1 ESO - UNIT 04 - INTEGER NUMBERS

1. Unit 04 December
1. POSITIVE AND NEGATIVE NUMBERS.
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).
Positive integers are all the whole numbers greater than zero:
𝐖𝐖𝐖𝐖𝐖𝐖𝐖𝐖𝐖𝐖 𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍 = {𝟎𝟎, 𝟏𝟏, 𝟐𝟐, … . }
Negative integers are all the opposites of the Natural numbers:
𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍 𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍 = {−𝟏𝟏, −𝟐𝟐, −𝟑𝟑, … . }
Negative numbers can be represented on a Number Line:
Axel Cotón Gutiérrez Mathematics 1º ESO 4.1

2. Unit 04 December
As you move to the left the number get smaller.
+8 > +1 > 0 > −1 > −8
You can use a number line to help solve problems involving negative numbers.
MATH VOCABULARY: To Deposit, To Withdraw, Forward, Backward, Integer Number,
Number Line.
2. INTEGER NUMBERS.
Integer Numbers are the set of whole numbers and the negative numbers. The
number line is used to represent integers.
Axel Cotón Gutiérrez Mathematics 1º ESO 4.2

3. Unit 04 December
This set is called ℤ = {… , −𝟐𝟐, −𝟏𝟏, 𝟎𝟎, +𝟏𝟏, +𝟐𝟐, … } (ℤ is from the word Zahlen
number in German). The number line goes on forever in both directions. This is
indicated by arrows:
Some facts:
• The Integer Zero is neutral. It is neither positive nor negative.
• The Sign of an Integer is either positive (+) or negative (-), except zero, which
has no sign. Positive integers can be written with or without a sign.
• Two integers are Opposites if they are each the same distance away from zero,
but on opposite sides of the number line. One will have a positive sign, the
other a negative sign. In the number line below, +3 and -3 are labeled as
opposites.
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 𝐧𝐧 by writing 𝒏𝒏 in between two vertical bars: |𝒏𝒏|.
Axel Cotón Gutiérrez Mathematics 1º ESO 4.3

4. Unit 04 December
|6| = 6 | − 10| = 10 |0| = 0 | − 3404| = 3404
MATH VOCABULARY: Opposite, Sign, Absolute Value.
3. ADDING AND SUBTRACTING INTEGERS.
There is a way to understand how to add integers. In order to add positive and
negative integers, we will imagine that we are moving along a number line. If we want
to add -2 and 5, we start by finding the number -2 on the number line, exactly one unit
to the left of zero. Then we would move five units to the right. Since we landed four
units to the right of zero, the answer must be 3.
If asked to add -3 and -2, we can start by finding the number -3 on the number
line. Then we move two units left from there because negative numbers make us move
to the left side of the number line. Since our last position is two units to the left of -3,
the answer is -5.
Axel Cotón Gutiérrez Mathematics 1º ESO 4.4

5. Unit 04 December
3.1. ADDITION RULES.
• When adding integers with the same sign: We add their absolute values, and
give the result the same sign.
(+10) + (+2) = +12
(−12) + (−4) = −16
• When adding integers with the opposite sign: We take their absolute values,
subtract the smallest from the largest, and give the result the sign of the
integer with the larger absolute value.
+8 + (−2) ⇒ |+8| = 8; |−2| = 2 ⟹ |+8| > |−2| ⇒ 𝑆𝑆𝑆𝑆 𝑆𝑆 𝑆𝑆 +⇒ +8 + (−2) = +6
8 + (−17) ⇒ |+8| = 8; |−17| = 17 ⟹ |+8| < |−17| ⇒ 𝑆𝑆𝑆𝑆 𝑆𝑆 𝑆𝑆 −⇒ 8 + (−17) = −9
Axel Cotón Gutiérrez Mathematics 1º ESO 4.5

6. Unit 04 December
3.2. SUBTRACTING INTEGERS.
To subtract an integer, add its opposite. Then apply the addition rules.
7 − 4 = 7 + (−4) = 3
12 − (−5) = 12 + (5) = 17
NOTE: When adding or subtracting more than two numbers we always start from the
left or we add the positive integers and the negative integers separately and the
subtract them.
4. ADDITION AND SUBTRACTION WITH BRACKETS.
Integer addition and integer subtraction can be easier if you know the rules for
removing brackets.
• If there is a ‘plus’ sign before a bracket, the sign of the numbers inside the
brackets hold the same.
Axel Cotón Gutiérrez Mathematics 1º ESO 4.6

7. Unit 04 December
3 + (−2) = 3 − 2 = 1
• If there is a ‘minus’ sign before a bracket, the sign of the numbers inside the
brackets is changed from ‘plus’ to ‘minus’ and from ‘minus’ to ‘plus’.
3 − (−2) = 3 + 2 = 5
12 − [8 − (7 − 10) + (2 − 6)] = 12 − [8 − (−3) + (−4)] =
= 12 − [8 + 3 − 4] = 12 − [+7] = 12 − 7 = 5
5. INTEGER MULTIPLICATION. INTEGER DIVISION.
We need to know the Rules for Multiplication of integers:
• The product of a positive integer and a negative integer is a negative integer.
3 ∙ (−2) = −6
• The product of two negative integers or two positive integers is a positive
integer.
(−3) ∙ (−2) = −6
Axel Cotón Gutiérrez Mathematics 1º ESO 4.7

8. Unit 04 December
The division rules are the same:
To know the Order of Operations when we have brackets, multiplication,
division, addition and subtraction we use the BEDMAS rules.
Axel Cotón Gutiérrez Mathematics 1º ESO 4.8

9. Unit 04 December
6. POWERS AND SQUARE ROOTS OF INTEGER NUMBERS.
6.1. POWERS OF INTEGER NUMBERS.
Remember the meaning of Powers: Instead of writing 2 𝑥𝑥 2 𝑥𝑥 2 𝑥𝑥 2 𝑥𝑥 2 we can
write 25
:
There are also Sign Rules for Powers:
• If the base of the power is a positive integer, the result is always a positive
integer.
(+2)4
= (+2) ∙ (+2) ∙ (+2) ∙ (+2) = +16
• If the base of the power is a negative integer: the result is a positive integer if
the exponent is an even number and the result is a negative integer if the
exponent is an odd number.
(−2)4
= (−2) ∙ (−2) ∙ (−2) ∙ (−2) = +16
(−2)3
= (−2) ∙ (−2) ∙ (−2) = −8
Axel Cotón Gutiérrez Mathematics 1º ESO 4.9

10. Unit 04 December
The Power Properties with whole numbers also work with integer numbers
(Unit 02):
𝒂𝒂𝒏𝒏
∙ 𝒂𝒂 𝒎𝒎
= 𝒂𝒂𝒏𝒏+𝒎𝒎 ( 𝒂𝒂 ∙ 𝒃𝒃)𝒏𝒏
= 𝒂𝒂𝒏𝒏
∙ 𝒃𝒃𝒏𝒏
𝒂𝒂𝒏𝒏
÷ 𝒂𝒂 𝒎𝒎
= 𝒂𝒂𝒏𝒏−𝒎𝒎 ( 𝒂𝒂 ÷ 𝒃𝒃)𝒏𝒏
= 𝒂𝒂𝒏𝒏
÷ 𝒃𝒃𝒏𝒏
( 𝒂𝒂𝒏𝒏) 𝒎𝒎
= 𝒂𝒂𝒏𝒏∙𝒎𝒎
𝒂𝒂𝟎𝟎
= 𝟏𝟏 (𝒂𝒂𝒍𝒍𝒍𝒍𝒍𝒍𝒍𝒍𝒍𝒍 𝒂𝒂 ≠ 𝟎𝟎)
𝒂𝒂𝟏𝟏
= 𝒂𝒂
6.2. SQUARE ROOTS OF INTEGER NUMBERS.
Remember now the meaning of Square Roots:
Looking that we can say:
• The Square Root of a positive integer has two solutions that are not always
integer numbers:
√+4 = �
−2 ⇔ (−2) ∙ (−2) = +4
+2 ⇔ (+2) ∙ (+2) = +4
• The Square Root of a negative integer does not exist in the set we will study in
ESO.
√−4 = ∄
Axel Cotón Gutiérrez Mathematics 1º ESO 4.10