Rational and Irrational Numbers: Background You’ll Need 1

  • Recognize different number types, such as whole numbers, integers, and counting numbers

Identify Counting Numbers and Whole Numbers

In elementary mathematics, we frequently use the most fundamental set of numbers, which we typically employ for counting objects: [latex]1, 2, 3, 4, 5, ...[/latex] and so forth. These numbers are referred to as the counting numbers.

The discovery of the number zero was a big step in the history of mathematics. Including zero with the counting numbers gives a new set of numbers called the whole numbers.

Whole and Counting Numbers

Counting numbers start with [latex]1[/latex] and continue.

 

[latex]1,2,3,4,5\dots[/latex]

 

Whole numbers are the counting numbers and zero.

[latex]0,1,2,3,4,5\dots[/latex]

 

The notation “…” is called an ellipsis, which is another way to show “and so on”, or that the pattern continues endlessly.

Opposites and Integers

On the number line, the negative numbers are a mirror image of the positive numbers with zero in the middle. Because the numbers [latex]2[/latex] and [latex]-2[/latex] are the same distance from zero, they are called opposites. The opposite of [latex]2[/latex] is [latex]-2[/latex], and the opposite of [latex]-2[/latex] is [latex]2[/latex] as shown in figure(a). Similarly, [latex]3[/latex] and [latex]-3[/latex] are opposites as shown in figure(b).

This figure shows two number lines. The first has points negative 2 and positive 2 labeled. Below the first line, the statement is the numbers negative 2 and 2 are opposites. The second number line has the points negative 3 and 3 labeled. Below the number line is the statement negative 3 and 3 are opposites.

Opposite

The opposite of a number is the number that is the same distance from zero on the number line but on the opposite side of zero.

 

The notation [latex]-a[/latex] is read the opposite of [latex]a[/latex].

Just as the same word in English can have different meanings, the same symbol in math can have different meanings. The specific meaning becomes clear by looking at how it is used. You have seen the symbol “[latex]-[/latex]” in three different ways.

[latex]10 - 4[/latex] Between two numbers, the symbol indicates the operation of subtraction.We read [latex]10 - 4[/latex] as [latex]10[/latex] minus [latex]4[/latex] .
[latex]-8[/latex] In front of a number, the symbol indicates a negative number.We read [latex]-8[/latex] as negative eight.
[latex]-x[/latex] In front of a variable or a number, it indicates the opposite.We read [latex]-x[/latex] as the opposite of [latex]x[/latex] .
[latex]-\left(-2\right)[/latex] Here we have two signs. The sign in the parentheses indicates that the number is negative [latex]2[/latex]. The sign outside the parentheses indicates the opposite.We read [latex]-\left(-2\right)[/latex] as the opposite of [latex]-2[/latex], which is [latex]2[/latex].

Integers

Integers are counting numbers, their opposites, and zero.

 

[latex]\dots{-3,-2,-1,0,1,2,3}\dots[/latex]