers called algebraic numbers. A number is called algebraic if it is the root of a polynomial (of any degree) with rational coefficients. Any number that is not algebraic is called transcendental. 2 and 2/3 are algebraic (they are also rational) because they are the roots of the rational polynomial 3x^2 - 8x + 4 (it is rational because the coefficients are rational numbers) 3x^2 - 8x + 4 = 0 or (3x - 2)(x - 2) = 0 so x = 2 or 2/3 Note that the square root of 2 is also algebraic (it is also irrational) because it is a solution of the rational polynomial x^2 - 2 = 0 A classic example of an transcendental number (that is, not algebraic) is the number pi (shown here to an accuracy of a thousand decimal digits) It goes on forever, just as the square root of 2. 3.14159265358979323846264338327950288419716939937510582097494459230781 6406286208998628034825342117067982148086513282306647093844609550582231 7253594081284811174502841027019385211055596446229489549303819644288109 7566593344612847564823378678316527120190914564856692346034861045432664 8213393607260249141273724587006606315588174881520920962829254091715364 3678925903600113305305488204665213841469519415116094330572703657595919 5309218611738193261179310511854807446237996274956735188575272489122793 8183011949129833673362440656643086021394946395224737190702179860943702 7705392171762931767523846748184676694051320005681271452635608277857713 4275778960917363717872146844090122495343014654958537105079227968925892 3542019956112129021960864034418159813629774771309960518707211349999998 3729780499510597317328160963185950244594553469083026425223082533446850 3526193118817101000313783875288658753320838142061717766914730359825349 0428755468731159562863882353787593751957781857780532171226806613001927 876611195909216420199.... It turns out that pi is also irrational (you cannot write it as the ratio of two integers). It is impossible to cover all possible aspects of these numbers in a single page, so I have kept their description at a bare minimum. If you are interested and want to find out more, you can find a lot of information in any good history of mathematics book. If you have any questions, feel free to reply. -Doctor Luis, The Math Forum Check out our web site! http://mathforum.org/dr.math/
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