Integers

The concept of integers is an integral part of mathematics, especially when we delve into understanding different number systems. In this comprehensive guide, we will understand the entire concept of integers in the simplest language possible. We will look at their definitions, properties, operations, and applications. With numerous examples and visual aids, this article aims to break down the potentially complex idea of integers into understandable material even for beginners.

What are integers?

Integers are a set of numbers that includes all whole numbers, both positive and negative, as well as zero. They can be placed on a number line extending to infinity in both directions. Integers can be defined as:

The number line given above shows a part of the integer set which consists of some positive numbers, negative numbers and zero. Any sign on this line is an integer. As we move to the right, the value of numbers increases, while their value decreases as we move to the left.

Types of integers

Integers can be classified into three main types:

Positive integers

These are numbers greater than zero, such as 1, 2, 3, 4, etc. These lie to the right of zero on the number line.

Negative integers

These are numbers less than zero, represented as -1, -2, -3, -4, etc. On the number line, they lie to the left of zero.

Zero

Zero is considered neutral; it is neither positive nor negative, and is the central point on the number line that separates positive integers from negative integers.

Properties of integers

Integers obey several fundamental properties that facilitate various calculations and operations. These properties include:

Closing assets

The set of integers is closed under addition, subtraction, and multiplication. This means that if you add, subtract, or multiply any two integers, the result will always be an integer.

Examples: 
Addition: 5 + (-3) = 2 
Subtraction: 7 - 10 = -3 
Multiplication: (-4) * 2 = -8

Commutative property

Integers are commutative under addition and multiplication. This means that the order of the numbers does not affect the result.

Examples: 
Addition: 3 + (-2) = (-2) + 3 = 1 
Multiplication: (-5) * 4 = 4 * (-5) = -20

Associative property

The associative property applies to addition and multiplication of integers, meaning the result will be the same no matter how the numbers are grouped.

Examples: 
Addition: (2 + 3) + 4 = 2 + (3 + 4) = 9 
Multiplication: (2 * 3) * 4 = 2 * (3 * 4) = 24

Distributive property

The distributive property connects multiplication and addition or subtraction, showing that multiplying a number by a group of numbers added together gives the same result as multiplying that number by each one separately and then adding or subtracting them.

Example: 
a * (b + c) = (a * b) + (a * c) 
3 * (4 + 5) = (3 * 4) + (3 * 5) = 12 + 15 = 27

Operations with integers

Operating with integers is similar to operating with whole numbers, but special attention has to be paid to the signs of the numbers.

Addition

The following rules apply when adding integers:

• The sum of two positive integers is a positive integer.
• The sum of two negative integers is a negative integer.
• To add a positive number and a negative number, ignore the sign and subtract the smaller number from the larger number, then take the sign of the larger number.

Examples: 
5 + 7 = 12 
(-3) + (-6) = -9 
8 + (-3) = 5 (since 8 is larger, the result is positive) 
(-5) + 12 = 7 (since 12 is larger, the result is positive)

Subtraction

Subtraction of integers can be understood as adding the opposites:

• To subtract a positive integer, add its negative counterpart.
• To subtract a negative integer, add its positive counterpart.

Examples: 
7 - 3 = 7 + (-3) = 4 
(-2) - 4 = (-2) + (-4) = -6 
6 - (-9) = 6 + 9 = 15

Multiplication

When multiplying integers, the following rules apply to the signs:

• The product of two positive integers is positive.
• The product of two negative integers is positive.
• The product of a positive integer and a negative integer is negative.

Examples: 
3 * 4 = 12 
(-2) * (-5) = 10 
7 * (-3) = -21

Division

The rules of division are the same as for multiplication with respect to signs:

• The quotient of two positive integers is positive.
• The quotient of two negative integers is positive.
• The quotient of a positive integer and a negative integer is negative.

Examples: 
10 / 2 = 5 
(-15) / (-3) = 5 
20 / (-4) = -5

Use of integers in real life scenarios

Integers are used in a variety of real-life scenarios and situations, often to represent positions or values relative to a baseline or datum. Examples include:

Temperature

Integers are often used in temperature readings, especially when measured on the Celsius or Fahrenheit scale. Temperatures below the freezing point are negative.

Financial transactions

In banking and finance, integers are used to represent loans and credits. A negative balance indicates a loan or outstanding balance.

Elevations

Geographic heights can use integers, with sea level being zero. Above sea level, heights are positive; below sea level, they are negative.

Sports scores

Many sports and competitions use points that can be expressed as integers. Negative points may be given as a penalty.

Further discoveries

In learning integers, one can explore additional concepts such as absolute value and how integers are integrated with other math topics, such as rational numbers and algebra. The absolute value of an integer is the distance from zero, regardless of direction, and is always non-negative.

Understanding the properties and operations involving integers is essential as a foundation for more advanced mathematical topics, including solving equations and inequalities, understanding limits, and working with different number systems and mathematical functions.

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