Positive and negative number activities

I've put together some math activities about positive and negative numbers and some basic exercises to the most advanced, I hope you like it.

RELATIVE WHOLE NUMBERS
INTRODUCTION:

Note that, in the set of natural numbers, the subtraction operation is not always possible.

examples:

a) 5 – 3 = 2 (possible: 2 is a natural number)
b) 9 - 9 = 0 (possible: 0 is a natural number)
c) 3 – 5 =? (impossible in natural numbers)

To make subtraction always possible, the set of relative integers was created,

-1, -2, -3,………

it reads: minus 1 or negative 1
it reads: minus two or two negative
it reads: minus three or three negative

Bringing together the negative numbers, zero and positive numbers, we form the set of relative integers, which will be represented by Z.

Z = { …..-3, -2, -1, 0, +1, +2, +3,……}

Important: Positive integers can be indicated without the + sign.

example

a) +7 = 7
b) +2 = 2
c) +13 = 13
d) +45 = 45

Since zero is neither positive nor negative

Temperature: We use positive and negative numbers to mark the temperature. If the temperature is 20 degrees above zero, we can represent it by +20 (positive twenty). If it reads 10 degrees below zero, that temperature is represented by -10 (negative ten).

Bank account: the expression negative balance is common. When we withdraw (debit) an amount greater than our credit in a bank account, we start to have a negative balance.

altitude level: when we are above sea level, we are at an elevation (positive altitude). When we are below sea level, we are in a depression (negative altitude).

Timezone: If the opening of a World Cup is taking place at 12 noon in London, you will be watching this ceremony broadcast live on television at a different time. If you are in São Paulo, it will be at 9 am. In Tokyo, it will be at 9 pm on the same day.

This occurs according to the location of each city in relation to a reference (in this case, London), considered the zero point.

EXERCISES and Answers

1) Look at the numbers and say:

-15, +6, -1, 0, +54, +12, -93, -8, +23, -72, +72

a) What are the negative integers?
R: -15,-1,-93,-8,-72

b) What are the positive integers?
R: +6,+54,+12,+23,+72

2) What is the integer that is neither positive nor negative?
A: It's zero

3) Write the reading of the following whole numbers:

a) -8 =(R: negative eight)
b)+6 = (R: six positive)
c) -10 = (R: negative ten)
d) +12 = (R: twelve positive)
e) +75 = (R: seventy-five positive)
f) -100 = (R: one hundred negative)

4) Which of the following sentences are true?

a) +4 = 4 = (V)
b) -6 = 6 = (F)
c) -8 = 8 = (F)
d) 54 = +54 = (V)
e) 93 = -93 = (F)

5) Temperatures above 0°C (zero degrees) are represented by positive numbers and temperatures below 0°C by negative numbers. Represent the following situation with relative integers:

a) 5° above zero = (R: +5)
b) 3rd below zero = (R: -3)
c) 9°C below zero = (R: -9)
d) 15° above zero = (+15)

REPRESENTATION OF THE WHOLE NUMBERS ON THE STRAIGHT

Let's draw a line and mark the point 0. To the right of point 0, with a certain unit of measure, mark the points that correspond to the numbers positive and to the left of 0, with the same unit, we will mark the points that correspond to the numbers negative.

_I___I___I___I___I___I___I___I___I___I___I___I___I___I_
-6.. -5…-4. -3,. -2,..-1,.. 0,.+1,.+2,.+3,.+4,..+5,.+6

Exercises

1) Write the whole numbers:

a) between 1 and 7 (R: 2,3,4,5,6)
b) between -3 and 3 (R: -2,-1.0,1,2)
c) between -4 and 2 (R: -3, -2, -1, 0, 1)
d) between -2 and 4 (R: -1, 0, 1, 2, 3 )
e) between -5 and -1 (R: -4, -3, -2)
f) between -6 and 0 (R: -5, -4, -3, -2, -1)

2) Answer:

a) What is the successor to +8? (R: +9)
b) What is the successor of -6? (R: -5)
c) What is the successor of 0? (R: +1)
d) What is the predecessor of +8? (R: +7)
e) What is the predecessor of -6? (R: -7)
f) What is the predecessor of 0? (R: -1)

3) Write in Z the predecessor and successor of the numbers:

a) +4 (R: +3 and +5)
b) -4 (R: -5 and - 3)
c) 54 (R: 53 and 55)
d) -68 (R: -69 and -67)
e) -799 (R: -800 and -798)
f) +1000 (R: +999 and +1001)

OPPOSITE AND SYMMETRICAL NUMBERS

On the numbered line, the opposite numbers are the same distance from zero.

-I___I___I___I___I___I___I___I___I___I___I___I___I___I_
-6.. -5…-4. -3,. -2,..-1,.. 0,.+1,.+2,.+3,.+4,..+5,.+6

Note that each integer, positive or negative, has a corresponding with different signs.

example

a) The opposite of +1 is -1.
b) The opposite of -3 is +3.
c) The opposite of +9 is -9.
d) The opposite of -5 is +5.

Note: The opposite of zero is zero itself.

EXERCISES

1) Determine:

a) The opposite of +5 = (R:-5)
b) The opposite of -9 = (R: +9)
c) The opposite of +6 = (R: -6)
d) The opposite of -6 = (R: +6)
e) The opposite of +18 = (R: -18)
f) The opposite of -15 = (R: +15)
g) The opposite of +234= (R: -234)
h) The opposite of -1000 = (R: +1000)

COMPARISON OF WHOLE NUMBERS,

Note the graphical representation of the integers on the line.

-I___I___I___I___I___I___I___I___I___I___I___I___I___I_
-6.. -5…-4. -3,. -2,..-1,.. 0,.+1,.+2,.+3,.+4,..+5,.+6

Given any two numbers, the one on the right is their largest, and the one on the left, the smallest one.

examples

a) -1 greater; -4, because -1 is to the right of -4.
b) +2 greater; -4, because +2 is to the right of -4
c) -4 minor -2, because -4 is to the left of -2.
d) -2 less +1, because -2 is to the left of +1.

Exercises

1) What is the biggest number?

a) +1 or -10 (R:+1)
b) +30 or 0 (R: +30)
c) -20 or 0 (R: 0)
d) +10 or -10 (R: +10)
e) -20 or -10 (R: -10)
f) +20 or -30 (R: +20)
g) -50 or +50 (R:+50)
h) -30 or -15 (R:-15)

2) compare the following pairs of numbers, saying if the first is greater, less than or equal

a) +2 and +3 (minor)
b) +5 and -5 (higher)
c) -3 and +4 (minor)
d) +1 and -1 (highest)
e) -3 and -6 (major)
f) -3 and -2 (minor)
g) -8 and -2 (minor)
h) 0 and -5 (highest)
i) -2 and 0 (smaller)
j) -2 and -4 (larger)
l) -4 and -3 (minor)
m) 5 and -5 (larger)
n) 40 and +40 (equal)
o) -30 and -10 (smaller)
p) -85 and 85 (minor)
q) 100 and -200 (greater)
r) -450 and 300 (minor)
s) -500 and 400 (smaller)

3) put the numbers in ascending order.

a) -9,-3,-7,+1.0 (R: -9,-7,-3,0.1)
b) -2, -6, -5, -3, -8 (R: -8, -6,-5, -3,-2)
c) 5,-3,1,0,-1.20 (R: -3,-1,0,1,5,20)
d) 25,-3,-18,+15,+8,-9 (R: -18,-9,-3,+8,+15,+25)
e) +60,-21,-34,-105,-90 (R: -105,-90,-34,-21, +60)
f) -400,+620,-840,+1000,-100 (R: -840,-400,-100,+620,+1000)

4) Put the numbers in descending order

a) +3,-1,-6,+5.0 (R: +5,+3.0,-1,-6)
b) -4.0,+4,+6,-2 (R: +6,+4.0,-2,-4)
c) -5.1,-3,4.8 (R: 8.4.1,-3,-5)
d) +10,+6,-3,-4,-9,+1 (R: +10,+6,+1,-3,-4,-9)
e) -18,+83.0,-172, -64 (R: +83.0,-18,-64,-172)
f) -286,-740, +827.0,+904 (R: +904,+827.0,-286,-740)

ADDITION AND SUBTRACTION WITH WHOLE NUMBERS

ADDITION

1) Addition of positive numbers

The sum of two positive numbers is a positive number.

EXAMPLE

a) (+2) + (+5) = +7
b) (+1) + (+4) = +5
c) (+6) + (+3) = +9

Simplifying the way to write

a) +2 +5 = +7
b) +1 + 4 = +5
c) +6 + 3 = +9

Note that we write the sum of the whole numbers without adding the plus sign and eliminate the parentheses from the parcels.

2) Addition of negative numbers

The sum of two negative numbers is a negative number.

Example

a) (-2) + (-3) = -5
b) (-1) + (-1) = -2
c) (-7) + (-2) = -9

Simplifying the way to write

a) -2 - 3 = -5
b) -1 -1 = -2
c) -7 – 2 = -9

Note that we can simplify the way of writing by leaving the + sign in the operation and eliminating the parentheses from the parcels.

EXERCISES

1) Calculate

a) +5 + 3 = (R:+8)
b) +1 + 4 = (R: +5)
c) -4 - 2 = (R: -6)
d) -3 - 1 = (R: -4)
e) +6 + 9 = (R: +15)
f) +10 + 7 = (R: +17)
g) -8 -12 = (R: -20)
h) -4 -15 = (R: -19)
i) -10 – 15 = (R: -25)
j) +5 +18 = (R: +23)
l) -31 - 18 = (R: -49)
m) +20 +40 = (R: + 60)
n) -60 - 30 = (R: -90)
o) +75 +15 = (R: +90)
p) -50 -50 = (R: -100)

2) Calculate:

a) (+3) + (+2) = (R: +5)
b) (+5) + (+1) = (R: +6)
c) (+7) + (+5) = (R: +12)
d) (+2) + (+8) = (R: +10)
e) (+9) + (+4) = (R: +13)
f) (+6) + (+5) = (R: +11)
g) (-3) + (-2) = (R: -5)
h) (-5) + (-1) = (R: -6)
i) (-7) + (-5) = (R: -12)
j) (-4) + (-7) = (R: -11)
l) (-8) + (-6) = (R: -14)
m) (-5) + ( -6) = (R: -11)

3) Calculate:

a) (-22) + (-19) = (R: -41)
b) (+32) + (+14) = (R: +46)
c) (-25) + (-25) = (R: -50)
d) (-94) + (-18) = (R: -112)
e) (+105) + (+105) = (R: +210)
f) (-280) + (-509) = (R: -789)
g) (-321) + (-30) = (R: -350)
h) (+200) + (+137) = (R: +337)

3) Addition of numbers with different signs

The sum of two integers with different signs is obtained by subtracting the absolute values, giving the sign of the number that has the greatest absolute value.

examples

a) (+6) + ( -1) = +5
b) (+2) + (-5) = -3
c) (-10) + (+3) = -7

simplifying the way of writing

a) +6 - 1 = +5
b) +2 – 5 = -3
c) -10 + 3 = -7

Note that the addition result has the same sign as the number with the greatest absolute value.

Observation:

When the parcels are opposite numbers, the sum equals zero.

Example

a) (+3) + (-3) = 0
b) (-8) + (+8) = 0
c) (+1) + (-1) = 0

simplifying the way of writing

a) +3 – 3 = 0
b) -8 + 8 = 0
c) +1 - 1 = 0

4) One of the given numbers is zero

When one of the numbers is zero, the sum equals the other number.

example

a) (+5) +0 = +5
b) 0 + (-3) = -3
c) (-7) + 0 = -7

Simplifying the way to write

a) +5 + 0 = +5
b) 0 - 3 = -3
c) -7 + 0 = -7

Exercises

1) Calculate:

a) +1 - 6 = -5
b) -9 + 4 = -5
c) -3 + 6 = +3
d) -8 + 3 = -5
e) -9 + 11 = +2
f) +15 - 6 = +9
g) -2 + 14 = +12
h) +13 -1 = +12
i) +23 -17 = +6
j) -14 + 21 = +7
l) +28 -11 = +17
m) -31 + 30 = -1

2) Calculate:

a) (+9) + (-5) = +4
b) (+3) + (-4) = -1
c) (-8) + (+6) = -2
d) (+5) + (-9) = -4
e) (-6) + (+2) = -4
f) (+9) + (-1) = +8
g) (+8) + (-3) = +5
h) (+12) + (-3) = +9
i) (-7) + (+15) = +8
j) (-18) + (+8) = -10
i) (+7) + (-7) = 0
l) (-6) + 0 = -6
m) +3 + (-5) = -2
n) (+2) + (-2) = 0
o) (-4) +10 = +6
p) -7 + (+9) = +2
q) +4 + (-12) = -8
r) +6 + (-4) = +2

PROPERTY OF THE ADDITION

1) Closing: the sum of two integers is always an integer

example (-4) + (+7) =( +3)

2) Commutative: the order of the parcels does not change the sum.

example: (+5) + (-3) = (-3) + (+5)

3) Neutral element: the number zero is the neutral element of addition.

example: (+8) + 0 = 0 + (+8) = +8

4) Associative: when adding three whole numbers, we can associate the first two or the last two, without altering the result.

example: [(+8) + (-3) ] + (+4) = (+8) + [(-3) + (+4)]

5) Opposite element: any whole number admits a symmetric or opposite.

example: (+7) + (-7) = 0

ADDING THREE OR MORE NUMBERS

To get the sum of three or more numbers we add the first two and then add that result with the third, and so on.

examples

1) -12 + 8 – 9 + 2 – 6 =
= -4 – 9 + 2 – 6 =
= -13 + 2 – 6 =
= -11 – 6 =
= -17

2) +15 -5 -3 +1 – 2 =
= +10 -3 + 1 – 2 =
= +7 +1 -2 =
= +8 -2 =
= +6

When adding whole numbers we can cancel out opposite numbers, because their sum is zero.

SIMPLIFIED NOMINATION

a) we can dispense with the + sign of the first installment when it is positive.

examples

a) (+7) + (-5) = 7 – 5 = +2

b) (+6) + (-9) = 6 – 9 = -3

b) We can dispense with the + sign of the sum when it is positive

examples

a) (-5) + (+7) = -5 + 7 = 2

b) (+9) + (-4) = 9 – 4 = 5

EXERCISES

1) Calculate

a) 4 + 10 + 8 = (R: 22)
b) 5 - 9 + 1 = (R: -3)
c) -8 - 2 + 3 = (R: -7)
d) -15 + 8 – 7 = (R: -14)
e) 24 + 6 – 12 = (R:+18)
f) -14 – 3 – 6 – 1 = (R: -24)
g) -4 + 5 + 6 + 3 - 9 = (R: + 1)
h) -1 + 2 – 4 – 6 – 3 – 8 = (R: -20)
i) 6 - 8 - 3 - 7 - 5 - 1 + 0 - 2 = (R: -20)
j) 2 – 10 – 6 + 14 – 1 + 20 = (R: +19)
l) -13 – 1 – 2 – 8 + 4 – 6 – 10 = (R: -36)

2) Make, canceling the opposite numbers:

a) 6 + 4 – 6 + 9 – 9 = (R: +4)
b) -7 + 5 – 8 + 7 – 5 = (R: -8)
c) -3 + 5 + 3 – 2 + 2 + 1 = (R: +6)
d) -6 + 10 + 1 – 4 + 6= (R: +7)
e) 10 - 6 + 3 - 3 - 10 - 1 = (R: -7)
f) 15 – 8 + 4 – 4 + 8 – 15 = (R: 0)

3) Put in simplified form (no parentheses)

a) (+1) + (+4) +(+2) = (R: 1 +4 + 2)
b) (+1) + (+8) + (-2) = (R: 1 + 8 - 2)
c) (+5) +(-8) + (-1) = (R: +5 – 8 – 1)
d) (-6) + (-2) + (+1) = (R: -6 - 2 + 1)

4) Calculate:

a) (-2) + (-3) + (+2) = (R: -3)
b) (+3) + (-3) + (-5) = (R: -5)
c) (+1) + (+8) +(-2) = (R: +7)
d) (+5) + (-8) + (-1) = (R: -4)
e) (-6) + (-2) + (+1) = (R: -7)
f) (-8) + (+6) + (-2) = (R: -4)
g) (-7) + 6 + (-7) = (R: -8)
h) 6 + (-6) + (-7) = (R: -7)
i) -6 + (+9) + (-4) = (R: -1)
j) (-4) +2 +4 + (+1) = (R: +3)

5) Determine the following sums

a) (-8) + (+10) + (+7) + (-2) = (R: +7)
b) (+20) + (-19) + (-13) + (-8) = (R: -20)
c) (-5) + (+8) + (+2) + (+9) = (R: +14)
d) (-1) + (+6) + (-3) + (-4) + (-5) = (R: -7)
e) (+10) + (-20) + (-15) + (+12) + (+30) + (-40) = (R: -23)
f) (+3) + (-6) + (+8) = (R: +5)
g) (-5) + (-12) + (+3) = (R: -14)
h) (-70) + (+20) + (+50) = (R: 0)
i) (+12) + (-25) + (+15) = (R: +2)
j) (-32) + (-13) + (+21) = (R: -24)
l) (+7) + (-5) + (-3) + (+10) = (R: +9)
m) (+12) + (-50) + (-8) + (+13) = (R: -33)
n) (-8)+(+4)+ (+8) + (-5) + (+3) = (R: +2)
o) (-36) + (-51) + (+100) + (-52) = (R: -39)
p) (+17) + (+13) + (+20) + (-5) + (-45) = (R: 0)

6) Given the numbers x= 6, y = 5 and z= -6, calculate

a) x + y = (R: +11)
b) y + z = (R: -4)
c) x + z = (R: -3)

SUBTRACTION

The subtraction operation is an inverse operation to the addition.

Examples

a) (+8) – (+4) = (+8) + (-4) = = +4
b) (-6) – (+9) = (-6) + (-9) = -15
c) (+5) – (-2) = (+5) + (+2) = +7

Conclusion: To subtract two relative numbers, just add the opposite of the second to the first.

Note: Subtraction on set Z has only the closure property (subtraction is always possible)

ELIMINATION OF PARENTHESES PRECEDING A NEGATIVE SIGN

To facilitate the calculation, we have eliminated the parentheses using the meaning of the opposite

Look:

a) -(+8) = -8 (means the opposite of +8 is -8)

b) -(-3) = +3 (means the opposite of -3 is +3)

analogically:

a) -(+8) – (-3) = -8 +3 = -5

b) -(+2) – (+4) = -2 – 4 = -6

c) (+10) – (-3) – +3) = 10 + 3 – 3 = 10

conclusion: we can eliminate parentheses preceded by a negative sign by changing the sign of the number that is inside the parentheses.

EXERCISES

1) Remove parentheses

a) -(+5) = -5
b) -(-2) = +2
c) - (+4) = -4
d) -(-7) = +7
e) -(+12) = -12
f) -(-15) = +15
g) -(-42) = +42
h) -(+56) = -56

2) Calculate:

a) (+7) – (+3) = (R: +4)
b) (+5) – (-2) = (R: +7)
c) (-3) – (+8) = (R: -11)
d) (-1) -(-4) = (R: +3)
e) (+3) – (+8) = (R: -5)
f) (+9) – (+9) = (R: 0 )
g) (-8) - (+5) = (R: -13)
h) (+5) – (-6) = (R: +11)
i) (-2) - (-4) = (R: +2)
j) (-7) – (-8) = (R: +1)
l) (+4) -(+4) = (R: 0)
m) (-3) – (+2) = (R: -5)
n) -7 + 6 = (R: -1)
o) -8 -7 = (R: -15)
p) 10 -2 = (R: 8)
q) 7 -13 = (R: -6)
r) -1 -0 = (R: -1)
s) 16 - 20 = (R: -4)
t) -18 -9 = (R: -27)
u) 5 - 45 = (R:-40)
v) -15 -7 = (R: -22)
x) -8 +12 = (R: 4)
z) -32 -18 = (R:-50)

3) Calculate:

a) 7 - (-2) = (R: 9)
b) 7 - (+2) = (R: 5)
c) 2 - (-9) = (R: 11)
d) -5 - (-1) = (R: -4)
e) -5 -(+1) = (R: -6)
f) -4 - (+3) = (R: -7)
g) 8 - (-5) = (R: 13)
h) 7 - (+4) = (R: 3)
i) 26 - 45 = (R: -19)
j) -72 -72 = (R: -144)
l) -84 + 84 = (R: 0)
m) -10 -100 = (R: -110)
n) -2 -4 -1 = (R: -7)
o) -8 +6 -1 = (R: -3)
p) 12-7 + 3 = (R: 8)
q) 4 + 13 – 21 = (R: -4)
r) -8 +8 + 1 = (R: 1)
s) -7 + 6 + 9 = (R: 8)
t) -5 -3 -4 - 1 = (R: -13)
u) +10 – 43 -17 = (R: -50)
v) -6 -6 + 73 = (R: 61)
x) -30 +30 – 40 = (R: -40)
z) -60 - 18 +50 = (R: -28)

4) Calculate:

a) (-4) -(-2)+(-6) = (R: -8)
b) (-7)-(-5)+(-8) = (R: -10)
c) (+7)-(-6)-(-8) = (R: 21)
d) (-8) + (-6) -(+3) = (R: -17)
e) (-4) + (-3) – (+6) = (R: -13)
f) 20 - (-6) - (-8) = (R: 34)
g) 5 - 6 - (+7) + 1 = (R: -7)
h) -10 - (-3) - (-4) = (R: -3)
i) (+5) + (-8) = (R: -3)
j) (-2) - (-3) = (R: +1)
l) (-3) -(-9) = (R: +6)
m) (-7) – (-8) =(R: +1)
n) (-8) + (-6) – (-7) = (R: -7)
o) (-4) + (-6) + (-3) = (R: -13)
p) 15 -(-3) - (-1) = (R: +19)
q) 32 - (+1) -(-5) = (R: +36)
r) (+8) – (+2) = (R:+6)
s) (+15) - (-3) = (R: +18)
t) (-18) - (-10) = (R: -8)
u) (-25) - (+22) = (R:-47)
v) (-30) - 0 = (R: -30)
x) (+180) - (+182) = (R: -2)
z) (+42) – (-42) = (R: +84)

5) Calculate:

a) (-5) + (+2) – (-1) + (-7) = (R: -9)
b) (+2) – (-3) + (-5) -(-9) = (R: 9)
c) (-2) + (-1) -(-7) + (-4) = (R: 0)
d) (-5) + (-6) -(-2) + (-3) = (R: -12)
e) (+9) -(-2) + (-1) - (-3) = (R: 13)
f) 9 - (-7) -11 = (R: 5 )
g) -2 + (-1) -6 = (R: -9)
h) -(+7) -4 -12 = (R: -23)
i) 15 -(+9) -(-2) = (R: 8 )
j) -25 - ( -5) -30 = (R: -50)
l) -50 - (+7) -43 = (R: -100)
m) 10 -2 -5 -(+2) - (-3) = (R: 4)
n) 18 - (-3) - 13 -1 -(-4) = (R: 11)
o) 5 -(-5) + 3 – (-3) + 0 – 6 = (R: 10)
p) -28 + 7 + (-12) + (-1) -4 -2 = (R: -40)
q) -21 -7 -6 -(-15) -2 -(-10) = (R: -11)
r) 10 -(-8) + (-9) -(-12)-6 + 5 = (R: 20)
s) (-75) - (-25) = (R: -50)
t) (-75) - (+25) = (R: -100)
u) (+18) - 0 = (R: +18)
v) (-52) - (-52) = (R: 0)
x) (-16)-(-25) = (R:+9)
z) (-100) - (-200) = (R: +100)

DISPOSAL OF RELATIVES

1) parentheses preceded by the + sign

When eliminating the parentheses and the + sign that precedes them, we must retain the signs of the numbers contained in those parentheses.

example

a) + (-4 + 5) = -4 + 5

b) +(3 +2 -7) = 3 +2 -7

2) Parentheses preceded by the sign -

When eliminating the parentheses and the - sign that precedes them, we must change the signs of the numbers contained in those parentheses.

example

a) -(4 - 5 + 3) = -4 + 5 -3

b) -(-6 + 8 – 1) = +6 -8 +1

EXERCISES

1) Eliminate the parentheses:

a) +(-3 +8) = (R: -3 + 8)
b) -(-3 + 8) = (R: +3 - 8)
c) +(5 - 6) = (R: 5 -6)
d) -(-3-1) = (R: +3 +1)
e) -(-6 + 4 - 1) = (R: +6 - 4 + 1)
f) +(-3 -2 -1) = (R: -3 -2 -1 )
g) -(4 -6 +8) = (R: -4 +6 +8)
h) + (2 + 5 - 1) = (R: +2 +5 -1)

2) Eliminate the parentheses and calculate:

a) + 5 + (7 – 3) = (R: 9)
b) 8 - (-2-1) = (R: 11)
c) -6 - (-3 +2) = (R: -5)
d) 18 - ( -5 -2 -3 ) = (R: 28)
e) 30 - (6 - 1 +7) = (R: 18)
f) 4 + (-5 + 0 + 8 -4) = (R: 3)
g) 4 + (3 - 5) + ( -2 -6) = (R: -6)
h) 8 -(3 + 5 -20) + (3 -10) = (R: 13)
i) 20 - (-6 +8) - (-1 + 3) = (R: 16)
j) 35 -(4-1) - (-2 + 7) = (R: 27)

3) Calculate:

a) 10 - (15 + 25) = (R: -30)
b) 1 - (25 -18) = (R: -6)
c) 40 -18 - (10 +12) = (R: 0)
d) (2 - 7) - (8 -13) = (R: 0 )
e) 7 - (3 + 2 + 1) - 6 = (R: -5)
f) -15 - (3 + 25) + 4 = (R: -39)
g) -32 -1 - ( -12 + 14) = (R: -35)
h) 7 + (-5-6) - (-9 + 3) = (R: 2)
i) -(+4-6) + (2 - 3) = (R: 1)
j) -6 - (2 -7 + 1 - 5) + 1 = (R: 4)

EXPRESSIONS WITH RELATIVE WHOLE NUMBERS

Remember that association signs are eliminated in the following order:

1°) PARENTHESES ( ) ;

2°) BRACKETS [ ] ;

3°) KEYS { } .

Examples:

1st) example

8 + ( +7 -1 ) – ( -3 + 1 – 5 ) =
8 + 7 – 1 + 3 – 1 + 5 =
23 – 2 = 21

2nd) example

10 + [ -3 + 1 – ( -2 + 6 ) ] =
10 + [ -3 + 1 + 2 – 6 ] =
10 – 3 + 1 + 2 – 6 =
13 – 9 =
= 4

3rd) example

-17 + { +5 – [ +2 – ( -6 +9 ) ]} =
-17 + { +5 – [ +2 + 6 – 9]} =
-17 + { +5 – 2 – 6 + 9 } =
-17 +5 – 2 – 6 + 9 =
-25 + 14 =
= – 11

EXERCISES

a) Calculate the value of the following expressions:

1) 15 -(3-2) + ( 7 -4) = (R: 17)
2) 25 – ( 8 – 5 + 3) – ( ​​12 – 5 – 8) = (R: 20)
3) ( 10 -2 ) – 3 + ( 8 + 7 – 5) = (R: 15)
4) ( 9 – 4 + 2 ) – 1 + ( 9 + 5 – 3) = (R: 17)
5) 18 - [ 2 + ( 7 - 3 - 8 ) - 10 ] = (R: 30 )
6) -4 + [ -3 + ( -5 + 9 – 2 )] = (R: -5)
7) -6 - [10 + (-8 -3 ) -1] = (R: -4)
8) -8 - [ -2 - (-12) + 3 ] = (R: -21)
9) 25 - { -2 + [ 6 + ( -4 -1 )]} = (R: 26)
10) 17 - { 5 - 3 + [ 8 - ( -1 - 3 ) + 5 ] } = (R: -2)
11) 3 - { -5 -[8 - 2 + ( -5 + 9 ) ] } = (R: 18)
12) -10 – { -2 + [ + 1 – ( – 3 – 5 ) + 3 ] } = (R: -20)
13) { 2 + [ 1 + ( -15 -15 ) – 2] } = (R: -29)
14) { 30 + [ 10 – 5 + ( -2 -3)] -18 -12} = (R: 0 )
15) 20 + { [ 7 + 5 + ( -9 + 7 ) + 3 ] } = (R: 33)
16) -4 – { 2 + [ – 3 – ( -1 + 7) ] + 2} = (R: 1)
17) 10 – { -2 + [ +1 + ( +7 – 3) – 2] + 6 } = (R: 3 )
18) -{ -2 - [ -3 - (-5) + 1 ]} - 18 = (R: -13)
19) -20 - { -4 -[-8 + ( +12 - 6 - 2 ) + 2 +3 ]} = (R: -15)
20) {[( -50 -10) + 11 + 19 ] + 20 } + 10 = (R: 0 )

MULTIPLICATION AND DIVISION OF WHOLE NUMBERS

MULTIPLICATION

1) multiplication of two numbers with equal signs

watch the example

a) (+5). (+2) = +10
b) (+3). (+7) = +21
c) (-5). (-2) = +10
d) (-3). (-7) = +21

conclusion: If the factors have equal signs, the product is positive

2) Multiplication of two different signal products

watch the examples

a) (+3). (-2) = -6
b) (-5). (+4) = -20
c) (+6). (-5) = -30
d) (-1). (+7) = -7

Conclusion: If two products have different signs, the product is negative

Practical rule of signs in multiplication

EQUAL SIGNS: the result is positive

a) (+). (+) = (+)

B) (-). (-) = (+)

DIFFERENT SIGNS: the result is negative -

a) (+). (-) = (-)

B) (-). (+) = (-)

EXERCISES

1) Perform the multiplications

a) (+8). (+5) = (R: 40)
b) (-8). ( -5) = (R: 40)
c) (+8) .(-5) = (R: -40)
d) (-8). (+5) = (R: -40)
e) (-3). (+9) = (R: -27)
f) (+3). (-9) = (R: -27)
g) (-3). (-9) = (R: 27)
h) (+3). (+9) = (R: 27)
i) (+7). (-10) = (R: -70)
j) (+7). (+10) = (R: 70)
l) (-7). (+10) = (R: -70)
m) (-7). (-10) = (R: 70)
n) (+4). (+3) = (R: 12)
o) (-5). (+7) = (R: -35)
p) (+9). (-2) = (R: -18)
q) (-8). (-7) = (R: 56)
r) (-4). (+6) = (R: -24)
s) (-2) .(-4) = (R: 8 )
t) (+9). (+5) = (R: 45)
u) (+4). (-2) = (R: -8)
v) (+8). (+8) = (R: 64)
x) (-4). (+7) = (R: -28)
z) (-6). (-6) = (R: 36)

2) Calculate the product

a) (+2). (-7) = (R: -14)
b) 13. 20 = (R: 260)
c) 13. (-2) = (R: -26)
d) 6. (-1) = (R: -6)
e) 8. (+1) = (R: 8)
f) 7. (-6) = (R: -42)
g) 5. (-10) = (R: -50)
h) (-8). 2 = (R: -16)
i) (-1). 4 = (R: -4)
j) (-16). 0 = (R: 0)

MULTIPLICATION WITH MORE THAN TWO NUMBERS

We multiply the first number by the second, the product obtained by the third and so on, until the last factor

examples

a) (+3). (-2). (+5) = (-6). (+5) = -30

b) (-3). (-4). (-5). (-6) = (+12). (-5). (-6) = (-60). (-6) = +360

EXERCISES

1) Determine the product:

a) (-2). (+3). (+4) = (R: -24)
b) (+5). (-1). (+2) = (R: -10)
c) (-6). (+5) .(-2) = (R: +60)
d) (+8). (-2) .(-3) = (R: +48)
e) (+1). (+1). (+1) .(-1)= (R: -1)
f) (+3) .(-2). (-1). (-5) = (R: -30)
g) (-2). (-4). (+6). (+5) = (R: 240)
h) (+25). (-20) = (R: -500)
i) -36) .(-36 = (R: 1296)
j) (-12). (+18) = (R: -216)
l) (+24). (-11) = (R: -264)
m) (+12). (-30). (-1) = (R: 360)

2) Calculate the products

a) (-3). (+2). (-4). (+1). (-5) = (R: -120)
b) (-1). (-2). (-3). (-4) .(-5) = (R: -120)
c) (-2). (-2). (-2). (-2) .(-2). (-2) = (R: 64)
d) (+1). (+3). (-6). (-2). (-1) .(+2)= (R: -72)
e) (+3). (-2). (+4). (-1). (-5). (-6) = (R: 720)
f) 5. (-3). (-4) = (R: +60)
g) 1. (-7). 2 = (R: -14)
h) 8. ( -2). 2 = (R: -32)
i) (-2). (-4) .5 = (R: 40)
j) 3. 4. (-7) = (R: -84)
l) 6 .(-2). (-4) = (R: +48)
m) 8. (-6). (-2) = (R: 96)
n) 3. (+2). (-1) = (R: -6)
o) 5. (-4). (-4) = (R: 80)
p) (-2). 5 (-3) = (R: 30)
q) (-2). (-3). (-1) = (R:-6)
r) (-4). (-1). (-1) = (R: -4)

3) Calculate the value of expressions:

a) 2. 3 - 10 = (R: -4)
b) 18 - 7. 9 = (R: -45)
c) 3. 4 - 20 = (R: -8)
d) -15 + 2. 3 = (R: -9)
e) 15 + (-8). (+4) = (R: -17)
f) 10 + (+2). (-5) = (R: 0)
g) 31 – (-9). (-2) = (R: 13)
h) (-4). (-7) -12 = (R: 16)
i) (-7). (+5) + 50 = (R: 15)
j) -18 + (-6). (+7) = (R:-60)
l) 15 + (-7). (-4) = (R: 43)
m) (+3). (-5) + 35 = (R: 20)

4) Calculate the value of expressions

a) 2 (+5) + 13 = (R: 23)
b) 3. (-3) + 8 = (R: -1)
c) -17 + 5. (-2) = (R: -27)
d) (-9). 4 + 14 = (R: -22)
e) (-7). (-5) - (-2) = (R: 37)
f) (+4). (-7) + (-5). (-3) = (R: -13)
g) (-3). (-6) + (-2). (-8) = (R: 34)
h) (+3). (-5) – (+4). (-6) = (R: 9)

MULTIPLICATION PROPERTIES

1) Closing: the product of two whole numbers is always a whole number.

example: (+2). (-5) = (-10)

2) Concurrent: the order of factors does not change the product.

example: (-3). (+5) = (+5). (-3)

3) Neutral Element: the number +1 is the neutral element of multiplication.

Examples: (-6). (+1) = (+1). (-6) = -6

4) Associative: in the multiplication of three whole numbers, we can associate the first two or the last two, without altering the result.

example: (-2). [(+3). (-4) ] = [ (-2). (+3) ]. (-4)

5) Distributive

example: (-2). [(-5) +(+4)] = (-2). (-5) + (-2). (+4)

DIVISION

You know that division is the inverse operation of multiplication.

Watch:

a) (+12): (+4) = (+3), because (+3). (+4) = +12
b) (-12): (-4) = (+3), because (+3). (-4) = -12
c) (+12): (-4) = (-3), because (-3). (-4) = +12
d) (-12): (+4) = (-3), because (-3). (+4) = -12

PRACTICAL RULE OF SIGNS IN THE DIVISION

The rules of signs in division are the same as in multiplication:

EQUAL SIGNS: the result is +

(+): (+) = (+)

(-): (-) = (-)

DIFFERENT SIGNS: the result is –

(+): (-) = (-)

(-): (+) = (-)

EXERCISES

1) Calculate the quotients:

a) (+15): (+3) = (R: 5)
b) (+15): (-3) = (R: -5)
c) (-15): (-3) = (R: 5)
d) (-5): (+1) = (R: -5)
e) (-8): (-2) = (R: 4)
f) (-6): (+2) = (R: -3)
g) (+7): (-1) = (R: -7)
h) (-8): (-8) = (R: 1)
f) (+7): (-7) = (R: -1)

2) Calculate the quotients

a) (+40): (-5) = (R: -8)
b) (+40): (+2) = (R: 20)
c) (-42): (+7) = (R: -6)
d) (-32): (-8) = (R: 4)
e) (-75): (-15) = (R: 5)
f) (-15): (-15) = (R: 1)
g) (-80): (-10) = (R: 8)
h) (-48 ): (+12) = (R: -4)
l) (-32): (-16) = (R: 2)
j) (+60): (-12) = (R: -5)
l) (-64): (+16) = (R: -4)
m) (-28): (-14) = (R: 2)
n) (0): (+5) = (R: 0)
o) 49: (-7) = (R: -7)
p) 48: (-6) = (R: -8)
q) (+265): (-5) = (R: -53)
r) (+824): (+4) = (R: 206)
s) (-180): (-12) = (R: 15)
t) (-480): (-10) = (R: 48)
u) 720: (-8) = (R: -90)
v) (-330): 15 = (R: -22)

3) Calculate the value of expressions

a) 20: 2 -7 = (R: 3 )
b) -8 + 12: 3 = (R: -4)
c) 6: (-2) +1 = (R: -2)
d) 8: (-4) – (-7) = (R: 5)
e) (-15): (-3) + 7 = (R: 12)
f) 40 - (-25): (-5) = (R: 35)
g) (-16): (+4) + 12 = (R: 8)
h) 18: 6 + (-28): (-4) = (R: 10)
i) -14 + 42: 3 = (R: 0)
j) 40: (-2) + 9 = (R: -11)
l) (-12) 3 + 6 = (R: 2)
m) (-54): (-9) + 2 = (R: 8)
n) 20+(-10). (-5) = (R: 70)
o) (-1). (-8) + 20 = (R: 28)
p) 4 + 6. (-2) = (R: -8)
q) 3. (-7) + 40 = (R: 19)
r) (+3). (-2) -25 = (R: -31)
s) (-4). (-5) + 8. (+2) = (R: 36)
t) 5: (-5) + 9. 2 = (R: 17)
u) 36: (-6) + 5. 4 = (R: 14)

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