INTRODUCTION TO INTEGERS

Transcription

1 1 INTRODUCTION TO INTEGERS A. WHAT IS AN INTEGER? Integers are numbers made from natural numbers. On a number line, the arrows on either side of it would extend in opposite directions to include integers. Numbers to the right of 0 are positive integers. These may be written with a plus sign or the plus sign can be left out. Numbers to the left of the 0 are negative integers and are written with a negative sign in front. Zero is neither positive nor negative. The symbol for integers is ZZ, e.g. ZZ= {-2; -1; 0; 1; 2} Example: B. THE HISTORY OF INTEGERS Counting was first done in the form of the notches carved onto a baboon s bone. However, as society developed and people s needs changed, a more sophisticated method of counting was needed. This led to other types of numbers being used, such as fractions and integers. Negative numbers were first used by the Babylonians and the Chinese. When they were first used, people found it a strange concept to use integers. In today s modern society, integers have become an important part of everyday life. Various fields like accounting and physics make use of negative numbers. Example: If you take a loan from a bank, the balance that you owe will be reflected as a minus balance. For example: -R If you deposit R into your bank account to pay the loan, the new balance will be reflected as a positive balance. For example: +R EXERCISE 1. Study the examples below: a) = 5 1

2 2 b) (-3) + (-4) = (-7) c) (+5) + (-11) = (-6) RULES WHEN ADDING OR SUBTRACTING INTEGERS From the above, we can conclude the following: 1. A positive integer + a positive integer = a positive integer 2. A negative integer + a negative integer = a negative integer 3. A positive integer + a negative integer = a positive or a negative integer, depending on which number is greater. For example: (+5) + (-11) = (-6); (-10) + (+25) = (+15) 1. Draw number lines and use them to answer the following: a. 6 8 b c d e Now, solve the problems below, without the use of number lines: a b c d e Complete the number sequences below: a. 8; 6; 4; 2 ; ; ;. b. 2 ½: 2; 1 ½; 1 ; ; ;. 2

3 3 c. -4; 3; 10 ; ; ;. d. -64; -56; -48 ; ; ;. e. 1; 2; 4; 7 ; ; ;. 4. According to the weather report, the temperature today is 9 0 C. What would the temperature be if it had to: a. rise by 8 0? b. drop by 12 0? c. drop by 9 0? 5. a. If a person was born years ago, in what year was he born? b. If someone was born in 500 BC, how many years ago was he born? 6. Write the calculations and the answers for the following: a. A diver descends 85m into the sea from a height of 55m above sea level. b. The current temperature of 12 0 C will decrease by 14 0 C by midnight. c. Lettie borrows R from a bank. She makes a loan of a further R How much does she owe the bank? d. If I climb into a lift on the 15 th floor and I descend 20 floors to the basement. What floor am I on? e. A miner descends 48m from a depth of 13m below sea level. What is his position? EXERCISE 2 1. Solve the problems below, without the use of a number line or a calculator: a k b l c m d n e o f p g q h r i s j t

4 4 2. Which is: a. Lower: 50m above sea level or 25m above sea level b. Earlier: 50 BC or 100 BC c. Warmer: -3 0 C or 0 0 C? d. Later: BC or 500 AD? e. Higher: 100m below sea level or 200m below sea level? 3. Arrange the numbers below in ascending order: a. 4; 8; -12; -9; 0; 12; -3; 6 b. 0; 13; -5; 2; -9; -2; 6; 1 c. 412; 0; 389; -245; 50; -16; -19; Arrange the numbers below in descending order: a. -3; -5; 0; 2; 7; -11; 5 b ; ; 1 500; 1 250; -75; 50; -50; 75 c. 10; -13; 15; -24; -2; 5; -12; 4 5. Fill in or = : a b c d e Fill in the missing numbers: a. 16; 12; 8 ; ; ;. b. -35; -20; -5 ; ; ;. c. -35; -32; -29 ; ; ;. d. 150; 100; -50; ; ;. e. -49; -42; -35 ; ; ;. 4

5 5 CONCEPT : MULTIPLYING AND DIVIDING WITH INTEGERS You have already discovered the rules for adding and subtracting integers i.e. + and + = + - and - = - + and - = + or (depending on the value of the numbers) Now, you need to learn the rules for multiplying and dividing integers 1. A positive multiplied by a positive gives a positive integer For example: (+) x (+) = (+) 2. A positive multiplied by a negative gives a negative integer For example: (-) x (+) = (-) or (+) x (-) = (-) 3. A negative multiplied by a negative gives a positive integer For example: (-) x (-) = (+) Examples: 1. 6 x 3 = x 3 = -18 ; 7 x (-2) = (-6) x (-3) = 18 EXERCISE 3 Complete the following: A x (-4) x (-8) 2. (-5) x (-9) 7. (-9) x (-7) 3. (-2) x 6 8. (-6) x x x 5 5. (-8) x (-9) x 9 B (-2) (-5) (-6) (-7) (-144) (-12) 10. (-32) (-4) 5

6 C. Write out the calculations and the answers for each of the problems below: a. A mother sees a bag of sweets costing R22. She needs to buy 5 of these bags for each member of her family, but does not have cash on her. How much money will she owe if she pays for the sweets by credit card? b. A company applies for a loan of R The loan is given and the 8 directors each undertake to pay back equal amounts. How much money will each director have to pay? c. Is [ (-3) x (-3) ] + [ (-4) x (-4) x (-4) ] equal to (-5) x (-5)? Show your working out. d. A child borrows R5 from a friend every day throughout the month of October. How much money will he have to pay back at the end of the month? 6 D. Complete: a. -5 x a + (-2 x 13) = 4 b. 4 x b + 20 = 4 c. 81 (-9) = c d. [(-5) x (-7)] + (-5) x (7) = d e. [(-40) 5] x [(-12) x (-10)] = e EXERCISE 4 CONCEPT : CONSOLODATION OF INTEGERS Complete the following on your own and without the use of a calculator 1. Complete: a f. -60 x 25 b (-896) g. 13 x (-10) c. ( ) + ( ) h. (-750) 4 d (-298) i (-5) e j. (-1 040) (-8) 2. Complete the following: a. 13 (11 +9) f b. 3 x ( ) g. 10 ( ) 2 c h x ( ) d. 40 ( ) i e. 13 x (8 19) + 5 j x (-5) 6

7 7 3. A golf score card looks as follows: Par 4 Hole 6 Tina -1 Rajesh +1 Lucy 0 Themba -1 Cassie -2 a. Who got the best score at his hole? b. What does -2 mean here? c. How many times did Rajesh hit the ball for this hole? 4. If you have R in your bank account: a. What would your balance be if you withdrew R 2 450? b. What would your balance be if you deposit R 625 after the withdrawal? 5. Thandi lends R 30,75 to a co-worker for each working day throughout the month of October so that the friend can pay for her taxi fare. How much money must Thandi s friend giver her at the end of the month? 6. An outstanding dept of R must be paid by a group of 8 people. How much money is each person responsible for? 7. Temperature can be measured in degrees Fahrenheit ( 0 F), degrees Celsius ( 0 C) or degrees Kelvin (K). The Kelvin scale is the international scientific temperature scale, but Celsius is more commonly used. Kelvin temperature can be converted to Celsius temperature by subtracting 273. i.e. Temperature in 0 C = K 273 a) Use this knowledge to complete the table and the questions below: Temperature in Kelvin Temperature in 0 Celsius Water (boiling or steam) Earth (deserts) The human body A Earth (at the Equator) B B Water (freezing or ice) C Earth (at the poles) D D Moon (at night) E E

8 b) Which temperature is higher: the earth at the Equator or the earth at the poles? Explain your answer. c) What is the difference in temperature between the earth at the Equator and the earth at the poles? 8 8. Study the time line and answer the questions that follow: BCE CE a. i) If someone was born years ago, what was the person s date of birth? ii) If that same person died 88 years later, what year would he have died in? b. i) If someone was born in 200BC, how many years ago was it? ii) If that same person died 94 years later, what year would he have died in? 7. Tabulate: a. The integers between -4 and 6. b. The five integers bigger than Write down: a. The integer that is 4 smaller than -5 b. The integer that is 11 larger than -1 c. The integer that is 11 larger than -1 d. The integer that is 6 smaller than -2 8