Real Number TerminologyDate: 12/04/96 at 20:25:04 From: Philip Silver Subject: Not existent over the reels What does it mean to be non-existent over the reals? Can you explain this so I can explain it to my 7th grade daughter? Thank you. Date: 12/10/96 at 11:56:38 From: Doctor Rob Subject: Re: Not existent over the reels This is a difficult idea for youngsters (or oldsters!) to grasp. The real number system, sometimes called "the reals" for short, consists of: all the whole numbers (0, 7, -3, and so on), all the fractions (1/2, -22/7, and so on), and all the decimal fractions, even with infinitely many digits (0.72, 3.333333333..., -24.380783... and the like, including Pi = 3.14159265358979...). They are called "real" numbers because they are used to measure real things like lengths, speeds, volumes, areas, forces, and so on. Often we are faced with the task of finding a real number that is a solution of an equation, such as the solution x of the equation 7*x - 13 = 8, or the solution A of the equation A^2 = 81. Students are taught how to deal with these equations in algebra class. Sometimes, however, there is no real number which works for a given equation. The usual example of this is x^2 = -1 (x^2 here means "x squared" or x times x). Why is there no real number x which works in this equation? It is because if x is positive, x^2 will be positive, but if x is negative, x^2 will still be positive, and if x is zero, x^2 will be zero. This covers every possible real number, since each one is either positive, negative, or zero. So x^2 is positive or zero, but -1 is negative, so the two can never be equal. The phrase that is used to express this situation is that a solution x does not exist in the real number system (or "over the reals"). Sometimes, however, we really do want to have a solution to x^2 = -1 that we can work with. This can be very useful in higher mathematics, for reasons I can't go into here. Since no real number will do, we have to invent a new kind of number, called an "imaginary" number. This name was chosen just to contrast them with the real numbers. Another case where there is no solution over the reals is an equation like 2^x = -4, since powers of 2 are all positive. The imaginary number which is the solution of x^2 = -1 is called "i", and it is also called the square root of -1. The creation of this new kind of number allows us to expand our number system by looking at all numbers of the form a + b*i, where a and b are real numbers, and i is as above. This system of numbers is called the complex number system. The real numbers are part of this system, being the complex numbers for which b is zero, that is a + 0*i = a. In the complex number system, every number has a square root, which isn't true in the real number system (as we saw above). If you want to learn more about all the different types of real numbers, take a look at this Web page: http://mathforum.org/dr.math/problems/chris.8.16.96.html I hope this helps. If I can explain further, please write back. -Doctor Rob, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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