INTEGERS

An integer is a complete whole number that can be positive, negative, or zero.

Integers do not include fractions or decimals.

Examples of integers are: -1 , 0 , 1, 20 , 200, -30

Identify which of the following numbers are integers: -5, 0, 3, 4.5, and 1 2 .

solution :

-5, 0, and 3 are integers because they are complete whole values.
4.5 is not an integer because it contains a decimal part.
1 2 is not an integer because it is a fraction representing a part of a whole.

Integers = -5, 0, 3

Integers are used in everyday life to represent values that go above or below a fixed starting point or zero mark:

Integers are used in everyday life when:

Measuring temperature below freezing point (e.g., -4°C) or above freezing point (e.g., 25°C).
Tracking money deposits as positive integers and bank withdrawals or debts as negative integers.
Recording geographical locations relative to sea level, where mountains are above sea level and deep ocean trenches are below sea level.
Counting floors in a building, where ground floors are positive numbers and basement levels are negative numbers.
Tracking scores in games or sports, such as being under par or over par in golf, or gaining and losing yards in football.

Types of Integers

Integers are classified into three distinct groups based on their position relative to zero:

Positive Integers: Whole numbers that are greater than zero. They are written with a plus sign (+) or with no sign at all. They represent gains, increases, or upward movement.
Negative Integers: Whole numbers that are less than zero. They are always written with a minus sign (-) in front of them. They represent losses, decreases, or downward movement.
Zero (0): A neutral integer that is neither positive nor negative. It acts as the reference point or starting value.

Example

Classify the following values into their correct integer types: -12, +7, 0, and 85.

solution :

-12 is a Negative Integer because it has a minus sign and is less than zero.
+7 and 85 are Positive Integers because they are greater than zero.
0 is a Neutral Integer because it has no positive or negative value.

Number Lines

A number line is a straight horizontal line that visually displays integers in order from left to right. It is a critical tool for comparing values and understanding distance between numbers.

The Center Point: Zero (0) sits exactly in the middle of the line and acts as the neutral starting reference anchor.
Right of Zero: Positive integers increase in size sequentially as you move further to the right (+1, +2, +3, ...).
Left of Zero: Negative integers decrease in size sequentially as you move further to the left (-1, -2, -3, ...).
Directional Value Rule: Any number that sits to the right of another number on a number line is always larger. Any number to the left is always smaller.

Example

Draw a balanced number line showing integers from -5 to +5 to compare the sizes of -3 and 1.

solution :

←
→
-5
-4
-3
-2
-1
0
1
2
3
4
5

Step 1: Locate -3 and 1 on the generated horizontal layout map.

The integer -3 is located 3 spaces to the left of the zero mark.

The integer 1 is located 1 space to the right of the zero mark.

Step 2: Compare their positions using the directional rule.

Since 1 sits to the right of -3 on the number line, 1 is the larger value.

Since -3 sits to the left of 1 on the number line, -3 is the smaller value.

Final Comparison = -3 < 1

Basic Math Rules for Integers

Operating with integers requires following specific sign combination laws. These rules govern how positive and negative values interact when performing basic operations.

1. Addition and Subtraction Rules

When adding or subtracting integers, think of moving along the number line or balancing positive gains against negative losses:

Same Signs: If two numbers have identical signs, add their absolute values together and keep the shared sign.
Different Signs: If two numbers have different signs, find the difference between them by subtracting the smaller value from the larger one, then keep the sign of the larger value.
Double Negatives: When two minus signs appear directly next to each other, they combine to form a positive sign: −(−) → +

Example 1

Evaluate: (−3) + (−5)

solution: Both signs are the same. Add 3 and 5 to get 8. Keep the negative sign.

Result = −8

Simple explanation :

Assume you had a debt of 3 dollars and then borrowed another 5 dollars from the same lender; your total debt would then be 8 dollars.

Evaluate: (−4) + 9

Solution: The signs are different. Subtract 4 from 9 to get 5. Keep the positive sign of the larger number (9).

Result = 5

Simple explanation :

Assume you had 9 dollars, used it to repay a debt of 4 dollars to a lender; you would then be left with 5 dollars.

Evaluate: 6 − (−2)

solution: The double negative turns into a plus sign: 6 + 2.

Result = 8

2. Multiplication and Division Rules

When multiplying or dividing two integers, calculate the numerical result normally, then apply the uniform sign combination law:

Like Signs Produce Positive: Multiplying or dividing two numbers with the same signs always results in a positive value.
Unlike Signs Produce Negative: Multiplying or dividing two numbers with different signs always results in a negative value.

Summary Reference Matrix:

Positive × Positive = Positive
Negative × Negative = Positive
Positive × Negative = Negative
Negative × Positive = Negative

Examples

Evaluate: (−4) × (−3)

solution: Multiply 4 by 3 to get 12. Since both signs are negative (like signs), the result is positive.

Result = 12

Evaluate: 15 ÷ (−3)

solution: Divide 15 by 3 to get 5. Since the signs are different (unlike signs), the result is negative.

Result = −5

Worked Examples

Example 1

Evaluate: −15 − (−8) + (−4)

solution :

Step 1: Simplify the double negative sign first. The term −(−8) becomes +8.

New Expression = −15 + 8 + (−4)

Step 2: Working from left to right, combine −15 and 8. Since the signs are different, subtract 8 from 15 to get 7, and keep the negative sign of the larger number.

−15 + 8 = −7

New Expression = −7 + (−4)

Step 3: Combine −7 and −4. Adding a negative number is the same as subtraction: −7 − 4. Since the signs are identical, add 7 and 4 together to get 11 and keep the negative sign.

−7 − 4 = −11

Final Answer = −11

Example 2

Evaluate: (−6) × 4 ÷ (−3)

solution :

Step 1: Working from left to right, perform the multiplication first. Multiply 6 by 4 to get 24. Since the signs are unlike (negative and positive), the product is negative.

(−6) × 4 = −24

New Expression = −24 ÷ (−3)

Step 2: Perform the division operation. Divide 24 by 3 to get 8. Since both signs are negative (like signs), the final result is positive.

−24 ÷ (−3) = 8

Final Answer = 8

Example 3

Evaluate using order of operations: −12 + (−18) ÷ 3 × (−2)

solution :

Step 1: Division and multiplication take priority over addition. Working from left to right, perform the division first. Divide −18 by 3 to get −6 because the signs are unlike.

−18 ÷ 3 = −6

New Expression = −12 + (−6) × (−2)

Step 2: Perform the multiplication next. Multiply −6 by −2 to get positive 12 because multiplying two negative numbers creates a positive result.

(−6) × (−2) = 12

New Expression = −12 + 12

Step 3: Perform the final addition. Adding 12 to its exact negative opposite results in zero.

−12 + 12 = 0

Final Answer = 0

Example 4

Arrange the following integers in ascending order (from the smallest to the largest value): −7, 3, 0, −2, +5, −10.

solution :

Step 1: Analyze the values based on their position on a horizontal number line. Numbers furthest to the left are the smallest.

Negative numbers are always smaller than 0 and positive numbers. Comparing negative numbers, −10 is further left than −7, and −7 is further left than −2.

Smallest negative group order: −10, −7, −2

Step 2: Place the neutral reference anchor, zero, after the negative values.

Updated order: −10, −7, −2, 0

Step 3: Order the positive integers from smallest to largest value on the right side of zero.

Positive group order: 3, 5

Step 4: Combine the strings for the final sorted sequence.

Final Answer = −10, −7, −2, 0, 3, 5

Example 5 (Word Problem)

The temperature in a town at midnight was −4°C. By noon, the temperature rose by 11°C, and by 10:00 PM it dropped by 5°C. Write an expression and calculate the final temperature.

solution :

Step 1: Translate the word steps into a single mathematical expression using positive integers for increases and negative integers for drops.

Expression = −4 + 11 − 5

Step 2: Calculate the first step from left to right. Combine −4 and 11. Since the signs are different, find the difference between 11 and 4, keeping the positive sign of the larger value.

−4 + 11 = 7

New Expression = 7 − 5

Step 3: Perform the final subtraction operation.

7 − 5 = 2

Final Answer = 2°C

Sample Questions

Attempt the following Questions:

(1.) Evaluate the expression: −22 − (−14)

(2.) Evaluate the expression: (−8) × 3 ÷ (−6)

(3.) Evaluate using the order of operations: −5 + (−4) × (−3) − 8

(4.) Arrange the following integers in descending order (from largest to smallest): −4, 9, −12, 0, −1, 6

(5.) A submarine cruises at a depth of 150 meters below sea level. It then dives down an additional 75 meters, and later rises up 40 meters. Express its final position as an integer relative to sea level.

Check The Answers Below:

Take Note:

No Fractions or Decimals: Integers are always complete whole numbers. Numbers with decimals like 2.5 or fractions like 1÷2 are never integers.
Zero is Neutral: The number 0 is an integer, but it is neither positive nor negative. It sits right in the middle of the number line.
Size on the Number Line: On a number line, any number to the right is always bigger than any number to the left. This means 0 is bigger than any negative number, and −1 is bigger than −100.
The Double Negative Rule: Whenever you see two minus signs right next to each other, they always join to become a plus sign. For example, 5 − (−3) becomes 5 + 3.
Multiplication and Division Signs: When you multiply or divide two numbers with the same signs, the answer is always positive. When you multiply or divide two numbers with different signs, the answer is always negative.

INTEGERS

Types of Integers

Number Lines

Basic Math Rules for Integers

1. Addition and Subtraction Rules

2. Multiplication and Division Rules

Worked Examples

Sample Questions

Check The Answers Below:

Answers:

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