Integers — Study Notes

1. Introduction

So far we have worked with whole numbers, which are 0, 1, 2, 3 and so on. But in daily life we also come across situations where numbers below zero are needed. When the temperature falls below freezing point, we say it is minus 3 degrees. When a lift goes below the ground floor, we say it is on floor minus 1. When a businessman loses money, we say his profit is negative. Such situations are handled with the help of integers.

Integers are the collection of all whole numbers and their negatives. The set of integers is written as ..., -3, -2, -1, 0, 1, 2, 3, ... The numbers 1, 2, 3, ... are called positive integers, the numbers -1, -2, -3, ... are called negative integers, and 0 is neither positive nor negative. Integers help us represent quantities that can increase and decrease on both sides of zero.

In this chapter, we learn to represent integers on the number line, compare them, and add and subtract them using the number line and by rules. We also study the important concept that subtraction of integers gives results different from whole numbers, and we learn the properties of integers under addition and subtraction.

2. Representation of Integers on the Number Line

To represent integers on a number line, we first mark zero. All positive integers are marked to the right of zero, and all negative integers are marked to the left of zero.

• The number line extends infinitely in both directions.
• Positive integers: 1, 2, 3, ... lie to the right of zero.
• Negative integers: -1, -2, -3, ... lie to the left of zero.
• Zero is neither positive nor negative.

On the number line, an integer is greater than all integers to its left and smaller than all integers to its right. For example, 2 > -3 and -1 > -5.

3. Ordering of Integers

We compare integers using their positions on the number line. The number on the right is always greater than the number on the left.

• Every positive integer is greater than every negative integer. For example, 1 > -10.
• Zero is greater than every negative integer and less than every positive integer.
• Among negative integers, the one with the smaller absolute value is greater. For example, -3 > -7 because -3 lies to the right of -7 on the number line.

We use the symbols greater than (>) and less than (<) to compare. For example, -5 < 3 and 0 > -2.

Successor and Predecessor of Integers

• The successor of an integer is the integer just after it, obtained by adding 1. Successor of -2 is -1.
• The predecessor of an integer is the integer just before it, obtained by subtracting 1. Predecessor of -2 is -3.
• Unlike whole numbers, every integer, including the negative ones, has both a successor and a predecessor.

4. Addition of Integers

We can add integers using the number line.

Adding a Positive Integer

To add a positive integer, move to the right on the number line. For example, 3 + 4 means starting at 3 and moving 4 steps to the right to reach 7.

Adding a Negative Integer

To add a negative integer, move to the left on the number line. For example, 3 + (-4) means starting at 3 and moving 4 steps to the left to reach -1.

Rules for Adding Integers

1. To add two positive integers, add them as usual. The result is positive.
2. To add two negative integers, add their absolute values and put a negative sign. For example, (-3) + (-5) = -8.
3. To add a positive and a negative integer, find the difference of their absolute values. The result takes the sign of the integer with the greater absolute value. For example, (-7) + 5 = -2 because 7 - 5 = 2 and the larger absolute value 7 has a negative sign.

5. Subtraction of Integers

Subtraction of integers can also be shown on the number line.

Subtracting a Positive Integer

To subtract a positive integer, move to the left. For example, 5 - 3 means starting at 5 and moving 3 steps to the left to reach 2.

Subtracting a Negative Integer

To subtract a negative integer, move to the right. For example, 5 - (-3) means starting at 5 and moving 3 steps to the right to reach 8.

Rules for Subtracting Integers

To subtract an integer, we add its additive inverse. - a - b = a + (-b) - a - (-b) = a + b

Examples: - 7 - 3 = 7 + (-3) = 4. - 7 - (-3) = 7 + 3 = 10. - -6 - 4 = -6 + (-4) = -10. - -6 - (-4) = -6 + 4 = -2.

6. Properties of Addition and Subtraction of Integers

• Closure property: Integers are closed under addition and subtraction. The sum or difference of any two integers is always an integer.
• Commutativity: Addition of integers is commutative: a + b = b + a. For example, -3 + 5 = 5 + (-3) = 2. Subtraction is NOT commutative.
• Associativity: Addition of integers is associative: (a + b) + c = a + (b + c). For example, (-2 + 3) + 4 = -2 + (3 + 4).
• Additive identity: 0 is the additive identity. a + 0 = a.
• Additive inverse: For every integer a, there is an integer -a such that a + (-a) = 0. The additive inverse of -7 is 7.

7. Integers in Real Life

Integers are used in many real-life situations: - Temperature above and below zero, measured in degrees. - Floors of a building above and below the ground. - Profit and loss in business. - Heights above and below sea level. - Deposits and withdrawals in a bank account.

A deposit of 500 rupees is written as +500 and a withdrawal as -500. A height of 100 metres above sea level is +100 and 50 metres below sea level is -50.

8. Simplifying Expressions with Integers

We can simplify expressions using the rules of addition and subtraction of integers.

• 10 - 4 - 3 = 10 + (-4) + (-3) = 10 - 7 = 3.
• (-8) + 3 + (-2) = (-8) + 3 - 2 = (-8) + 1 = -7.
• Using the idea of additive inverse makes calculations systematic.

Quick Revision Tables

Rule	Example
Two positive integers: add, result positive	4 + 6 = 10
Two negative integers: add absolute values, put negative sign	(-4) + (-6) = -10
Positive and negative: subtract absolute values, take sign of bigger	(-7) + 5 = -2
Subtract an integer: add its additive inverse	7 - (-3) = 7 + 3 = 10

Integer	Additive Inverse
5	-5
-8	8
0	0
-1	1

Mind Map

flowchart TD
    A["Integers: ..., -3, -2, -1, 0, 1, 2, 3, ..."] --> B["Number Line"]
    A --> C["Comparison and Ordering"]
    A --> D["Addition"]
    A --> E["Subtraction"]
    A --> F["Properties"]
    B --> B1["Negatives to the left of zero"]
    B --> B2["Positives to the right of zero"]
    D --> D1["Move right for positive"]
    D --> D2["Move left for negative"]
    E --> E1["Subtract positive: move left"]
    E --> E2["Subtract negative: move right"]
    F --> F1["Closure and commutativity"]
    F --> F2["Associativity"]
    F --> F3["Additive identity 0"]
    F --> F4["Additive inverse"]
    A --> G["Real life: temperature, sea level, profit and loss"]

Important Diagrams (SVG)

Integer Number Line

Addition of Integers 3 + (-5) = -2

Common Mistakes

• Thinking -3 is greater than -1. On the number line -1 is to the right of -3, so -1 > -3.
• Confusing the addition of two negatives. (-4) + (-6) = -10, not 10.
• When adding a positive and negative integer, forgetting that the result takes the sign of the number with the greater absolute value.
• Treating a - (-b) as a - b. Remember minus of a negative is plus: a - (-b) = a + b.
• Forgetting that zero has no sign; it is neither positive nor negative.
• Confusing the successor and predecessor of negative integers.

Exam Tips

• Always draw a number line to solve addition and subtraction questions involving integers.
• Memorise the sign rules: like signs add and keep the sign; unlike signs subtract and take the sign of the bigger absolute value.
• Remember a - b = a + (-b) for every subtraction.
• Write the additive inverse correctly: the additive inverse of -5 is 5.
• Use real-life contexts like temperature to check whether your answers are sensible.
• Practise comparing negative integers: the one closer to zero is greater.

Conclusion

Integers extend the number system to include negative numbers, allowing us to describe temperatures below zero, depths below sea level and losses in business. We learnt to represent integers on the number line, compare them, and add and subtract them using clear rules. The properties of integers under addition and subtraction, especially the additive inverse, will be used extensively in algebra and equations in higher classes.