UNIT 3: INTEGERS. Mónica Cárceles Alemán. IES Rector don Francisco Sabater García Positive and negative integers

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1 UNIT 3: INTEGERS 3.1. Positive and negative integers There are many situations in which you need to use numbers below zero, one of these is temperature, others are money that you can deposit (positive) or withdraw (negative) in a bank, steps that you can take forwards (positive) or backwards (negative). An integer is a whole number (not a fractional number) that can be positive, negative, or zero. Examples of integers are: -5, 1, 5, 8, 97 Positive integers are all the whole numbers greater than zero: 1, 2, 3, 4, 5,... Negative integers are all the opposites of these whole numbers: -1, -2, -3, -4, The set of integers, denoted Z, is formally defined as follows: Z = {..., -3, -2, -1, 0, 1, 2, 3,...} The number line is a line labelled with the integers in increasing order from left to right, that extends in both directions: For any two different places on the number line, the integer on the right is greater than the integer on the left. 9 > 4 Is read: nine is greater than four, -7 < 9 Is read: minus seven is less than nine Absolute Value of an integer The absolute value of any number is the distance between that number and zero on the number line. If the number is positive, the absolute value is the same number. If the number is negative, the absolute value is the opposite. The absolute value of a number is always a positive number (or zero). We specify the absolute value of a number n by writing n in between two vertical bars: n. Examples: 6 = 6-10 = 10 0 = = =

2 3.3. How to compare integers - Every positive integer is bigger than any negative integer. 5 > If we have two positive integers, the biggest is the one with bigger absolute value. 18 > 3 - If we have two negative integers, the biggest is the one with smaller absolute value. 5> How to add and subtract integers When we are adding or subtracting two integers we can find two situations: If both integers have the same sign, we add them and write the same sign in the solution. Examples: = = - 7 If one of the integers is positive and the other is negative, we subtract them and the solution will have the same sign as the biggest of the initial integers. Examples: 3 1 = = = = - 6 When we find parentheses we proceed like that: If there is a + before the parentheses, the numbers inside it keep their signs. If there is a - before the parentheses, the numbers inside it change their signs to the opposite. Examples: 5 + ( ) = = = 6-9 (2 10) = = = How to multiply and divide integers To multiply or divide integers, we multiplicate or divide their absolute values and add the sign + if they have the same sign, and the sign - if they have different signs. Examples: 2 (-3) = (-8) = = = : 5 = 3 20 : (-2) = : 8 = -2-9 : (-3) = Combined operations The order we have to follow to do combined operations with integers is the same we have with natural numbers: 1º) Parentheses. 2º) Powers and roots. 3º) Multiplications and divisions, left to right. 4º) Additions and subtractions, left to right. 2

3 Example: (-10) : 2 (-4) (+1) = -5 (-4) = = - 1 EXERCISES 1. What is the temperature which is: a. 7 degrees lower than 5º C b. 6 degrees lower than 4º C c. 16 degrees higher than - 4º C d. 9 degrees lower than - 6º C 2. Represent the following number in the number line: Calculate the absolute value of the following numbers: a. +3 b. 2 c. 5 d Organize these numbers from the smallest to the largest: Calculate: a) = b) 5 4 = c) +8 2 = d) 5 +9 = 6. Calculate: a) = b ) = c) = d) = 7. Write without parentheses: a) + (+3) = b) (+4)= c) (-5)= d) + (-2) = 8. Calculate: a) +(+3) + (-5) = b) (+4) (+6) = c) (-5) + (+7) = 3

4 d) - (+3) + (+1) (-4) = e) -(+2) - (+1) (+5) = f) -(+2) + (-1) + (-4) (-5)= g) -(+1) - (+3) - (-4) (-5)= 9. Calculate: A) 4 + (5 8) + 3 = B) 9 - (7 + 1) - 10 = C) (-7 3) = D) - (7+ 2 9) + 12 = E) (9 + 3) - (12 15) - 2 = 10. Calculate: a) ( 1 2 3) ( ) = d) ( 1+2 9) (5 5) 4+5 = e) ( 1 9) ( ) (8 7) = f) 4 (4+5) (8 9)+1+6 = 11. Calculate: a) (+7) (+2) = b) (+12) ( 3) = c) ( 10) (+10) = d) ( 5) (+8) = e) ( 1) ( 1) = f) (+5) (+20) = g) (+16) : (+2) = h) ( 8):( 1)= i) ( 25):(+5)= j) ( 100) : (+10) = k) (+12):( 3)= l) (+45):(+9)= 12. Complete: a) (+9)... = 36 b) ( 7)... = +21 c)... ( 8) = 40 d)... (+10) = 100 e) ( 30)... = +30 f) (+6)... = 0 g) (+42) :... = 7 h) ( 8) :... = +1 i)... : ( 9) = +6 j) ( 20) :... = 20 k)... : ( 6) = +5 l) (+9) :... = Calculate: a) : 4 7 4

5 b) c) d) : e) f) ( 4 2) g) ( 7 8) h) : 3 i) 40: Calculate the value of the following powers: a) 5 1 b) 5 1 c) 5 2 d) On the second of January, the temperature dropped from 3ºC at two o clock in the afternoon to -11 ºC at 8 a.m. the next day. How many degrees did the temperature fall? 16. On the 1 st of December, the level of the water in a reservoir was 130 cm above its average level. On the 1 st of July it was 110 cm below its average level. How many cm did the water level drop in this time? 17. What is the change in temperature a customer in a grocery store experiences when they walk from the chilled vegetable section at 4 ºC to the frozen fish section which is set to 18 ºC? 18. In an experiment, Carlos has to manipulate some liquids which temperature is 17ºC minus zero. He has to decrease the temperature 15ºC. After 30 minutes, he has to increase the temperature 4ºC, and finally, after another 30 minutes, he has to decrease the temperature 13ºC. 5