Integers, Rational & Irrational Numbers

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#### What is an integer?   { ... -3, -2, -1, 0, 1, 2, 3, ... }

Integers are the whole numbers, negative whole numbers, and zero. For example, 43434235, 28, 2, 0, -28, and -3030 are integers, but numbers like 1/2, 4.00032, 2.5, Pi, and -9.90 are not. We can say that an integer is in the set: {...3,-2,-1,0,1,2,3,...} (the three dots mean you keep going in both directions.)

It is often useful to think of the integers as points along a 'number line', like this:

Note that zero is neither positive nor negative.

The terms even and odd only apply to integers; 2.5 is neither even nor odd. Zero, on the other hand, is even since it is 2 times some integer: it's 2 times 0. To check whether a number is odd, see whether it's one more than some even number: 7 is odd since it's one more than 6, which is even.

Another way to say this is that zero is even since it can be written in the form 2*n, where n is an integer. Odd numbers can be written in the form 2*n + 1. Again, this lets us talk about whether negative numbers are even and odd: -9 is odd since it's one more than -10, which is even.

Every positive integer can be factored into the product of prime numbers, and there's only one way to do it for every number. For instance, 280 = 2x2x2x5x7, and there's only one way to factor 280 into prime numbers. This is an important theorem: the Fundamental Theorem of Arithmetic. See Notes and Literature on Prime Numbers from Understanding Mathematics by Peter Alfeld, and the Largest Known Primes page.

Most mathematicians, at least when they're talking to each other, use Z to refer to the set of integers. In German the word "zahlen" means "to count" and "Zahl" means "number." Mathematicians also use the letter N to talk about the set of positive integers, in other words the set {1,2,3,4,5,6, ...}.

#### Rational Numbers     5/1, 1/2, 1.75, -97/3 ...

A rational number is any number that can be written as a ratio of two integers (hence the name!). In other words, a number is rational if we can write it as a fraction where the numerator and denominator are both integers.

The term "rational" comes from the word "ratio," because the rational numbers are the ones that can be written in the ratio form p/q where p and q are integers. Irrational, then, just means all the numbers that aren't rational.

Every integer is a rational number, since each integer n can be written in the form n/1. For example 5 = 5/1 and thus 5 is a rational number. However, numbers like 1/2, 45454737/2424242, and -3/7 are also rational, since they are fractions whose numerator and denominator are integers.

So the set of all rational numbers will contain the numbers 4/5, -8, 1.75 (which is 7/4), -97/3, and so on.

Is .999 repeating a rational number? Well, a number is rational if it can be written as A/B (A over B): .3 = 3/10 and .55555..... = 5/9, so these are both rational numbers. Now look at .99999999..... which is equal to 9/9 = 1. We have just written down 1 and .9999999 in the form A/B where A and B are both 9, so 1 and .9999999 are both rational numbers. In fact all repeating decimals like .575757575757... , all integers like 46, and all finite decimals like .472 are rational.

#### Irrational Numbers     sqrt(2), pi, e, the Golden Ratio ...

Irrational numbers are numbers that can be written as decimals but not as fractions.

An irrational number is any real number that is not rational. By real number we mean, loosely, a number that we can conceive of in this world, one with no square roots of negative numbers (such a number is called complex.)

A real number is a number that is somewhere on a number line, so any number on a number line that isn't a rational number is irrational. The square root of 2 is an irrational number because it can't be written as a ratio of two integers.

Other irrational numbers include the square root of 3, the square root of 5, pi, e, and the golden ratio. (For more information about pi and e, see Pi = 3.14159... and E = 2.71828..., also from the Dr. Math FAQ.)

Pi is an irrational number because it cannot be expressed as a ratio (fraction) of two integers: it has no exact decimal equivalent, although 3.1415926 is good enough for many applications. The square root of 2 is another irrational number that cannot be written as a fraction.

In mathematics, a name can be used with a very precise meaning that may have little to do with the meaning of the English word. ("Irrational" numbers are NOT numbers that can't argue logically!)

#### On the Web:

More information can be found by searching the Dr. Math archives or the Math Forum site for "integer" or "rational" or "irrational" (just the word, not the quotes).