# Integers

“Integers” is a mathematical term used to describe a set of numbers that includes all whole numbers, both positive and negative, as well as zero. In other words, integers are a collection of numbers that do not have fractional or decimal parts. They are typically represented by the symbol “Z.”

Here are some examples of integers:

• Positive integers: 1, 2, 3, 4, 5, …
• Negative integers: -1, -2, -3, -4, -5, …
• Zero: 0

Integers are used in various mathematical operations, including addition, subtraction, multiplication, and division. They play a fundamental role in mathematics and are used in many practical applications, such as counting, measuring, and solving equations.

## What are integers in Math?

In Mathematics, the term “integers” refers to a set of whole numbers that includes both positive and negative numbers, along with zero. Integers are numbers that do not have any fractional or decimal parts.

In formal mathematical notation, integers are often represented by the symbol “Z,” and the set of integers can be expressed as:

Z = {…, -3, -2, -1, 0, 1, 2, 3, …}

This set includes all positive whole numbers (1, 2, 3, …), all negative whole numbers (-1, -2, -3, …), and zero (0). Integers are used in a wide range of mathematical calculations and have many practical applications in fields like arithmetic, algebra, and number theory.

## Integers symbol

The symbol used to represent the set of integers in mathematics is “ℤ”. It is a script uppercase letter “Z” and is often written in a bold or fancy font to distinguish it from other variables or numbers. This symbol represents the set of all integers, including positive integers, negative integers, and zero. In mathematical notation, you may see expressions like “x ∈ ℤ,” which means that the variable “x” is an integer.

Z = {…, -3, -2, -1, 0, 1, 2, 3, …}

## Integers examples

Positive Integers (greater than zero):

1
2
3
4
5

Negative Integers (less than zero):

-1
-2
-3
-4
-5

Zero:

0

These examples cover the three main categories of integers: positive integers, negative integers, and zero. Integers are used in various mathematical operations and have practical applications in many aspects of science, engineering, and everyday life.

## Types of integers

Integers are classified into three types:

### Positive integers

Positive integers are a set of whole numbers that are greater than zero. In other words, positive integers are the counting numbers. They do not include any fractions, decimals, or negative numbers. Here are some examples of positive integers:

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …

These numbers continue indefinitely in a sequence, with each number being one greater than the previous one. Positive integers are used for counting, ordering, and representing quantities that are greater than zero. They are an essential concept in mathematics and have many practical applications in various fields.

### Negative integers

Negative integers are a set of whole numbers that are less than zero. In other words, negative integers are whole numbers that are represented with a minus sign (-) in front of them. They are used to describe values or quantities that are below zero.

Here are some examples of negative integers:

-1, -2, -3, -4, -5, -6, -7, -8, -9, -10, …

Negative integers are important in mathematics and various real-world applications. They represent values that are lower than zero on the number line and play a crucial role in calculations involving debt, temperature below freezing, and other situations where values are less than zero.

### Zero

Zero is considered an integer and is represented as 0. It is neither positive nor negative and serves as a reference point between positive and negative numbers.

## Arithmetic Operations on integers

Adding integers involves following specific rules to combine positive and negative whole numbers accurately. Here are the rules for adding integers:

Rule 1: Adding Integers with the Same Sign

When you’re adding integers with the same sign (both positive or both negative), follow these steps:

Add the absolute values (ignoring the signs) of the numbers.

Keep the common sign for the result.

Examples:

(+3) + (+5) = +8 (3 + 5 = 8, and both numbers are positive, so the result is positive).
(-2) + (-7) = -9 (2 + 7 = 9, and both numbers are negative, so the result is negative).

Rule 2: Adding Integers with Different Signs

When you’re adding integers with different signs (one positive and one negative), follow these steps:

Subtract the absolute value of the smaller number from the absolute value of the larger number.

Assign the sign of the number with the larger absolute value to the result.

Examples:

(+4) + (-2) = +2 (4 – 2 = 2, and the number with the larger absolute value is positive, so the result is positive).
(-8) + (+6) = -2 (8 – 6 = 2, and the number with the larger absolute value is negative, so the result is negative).

Adding zero to any integer doesn’t change the value of that integer.

Examples:

(+7) + 0 = +7
(-3) + 0 = -3

By following these rules, you can accurately add integers and determine the correct sign of the result based on the signs of the numbers you’re adding.

### Subtracting integers

Subtracting integers involves following specific rules to find the difference between two positive or negative whole numbers. Here are the rules for subtracting integers:

Rule 1: Subtracting Integers with the Same Sign

When you’re subtracting integers with the same sign (both positive or both negative), follow these steps:

1. Subtract the absolute value (ignoring the sign) of the smaller number from the absolute value of the larger number.
2. Assign the sign of the number with the larger absolute value to the result.

Examples:

• (+7) – (+3) = +4 (7 – 3 = 4, and both numbers are positive, so the result is positive).
• (-8) – (-2) = -6 (8 – 2 = 6, and both numbers are negative, so the result is negative).

Rule 2: Subtracting Integers with Different Signs

When you’re subtracting integers with different signs (one positive and one negative), it’s like changing the subtraction into addition. Follow these steps:

1. Keep the sign of the first number.
2. Change the subtraction sign to addition.
3. Change the sign of the second number to its opposite (positive becomes negative or vice versa).

Examples:

• (+5) – (-2) is equivalent to (+5) + (+2) = +7 (changing subtraction to addition and keeping the sign of the first number).
• (-4) – (+1) is equivalent to (-4) + (-1) = -5 (changing subtraction to addition and keeping the sign of the first number).

Rule 3: Subtracting Zero

Subtracting zero from any integer doesn’t change the value of that integer.

Examples:

• (+9) – 0 = +9
• (-3) – 0 = -3

By following these rules, you can subtract integers accurately and determine the correct sign of the result based on the signs of the numbers you’re subtracting.

### Multiplying integers

Multiplying integers involves specific rules for combining positive and negative whole numbers. Here are the rules for multiplying integers:

Rule 1: Multiplying Integers with the Same Sign

When you’re multiplying integers with the same sign (both positive or both negative), the result is always positive:

• Multiply their absolute values (ignore the signs).
• The result is positive.

Examples:

• (+3) × (+5) = +15 (3 × 5 = 15, and both numbers are positive, so the result is positive).
• (-2) × (-7) = +14 (2 × 7 = 14, and both numbers are negative, so the result is positive).

Rule 2: Multiplying Integers with Different Signs

When you’re multiplying integers with different signs (one positive and one negative), the result is always negative:

• Multiply their absolute values (ignore the signs).
• The result is negative.

Examples:

• (+4) × (-2) = -8 (4 × 2 = 8, and one number is positive and the other is negative, so the result is negative).
• (-8) × (+6) = -48 (8 × 6 = 48, and one number is negative and the other is positive, so the result is negative).

Rule 3: Multiplying by Zero

Multiplying any integer by zero results in zero:

• Any integer × 0 = 0

Examples:

• (+7) × 0 = 0
• (-3) × 0 = 0

These rules help you accurately multiply integers and determine the sign of the result based on the signs of the numbers you’re multiplying.

### Dividing integers

Dividing integers involves specific rules for dividing positive and negative whole numbers. Here are the rules for dividing integers:

Rule 1: Dividing Integers with the Same Sign

When you’re dividing integers with the same sign (both positive or both negative), the result is always positive:

• Divide their absolute values (ignore the signs).
• The result is positive.

Examples:

• (+12) ÷ (+3) = +4 (12 ÷ 3 = 4, and both numbers are positive, so the result is positive).
• (-24) ÷ (-6) = +4 (24 ÷ 6 = 4, and both numbers are negative, so the result is positive).

Rule 2: Dividing Integers with Different Signs

When you’re dividing integers with different signs (one positive and one negative), the result is always negative:

• Divide their absolute values (ignore the signs).
• The result is negative.

Examples:

• (+15) ÷ (-3) = -5 (15 ÷ 3 = 5, and one number is positive and the other is negative, so the result is negative).
• (-18) ÷ (+6) = -3 (18 ÷ 6 = 3, and one number is negative and the other is positive, so the result is negative).

Rule 3: Dividing by Zero (Undefined)

Division by zero is undefined in mathematics. You cannot divide any integer by zero. It is important to avoid attempting such divisions as they have no meaningful mathematical interpretation.

These rules help you accurately divide integers and determine the sign of the result based on the signs of the numbers you’re dividing.

### Question 1: Can integers be negative?

Answer: Yes, integers can indeed be negative. In mathematics, the set of integers includes both positive and negative whole numbers, as well as zero. Integers encompass a wide range of values, including:

• Positive integers: These are whole numbers greater than zero, such as 1, 2, 3, 4, 5, and so on.
• Negative integers: These are whole numbers less than zero, indicated by a minus sign, such as -1, -2, -3, -4, -5, and so on.
• Zero: Zero is considered an integer, and it lies between the positive and negative integers, often denoted as 0.

So, to sum it up, integers can definitely be negative, and they represent a fundamental set of numbers in mathematics that includes both positive and negative values, as well as zero.

### Question 2: Are fractions integers?

Answer: No, fractions are not integers. Fractions and integers are two distinct categories of numbers in mathematics.

Integers are whole numbers, both positive and negative, as well as zero, and they do not have fractional or decimal parts. Examples of integers include -3, -2, -1, 0, 1, 2, 3, and so on.

Fractions, on the other hand, represent numbers that are not whole. They consist of two parts: a numerator (the top number) and a denominator (the bottom number), separated by a horizontal line. Fractions can represent parts of a whole or values that fall between integers. Examples of fractions include 1/2, 3/4, and 5/6.

In summary, integers are a specific type of whole number, while fractions represent values that can be expressed as parts of a whole and typically include non-integer values.

### Question 3: Are all integers whole numbers?

Answer: Yes, all integers are indeed whole numbers. In mathematics, the term “whole numbers” refers to a set of numbers that includes zero and all the positive integers (natural numbers). In other words, whole numbers are a subset of integers.

Here’s the relationship:

• Integers: This set includes all positive integers (1, 2, 3, 4, …) and all their negative counterparts (-1, -2, -3, -4, …), along with zero (0).
• Whole Numbers: Whole numbers are a subset of integers and include all the positive integers (1, 2, 3, 4, …) and zero (0). Whole numbers do not include negative integers.

So, in summary, every integer is a whole number because it includes zero and all the positive and negative integers, while whole numbers only include zero and the positive integers.

### Question 4: Are integers rational numbers?

Answer: Yes, integers are indeed rational numbers. In mathematics, rational numbers are numbers that can be expressed as a ratio (fraction) of two integers, where the denominator is not zero. Since integers can be expressed as fractions with a denominator of 1 (e.g., 5 can be written as 5/1), they are considered a subset of rational numbers.

To be more precise:

Integers: These are whole numbers, both positive and negative, as well as zero. They can be written as fractions with a denominator of 1. For example, 5 is the same as 5/1, and -3 is the same as -3/1.

Rational Numbers: These are numbers that can be expressed as fractions, where both the numerator and denominator are integers, and the denominator is not zero. Rational numbers include integers, fractions, and decimals that terminate or repeat (such as 0.5 = 1/2 or 0.333… = 1/3).

So, in summary, integers are a specific type of rational number, and they fit within the broader category of rational numbers in mathematics.

### Question 5: Can integers be decimals?

Answer: No, integers cannot be decimals. Integers are a specific category of whole numbers that do not have any fractional or decimal parts. They are whole numbers that include both positive and negative numbers, as well as zero.

Examples of integers:

• Positive integers: 1, 2, 3, 4, 5, …
• Negative integers: -1, -2, -3, -4, -5, …
• Zero: 0

Decimals, on the other hand, represent numbers with fractional parts. Decimals include digits after the decimal point, such as 0.5, -3.14, 2.75, and so on. Integers and decimals are distinct categories of numbers, with integers being whole numbers and decimals being numbers that include fractional or decimal components.

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