Example 3 Of which set(s) is 0an element: integers, non-negative integers or positive integers?
Solution: Since 0 is in the listings {0,1,2,3, ...}and {..., −2,−1,0,1,2, ...}but not in {1,2,3, ...}, it is an element of the
integers and the non-negative integers.
Practice 4 Of which set(s) is 5an element: integers, non-negative integers or positive integers?
Solution: Click here to check your answer.
When it comes to sharing a pie or a candy bar we need numbers which represent a half, a third, or any partial amount
that we need. A fraction is an integer divided by a nonzero integer. Any number that can be written as a fraction is called
arational number. For example, 3 is a rational number since 3 = 3 ÷1 = 3
1. All integers are rational numbers. Notice
that a fraction is nothing more than a representation of a division problem. We will explore how to convert a decimal to a
fraction and vice versa in section 1.9.
Consider the fraction 1
2. One-half of the burgandy rectangle below is the gray portion in the next picture. It represents
half of the burgandy rectangle. That is, 1 out of 2 pieces. Notice that the portions must be of equal size.
Rational numbers are real numbers which can be written as a fraction and therefore can be plotted on a number line. But
there are other real numbers which cannot be rewritten as a fraction. In order to consider this, we will discuss decimals. Our
number system is based on 10. You can understand this when you are dealing with the counting numbers. For example, 10
ones equals 1 ten, 10 tens equals 1 one-hundred and so on. When we consider a decimal, it is also based on 10. Consider the
number line below where the red lines are the tenths, that is, the number line split up into ten equal size pieces between 0
and 1. The purple lines represent the hundredths; the segment from 0 to 1 on the number line is split up into one-hundred
equal size pieces between 0 and 1.
As in natural numbers these decimal places have place values. The first place to the right of the decimal is the tenths
then the hundredths. Below are the place values to the millionths.
tens: ones: . : tenths: hundredths: thousandths: ten-thousandths: hundred-thousandths: millionths
The number 13.453 can be read “thirteen and four hundred fifty-three thousandths”. Notice that after the decimal
you read the number normally adding the ending place value after you state the number. (This can be read informally as
“thirteen point four five three.) Also, the decimal is indicated with the word “and”. The decimal 1.0034 would be “one and
thirty-four ten-thousandths”.
Real numbers that are not rational numbers are called irrational numbers. Decimals that do not terminate (end) or
repeat represent irrational numbers. The set of all rational numbers together with the set of irrational numbers is called
the set of real numbers. The diagram below shows the relationship between the sets of numbers discussed so far. Some
examples of irrational numbers are √2, π, √6 (radicals will be discussed further in Section 1.10). There are infinitely many
irrational numbers. The diagram below shows the terminology of the real numbers and their relationship to each other.
All the sets in the diagram are real numbers. The colors indicate the separation between rational (shades of green) and
irrational numbers (blue). All sets that are integers are in inside the oval labeled integers, while the whole numbers contain
the counting numbers.
3