Kinds of Numbers

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     There are different kinds of numbers used in Mathematics. Here we list some of the key number sets and give a few major properties of each.

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Number Sets

The natural numbers.   These are the counting numbers; the first numbers you learn as a child.

N = {1, 2, 3, ... }

The Integers.   The integers include the zero as well as the negative whole numbers. Although obvious to us, the negative numbers and zero only came into acceptance slowly in the late middle ages. Without these added in to the naturals, many subtractions would go unanswered

z = { ... -2, -1, 0, 1, 2, ... }

The Rational Numbers.   Here we have the fractions. Just as we need negatives to answer some subtraction problems (3 - 5 = ?) we must introduce fractons if we want to answer some division problems. What is 5 divided by 3? We need a fraction 5/3. The rationals are all of the ratios(divisions) of integers, but you cannot divide by zero.

Q = { a/b | a, b Z and b 0 }

It would be nice if we could just stop right here. Unfortunately we cannot. For a number as simple as 2 is not in the set Q.

2 Q.   Well, what if 2 were in Q. Then 2 = a/b where a and b would be integers. So, 2 = (a/b)2 or 2 = a2/b2. This means that

2b2 = a2

Already something looks fishy about this. It is well known that an integer can be factored into primes in just one way. Apparently we have an even number of prime factors on the right-hand side and an odd number of prime factors on the left-hand side. This cannot be. Ah, a contradiction! So it can only be that 2 a/b for any a or b in the integers. And 2 cannot be a rational number.
The Irrational Numbers   How long is the diagonal of a square one unit on a side? Well, it's 2 . (Recall that the hypoteneuse of a right triangle is the sqare root of the sum the squares of the other two sides by The Pythagorean Theorem.) So 2 seems to correspond to a definite point on the line. If we want to have a number for each point on the line we have no choice but to include 2 in our numbers. We now define the irrational numbers

Q' = {x | (x corresponds to a point on the line) and (x Q) }

Examples are p, e, 2, 3, 5, and 1/2 amongst many more.

The Real Numbers   The real numbers are the numbers that we deal with most frequently in cegep level mathematics courses. They correspond in a one-to-one fassion with the points on a line. In fact this correspondence will serve as our "definition" of the real numbers. We can also state that

R = Q Q'

That is the reals are made up of two important subcollections; those numbers that can be expressed as ratios of integers and those that cannot.

The Complex Numbers   Try to solve x2 + 1 = 0. You end up trying to write (-1) but no such real number exists. This is why mathematicians define i = (-1) to be an imaginary number. The complex numbers are numbers built up from a real part and an imaginary part.

C = { a + bi | a, b R and i = (-1) }

Although they play only a minor role in cegep level mathematics, these numbers are extremely important in more advanced mathematics and science.

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Open intervals.   Open intervals are subsets of the reals whose elements lie between two endpoints, but do not include those endpoints. They are represented by using parentheses. In set notation we write

(a, b) = { x R | a < x < b }

Notice that any number in an open interval always has at least a little bit of the inteval on either side.

Closed Intervals   Closed intervals are subsets of the reals whose elements are between two endpoints and may be equal to those endpoints. They are represented by using square brackets. We write

[a, b] = { x R | a x b }

Since points have no length the open interval (a, b) and the corresponding closed interval [a, b] have the same lenth given by b - a.

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Density Properties

  1. Between any two rationals there is another rational.

    Let p and q be two ratioal numbers. Then their average r = (p + q)/2 is half way between them.

  2. Beween any two rationals there are infinitely many rationals.

    Just reapply (1) above on the interval (p, q) over and over again.

  3. No rational number has a "next" rational number.

    Notice that a natural number n is followed by a next natural n + 1. Not so for rationals -- let p be a rational number and suppose q is the "next" rational number. But r = (p + q)/2 is between p and q. This is contradictory.

  4. Open intervals of rationals have neither "first" nor "last" elements.

    suppose r is the "first" element in (a,b). Where is (a + r)/2? Again, we have a contradiction.

  5. Properties (1) to (4) above remain true if the word rational is replaces by irrational

    No argument is supplied.

  6. In any interval (a, b), for a and b real, there are infinitely many rationals, and infinitely many irrationals.

    We say that the rationals are dense in the irrationals, and vice versa.

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Representation Properties

  1. The rationals are represented by terminating or repeating dcimals.

    Notice that 3/5 = .6 is a termainating deimal; 1/3 = .333... is a rpeating decimal; 11/7 = 1.571428571428... is also a repeating decimal.

  2. The irrational numbers are represented by nonrepeating-nonterminationg decimals.

    For example 1.01001000100001... never repeats the same sequence of numbers.

  3. The decimal representation for rationals is not unique.

    What is the difference between 0.5 and 4.999...? Let d = 0.5 - 4.999... be, say, 0.0...01 and add this back to 4.999.... We get a result bigger than 5. Thus d = 0, and 5 = 4.999.... As we see below



    which is larger than 5.

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Transcendental Numbers

A number that is not the solution of any polynomial with rational coefficients is called a transcendental number. Both p and e are transcendental numbers. Numbers that are solutions of polynomials with rational coefficients are called algebraic numbers. 2 and 3 are algebraic numbers. For example, the solution of x2 - 3 = 0 is 3.

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