Sets

Although set theory and operations are not now included in this course it is useful to know what a set is, and about the various number sets.

Notation

A set is a collection of objects. These objects, called the members or elements of the set, are enclosed in braces {...} and separated by commas.

A set can be described in words: A = { the first three natural numbers}

or by listing each member: A = {1, 2, 3}


 

Number sets

The numbers we use can be chosen from several sets.

  • The set of natural numbers. 
  • N = {1, 2, 3, 4, ...}

  • The set of whole numbers.

    W = {0, 1, 2, 3, ...}

  • The set of integers.

    I = {... -2, -1, 0, 1, 2, 3,...}

  • The set of rational numbers

    Q =Y8_Integers_01.gif

    e.g. Y8_Integers_02.gif

  • The set of irrational numbers

    Irrational numbers are numbers that cannot be written as rational numbers.

    e.g. {infinite, non-recurring decimals}

    Y8_Integers_03.gif

  • The set of real numbers (rational numbers + irrational numbers)

    R = {all numbers}



Number lines

The number sets can be shown on number lines.

Natural numbers, N (excludes zero)

Whole numbers, W (includes zero)

Integers, (positive and negative whole numbers)

Rational numbers (all except irrationals)

Real numbers, R (all numbers)

Y8_Integers_04.gif


 

Integers

The integers are the positive and negative numbers and zero.

For practice with integer calculations 

 

Addition

Use a number line when adding and subtracting integers. The first integer is the starting point on the number line.

  • When adding a positive integer, move to the right.

e.g. 4 + 2 = 6

Y8_Integers_05.gif
  • When adding a negative integer, move to the left.

    e.g. 4 + − 2 = 2

Y8_Integers_06.gif

 

Subtraction

  • When subtracting a positive integer, move to the left.
 

e.g. 4 − 2 = 2

Y8_Integers_07.gif
  • When subtracting a negative integer, move to the right
 

e.g. 4 − − 2 = 6

Y8_Integers_08.gif

 

 

Multiplication and division

The rule is the same for both multiplying and dividing two integers.

  • If the signs of the two integers are the same, the answer is positive.
  • If the signs of the two integers are different, the answer is negative.

The table illustrates these rules:

×/÷
+ve
-ve
+ve
+
-
-ve
-
+

 

 

Examples:

For multiplication
For division
4 × 2 = 8
4 ÷ 2 = 2
- 4 × − 2 = 8
- 4 ÷ − 2 = 2
4 × − 2 = − 8
4 ÷ − 2 = − 2
- 4 × 2 = − 8
- 4 ÷ 2 = − 2