Sets of Numbers

The table below gives the different sets of numbers that we will be using. Note that the table is not complete. It refers to the numbers that most of us have experienced up to this course.
Name Set DescriptionVerbal Description
Counting Numbers
Natural Numbers
N={1,2,3,4,5,6,7,8,9,10,...} These are the numbers that we use to count things. There are no fractions here, and no negative numbers. Notice that the set N does not include the number zero.

Also, note that if we add any two natural numbers then the sum is a natural number. If we multiply any two natural numbers, then the product is also a natural number. However, we can not always subtract or divide natural numbers and get an answer that is also a natural number. Thus, 3-7 is not a natural number, nor is 3/7.

The number 1 is particularly important to multiplication because the product of 1 and any natural number is that natural number. We say that 1 is the identity element for multiplication. Addition does not have a similar special identity value in the set of natural numbers.

Whole Numbers W={0,1,2,3,4,5,6,7,8,9,...}
or W=N{0}
Whole numbers are all of the natural numbers and 0. Zero is an extremely important number. It is the special identity element for addition that was missing in the natural numbers. Thus, the sum of 0 and any whole number is that whole number. Multiplication has not changed (we can multiply any whole numbers and the result will be a whole number, and 1 remains the identity element for multiplication). Subtraction and division still do not necessarily give whole number answers.
Integers Z={...,-3,-2,-1,0,1,2,3,...} The integers are formed by taking the whole numbers and their opposites. This introduction of the negative numbers means that the integers can get a positive or as negative as we want. They go in both directions. Furthermore, with integers, we can be sure that we can subtract any integer from any integer and we will get an integer as an answer. Both 0 and 1 retrain their respective identity properties. Division between integers is still not sure to give us an integer answer.
Rational Numbers Q={x| x=p/q, where pZ,
qZ, and q0}
Q={x| x is a repeating or terminating decimal}
The set definition is pretty strange. Where the Natural numbers (N), the Whole numbers (W), and the Integers (Z) could be listed, the Rational numbers (Q for Quotient) need to use a set description. Furthermore, that description has two different forms.

The first set description says that a rational number is any number that can be expressed as a fraction, where both the numerator and the denominator are integers, and where the denominator can not be 0. This means that the number 5/7 is a rational number: it fits that definition. So do -4/9, -5/2, and 234/53. The number 8 is rational because it can be written as 8/1, or as 16/2, or as 1200/150. Note that the value /5 does not pass the test because is not an integer.

The second set description says that rational numbers are all of the repeating and termination decimal numbers. In this case, when we say repeating we mean that the number repeats some set pattern of digits. Thus, 6.38 terminates at the hundredths place, whereas 6.3473247347347... repeats the sequence 347 forever. Both of these numbers are rational numbers. There are numbers that do not repeat the same pattern of digits, and they are not rational numbers. We have special names for some of them, such as . That is a number whose deciaml representation can be calculated for as many digits as we want. However, the value of never falls into a situation where it is a set pattern of digits that just keeps repeating. We can even create decimal numbers that do not repeat a specific set of digits. For example, the number

has a pattern so we can write it out for as many digits as we want, but it does not repeat a specific set of digits. Therefore, it is not a rational number.

It is important to note that the two descriptions are equivalent. They specify exactly the same elements of the Rational numbers. Any fraction p/q where p and q are integers and q is not zero, can be represented as a terminationg or repeating decimal. Thus, 5/8=0.625, a terminating decimal, and 4/37=0.108108108108..., a repeating decimal. We can get that decimal representation by dividing the numerator bu the denominator. Furthermore, any terminating or repeating decimal can be written in the form of p/q where p and q are integers.

Note that we can add, subtract, multiply, and divide any two rational numbers (except division by 0) and the answer will be another rational number.

Irrational Numbers L={x|x is a decimal number that neither terminates nor repeats} In short, irrational numbers are the decimal values that are not rational. is one of those values, as is the strange number that we constructed above, namely,
At first glance there may seem to be just a few irrational numbers but this is not the case. We know and use rational numbers all the time. We rarely deal with irrational numbers. But they are there, and there are lots of them. In fact, in a very real sense there are more irrational numbers than there are rational numbers. You might be able to glimpse this if you note that not only is irrational, so are +4, +3.2, 5, -7/3, and 8/21+13. And each of these is a different irrational number.

One might be think that the sum, product, difference, and quotient of irrational numbers should also be irrational, but this need not be the case. Using our friendly value of we see that - is 0, a rational number, and that /=1, another rational number.

Another way to generate an irrational number is to look at the square root of numbers. The is the number that can be multiplied by itself to produce 2. The number is an irrational number. And yet ()()=2 by definition, so again, we have the product of two irrational numbers giving a rational number.

Note that square roots do not have to be irrational. For example, is rational because its value is 2 (since 2 times 2 is 4).

Real Numbers R={all decimal numbers}
Real numbers are all the decimal numbers, terminating, repeating, and non-terminating. Therefore, R is the set of all the rational numbers, Q, and all the irrational numbers, L.

©Roger M. Palay
Saline, MI 48176
January, 1999