1 CHAPTER 1 NUMBERS AIM: To give a background on different kinds of numbers and the number line. 1.1 INTEGERS When starting to count we used the set of counting numbers, also called positive integers. Counting numbers 7 1 ; 2 ; 3 ; 4 ; 5............? A zero was added to the set of counting numbers and then it was called whole numbers. Whole numbers 7 0 ; 1 ; 2 ; 3 ; 4 ; 5........? Soon it became evident that every problem could not be solved with the above whole numbers. EXAMPLE 1.1 Solve the following problems: 1.1.1 4+3 This is no problem. Add the whole numbers 4 and 3 to get an answer of 7. 4+3=7 1.1.2 5-8 When subtracting a larger whole number from a smaller one there is no solution in the set of whole numbers. The set of numbers will have to be extended to include negative integers. Negative integers 7 ......... -6 ; -5 ; -4 ; -3 ; -2 ; -1 ? The answer for example 1.1.2 will be -3 and that will come from the set of negative integers. If the set of whole numbers is combined to the set of negative integers, the new set of numbers is called the set of integers. Integers 7 ........ -4; -3; -2; -1; 0; 1; 2; 3; 4; 5 .........? An integer can be odd (uneven) or even. Odd integers 7 ........ -3 ; -1 ; 1 ; 3 ; 5 .........? Even integers 7 ........ -4 ; -2 ; 0 ; 2 ; 4..........? 2 1.2 NUMBER LINE FOR INTEGERS To construct the number line we have to construct a straight line with a reference point on it. The reference point will normally be 0 . }~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~| 0 Divide the line in convenient unit lengths. }~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~| 0 We shall now assign positive values to the units to the right of zero and negative values to the left of zero. }~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~| -4 -3 -2 -1 0 1 2 3 4 5 We refer to any position on the number line as a certain number of units away from 0. 1.3 RATIONAL NUMBERS EXAMPLE 1.2 Solve the following problems: 1.2.1 4×3 This is no problem. Multiply integer 4 by integer 3 to get an answer of 12. 4 × 3 = 12 1.2.2 4÷8 When dividing the integer 4 by the integer 8 the answer will not be an integer. The answer will be the number 4/8. That is an example of a rational number. In general we define rational numbers as any number of the form where c and d are integers (d not zero). EXAMPLE 1.3 The following are all examples of rational numbers: 3 1.4 NUMBER LINE FOR RATIONAL NUMBERS Rational numbers can also be represented on the number line. EXAMPLE 1.4 Represent the following rational numbers on the number line: C C C C CC C C C }~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~| -4 -3 -2 -1 0 1 2 3 4 5 1.5 REAL NUMBERS The set of counting numbers, whole numbers, integers and rational numbers is called real numbers. EXERCISE 1.1 Draw a number line and represent the following numbers on it: 4 1.6 RELATIONSHIPS BETWEEN NUMBERS Between any two real numbers c and d , one of the following relationships must exist: c=d c>d c<d number c equals number d number c is greater than number d number c is less than number d EXAMPLE 1.5 Determine the relationship between the following numbers: 1.5.1 1.5.2 1.5.3 1.7 8,9 8,0 -4 , +4 8 < 9 8 > 0 -4 < +4 NUMBER LINE FOR REAL NUMBERS There are more than one real number that will satisfy the relationship y > 2. That is called an inequality. The set of all the possibilities that will satisfy the inequality is called the solution set. It will not be possible to write down all the possibilities, therefore it will be represented on the number line. Note that there is a difference between the next two inequalities: y>2 and y $ 2. In the first inequality 2 is not included in the solution set. Only values more than 2 are included. In the next inequality 2 and everything greater than 2 are included in the solution set. ( > - more than ; $ - more and equal to) 5 Consider: y < -1 and y # -1. In the first inequality -1 is not included in the solution set. Only values less than -1 are included. In the next inequality -1 and everything less than -1 are included in the solution set ( < - less than # - less and equal to). EXAMPLE 1.6 Represent the following inequalities on the number line: 1.6.1 y=3 C wxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxv -4 -3 -2 -1 0 1 2 3 4 5 6 7 1.6.2 y>0 B——————————< wxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxv -4 -3 -2 -1 0 1 2 3 4 5 6 7 1.6.3 y<3½ =——————————B wxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxv -4 -3 -2 -1 0 1 2 3 4 5 6 7 1.6.4 y # -2 =——————C wxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxv -4 -3 -2 -1 0 1 2 3 4 5 6 7 1.6.5 y $ -1 ½ C——————————< wxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxv -4 -3 -2 -1 0 1 2 3 4 5 6 7 1.6.6 -1 < y < 2 B————B wxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxv -4 -3 -2 -1 0 1 2 3 4 5 6 7 1.6.7 1.6.8 1.6.9 0#y#4 C—–————C wxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxv -4 -3 -2 -1 0 1 2 3 4 5 6 7 y # -2 or y $ 2 =———————C C———————< wxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxv -4 -3 -2 -1 0 1 2 3 4 5 6 7 y < 0 or y $ 3 =————————B C———————< wxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxv -4 -3 -2 -1 0 1 2 3 4 5 6 7 6 Note that B indicates that the value is not included in the solution set and C indicates that the value is included in the solution set. EXERCISE 1.2 Illustrate the solution set for each of the following on a number line: 1.2.1 1.2.3 1.2.5 1.2.7 1.2.9 y<2 y $ -2 -2 < y < 4 0<y#3 y # 1 or y $ 2 1.2.2 1.2.4 1.2.6 1.2.8 1.2.10 1.8 PROPERTIES OF ZERO y>4 -3 # y -1 # y # 3 y < -3 or y $ 3 y=3 Zero can cause a lot of errors in your calculations, if you are not aware of the different properties of zero. For any real number c : 1.7.1 1.7.2 1.7.3 c+0=c c -0=c c×0=0 6+0 =6 -5 - 0 = -5 2×0=0 EXERCISE 1.3 Perform the following operations: 1.3.1 3+(6-6) 1.3.2 7( 2 - 2 )
