Integers

• What is an Integer
• Integers Definition
• Types of Integers
• Set of Integers
• Integers on Number Line
• Integer Operations
• Addition of Integers
• Subtraction of Integers
• Multiplication of Integers
• Division of Integers
• Properties of Integers
• Solved Examples
• Practice Problems
• Frequently Asked Questions

What is an Integer

The term "integer" originates from the Latin word "integer," which means "whole" or "untouched." In mathematics, integers refer to whole numbers, both positive and negative, including zero.

Integers Definition

Integers are fundamental elements in mathematics, comprising positive numbers, negative numbers, and zero. Unlike fractions or decimals, integers are whole numbers without any fractional parts.

On a number line, zero lies at the center, positive integers extend to the right and negative integers to the left. They are essential in various mathematical operations such as addition, subtraction, multiplication, division and many more. Additionally, integers find applications in real-life situations like temperature readings, financial transactions, and counting objects. Some examples of integers are `7`, `-2`, `0`, `15`, and `-10`.

Types of Integers

Integers can be categorized into three types: positive integers, negative integers, and zero. 

`1`. Positive Integers: These are whole numbers greater than zero. They are represented on the number line by points extending to the right from zero. Examples include `1`, `2`, `3`, `4`, and so on.

`2`. Negative Integers: Negative integers are whole numbers less than zero. They are represented on the number line by points extending to the left from zero. Examples include `-1`, `-2`, `-3`, `-4`, and so forth.

`3`. Zero: Zero is a unique integer that represents the absence of quantity or value. It lies at the center of the number line and acts as a reference point. It is neither positive nor negative. 

Set of Integers

The set of integers, denoted by `ℤ`, encompasses all positive and negative whole numbers, along with zero. It is represented as `{..., -3, -2, -1, 0, 1, 2, 3, ...}`. 

In set notation, it can be written as `ℤ = {..., -3, -2, -1, 0, 1, 2, 3, ...}`. 

This set is infinite and includes numbers that extend infinitely in both the positive and negative directions. Integers are closed under addition, subtraction, and multiplication, meaning that performing these operations on integers will always yield another integer. 

The set of integers serves as a fundamental concept in mathematics, forming the basis for various mathematical theories and calculations.

Integers on Number Line

Integers are positioned on a number line, with positive integers to the right of zero and negative integers to the left. Zero is at the center, acting as a reference point. This visualization helps understand their placement and relationships.

Integer Operations

Integer operations involve mathematical calculations performed on integers. Some of these basic operations include addition, subtraction, multiplication, and division.

Addition of Integers

When adding integers, if the integers have the same sign, we add their absolute values and keep the common sign. If the integers have different signs, we subtract the smaller absolute value from the larger one and use the sign of the integer with the larger absolute value.

`1`. Adding Integers with the Same Sign: 

When adding integers with the same sign (either both positive or both negative), add their absolute values and keep the common sign. 

Example: What is the sum of `-5` and `-3`? 

Solution: Both integers are negative. So, we add their absolute values: `|-5| + |-3| = 5 + 3 = 8`. Since both integers are negative, the sum is also negative. Therefore, `-5 + (-3) = -8`.

`2`. Adding Integers with Different Signs:

When adding integers with different signs (one positive and one negative), subtract the smaller absolute value from the larger one and keep the sign of the integer with the larger absolute value.

Example: What is the sum of `-7` and `4`?

Solution: One integer is negative and the other is positive. So, we subtract the smaller absolute value from the larger one: `|-7| - |4| = 7 - 4 = 3`. Since `7` has a larger absolute value and is positive, the sum is positive. Therefore, `-7 + 4 = -3`.

Subtraction of Integers

The subtraction of integers is similar to addition but with a different rule for subtracting integers with different signs. To subtract integers, we change the subtraction to addition and change the sign of the integer being subtracted. Then, we proceed with the addition as usual.

`1`. Subtracting Integers with the Same Sign:

When subtracting integers with the same sign (either both positive or both negative), change the subtraction to addition and subtract the smaller absolute value from the larger one. Keep the sign of the integer with the larger absolute value.

Example: What is the result of `-8 - (-3)`?

Solution: Both integers have the same sign (both negative). So, we change the subtraction to addition: `-8 + 3`. Then, we subtract the smaller absolute value `(3)` from the larger one `(8): |(-8)| - |(-3)| = 8 - 3 = 5`. Since `8` has a larger absolute value and is negative, the result is negative. Therefore, `-8 - (-3) = -5`.

`2`. Subtracting Integers with Different Signs:

When subtracting integers with different signs (one positive and one negative), change the subtraction to addition and keep the sign of the first integer. Then, add the absolute values.

Example: What is the result of `5 - (-2)`?

Solution: One integer is positive and the other is negative. So, we change the subtraction to addition: `5 + 2`. Then, we add their absolute values: `|5| + |(-2)| = 5 + 2 = 7`. Since the first integer `(5)` is positive, the result is positive. Therefore, `5 - (-2) = 7`.

Multiplication of Integers

In integer multiplication, if the integers have the same sign, the product is positive. If the integers have different signs, the product is negative.

`1`. Multiplying Integers with the Same Sign:

When multiplying integers with the same sign (either both positive or both negative), the product is positive.

Example: What is the result of `(-4) × (-3)`?

Solution: Both integers are negative. So, the product of `(-4) × (-3)` is positive. `(-4) × (-3) = 12`.

`2`. Multiplying Integers with Different Signs:

 When multiplying integers with different signs (one positive and one negative), the product is negative.

Example: What is the result of `(-5) × 2`?

Solution: One integer is negative and the other is positive. So, the product of `(-5) × 2` is negative: `(-5) × 2 = -10`.

Division of Integers

Integer division involves dividing one integer by another. When dividing integers, if both integers have the same sign, the quotient is positive. If the integers have different signs, the quotient is negative.

`1`. Dividing Integers with the Same Sign:

When dividing integers with the same sign (either both positive or negative), the quotient is positive.

Example: What is the result of `(-12) ÷ (-3)`?

Solution: Both integers are negative. So, the quotient of `(-12) ÷ (-3)` is positive: `(-12) ÷ (-3) = 4`.

`2`. Dividing Integers with Different Signs:

When dividing integers with different signs (one positive and one negative), the quotient is negative.

Example: What is the result of `(-15) ÷ 3`?

Solution: One integer is negative and the other is positive. So, the quotient of `(-15) ÷ 3` is negative: `(-15) ÷ 3 = -5`.

Properties of Integers

`1`. Closure Property: Integers are closed under addition, subtraction, and multiplication. This means that when you add, subtract, or multiply any two integers, the result is always an integer.

`2`. Associative Property: The addition and multiplication of integers are associative, meaning that changing the grouping of the numbers being added or multiplied does not change the result. For example, `(a + b) + c = a + (b + c)` and `(a × b) × c = a × (b × c)`.

`3`. Commutative Property: The addition and multiplication of integers are commutative, meaning that changing the order of the numbers being added or multiplied does not change the result. For example, `a + b = b + a` and `a × b = b × a`.

`4`. Identity Property: The identity element for addition in integers is zero, meaning that adding zero to any integer leaves the integer unchanged. Similarly, the identity element for multiplication in integers is one, meaning that multiplying any integer by one leaves the integer unchanged.

`5`. Inverse Property: Every integer has an additive inverse, which is the integer that when added to it, gives zero. For example, the additive inverse of `5` is `-5`, and the additive inverse of `-3` is `3`.

`6`. Distributive Property: Multiplication distributes over addition in integers, meaning that `a × (b + c) = (a × b) + (a × c)` for any integers `a`, `b`, and `c`.

Solved Examples

Example `1`: Evaluate the expression: \( (-8) + (-5) \times (-3) \).

Solution:

First, perform the multiplication operation, \( (-5) \times (-3) \), which results in \( 15 \).

Now, rewrite the expression with the product: \( (-8) + 15 \)

Finally, perform the addition: \( (-8) + 15 = 7 \)

So, \( (-8) + (-5) \times (-3) = 7 \).

Example `2`: Find the value of \( x \) that satisfies the equation: \( -2x + 7 = 19 \).

Solution:

First, isolate \( x \) by subtracting `7` from both sides of the equation:

\( -2x + 7 - 7 = 19 - 7 \)

which simplifies to \( -2x = 12 \).

Next, divide both sides by `-2` to solve for \( x \):

\( \frac{-2x}{-2} = \frac{12}{-2} \)

resulting in \( x = -6 \).

So, the value of \( x \) that satisfies the equation \( -2x + 7 = 19 \) is \( x = -6 \).

Practice Problems

Q`1`: What is the result of \( (-9) \times (-4) \)?

a) `-36` 

b) `36`  

c) `-45`  

d) `45`  

Answer: b

Q`2`. Solve for \( x \) in the equation \( 5x - 12 = -7 \).

a) \( x = -1 \)  

b) \( x = 1 \)  

c) \( x = 5 \)  

d) \( x = -5 \)  

Answer: b 

Q`3`. What is the value of \( |-7| + |-15| \)?

a) `-22`  

b) `8`  

c) `22`  

d) `-8`  

Answer: c

Q`4`. Evaluate \( (-20) - (-8) \).

a) `-12`  

b) `12` 

c) `-28`  

d) `28`  

Answer: a

Frequently Asked Questions

Q`1`. What are integers?

Answer: Integers are whole numbers that can be positive, negative, or zero. They do not contain any fractional or decimal parts.

Q`2`. How are integers represented on a number line?

Answer: Positive integers are represented to the right of zero, with each successive positive integer placed at an equal distance from the previous one. Negative integers are represented to the left of zero in a similar manner, with each successive negative integer placed at an equal distance from the previous one.

Q`3`. What are the rules for adding and subtracting integers?

Answer: When adding integers, if they have the same sign, add their absolute values and keep the common sign. If they have different signs, subtract the smaller absolute value from the larger one and use the sign of the integer with the larger absolute value. Subtraction is similar but involves changing subtraction to addition and adjusting signs accordingly.

Q`4`. How do you multiply and divide integers?

Answer: When multiplying integers, if they have the same sign, the product is positive. If they have different signs, the product is negative. The division follows a similar pattern.

Q`5`. What are some real-world applications of integers?

Answer: Integers are used in various real-life scenarios such as temperature readings (positive for hot, negative for cold), financial transactions (positive for deposits, negative for withdrawals), representing scores in games (positive for points earned, negative for points lost), and many other situations where quantities can be counted or compared.