You can count, measure, and compare gains and losses with integers, which are fundamental building blocks in math. When we need to understand numbers in the real world or in school, integers help us. They are helpful for daily tasks, such as keeping track of temperatures, and for more complicated mathematical problems. Let’s discuss what an integer is, its types, characteristics, cases, and common uses in this article.
What are Integers?
What is an integer in mathematics? It can be any of the following:
- Positive (e.g., 1, 2, 3…)
- Negative (e.g., -1, -2, -3…)
- Zero (0)
An integer is not a number, a fraction, or a percentage. The set of integers is shown in math by the symbol ℤ, which comes from the German word Zahlen, which means “numbers”:
ℤ = {… -3, -2, -1, 0, 1, 2, 3 …}
We can use integers to do sums involving gain/loss, profit/deficit, weather change, altitude, etc. What is an integer, though? In the simplest words, it is any number that can be counted without being divided.
Examples of Integers
Now, to help you understand better, let’s look at some real-life and scientific examples of integers.
Integers on a Number Line:
A number line provides a visual representation of integers:
… -4 -3 -2 -1 0 1 2 3 4 …
Each number always stays one unit away from its neighbors. An integer that is positive is to the right of zero, and an integer that is negative is to the left of zero.
Real-Life Examples:
- Banking: ₹500 deposited = +500; ₹200 withdrawn = -200
- Temperature: +25°C (warm day), -10°C (cold day)
- Gaming: Score difference = +3 (win), -4 (loss)
- Geography: 100m below sea level = -100; 300m above = +300
- Finance: Stock gains = +15%; Losses = -8%
These examples of integers show how frequently they’re used in daily life.
What are the Types of Integers
There are three main types of integers: positive, negative, and zero. Each type has a specific use in mathematics and in real life.
Positive Integers:
These are the whole numbers that are not zero. In most real-life situations, they stand for gain, growth, or number. One example is counting from 1 to 100, or counting down from 1 to 2.
Negative Integers:
These are whole numbers that are less than zero. There is always a minus sign (-) before them. These words mean loss, debt, or a drop. -1, -5, -100, and so on are some examples.
Zero:
Zero (0) is independent because it is neither positive nor negative. It is located in the middle of the number line and often represents balance, neutrality, or a point of reference.
Knowing these three kinds of integers is essential for calculating and correctly handling math questions.
Properties of Integers
To add, multiply, or otherwise change a number, you must follow a set of mathematical rules, or properties of integers.
Closure Property:
When you add or multiply two numbers, you always get another integer. This rule ensures that the results are all whole numbers when you add or multiply integers.
Example:
7 + (-3) = 4,
5 × (-2) = -10
Commutative Property:
When adding or multiplying integers, changing the order of the numbers doesn’t change the answer. This only works for adding and multiplying. It doesn’t work for taking away or dividing.
Example:
(-4) + 9 = 5 and 9 + (-4) = 5
(-3) × 6 = -18 and 6 × (-3) = -18
Associative Property:
The order in which three or more numbers are grouped does not change the answer when adding or multiplying them.
Example:
[(-2 + 4) + 6] = –2 + (4 + 6) = 8
[(-3 × 2) × 5] = –3 × (2 × 5) = -30
Distributive Property:
When you multiply a number by a sum or difference, you get the same answer as when you multiply each term separately and then add or subtract.
Example:
2 × (3 + 4) = (2×3) + (2×4) = 6 + 8 = 14
Identity Property:
Adding 0 to any integer keeps its value (additive identity). Multiplying any integer by 1 also keeps its value (multiplicative identity).
Example:
(-5) + 0 = -5
7 × 1 = 7
Inverse Property:
Every integer has an opposite (additive inverse) that results in zero when added to it.
Example:
+9 and -9 are additive inverses
9 + (-9) = 0
These properties of integers are necessary to solve problems and make mathematical models.
Integers Number Line
A number line is a simple tool to visualize integers and understand their relationship.
<– -5 -4 -3 -2 -1 0 1 2 3 4 5 –>
Observations:
- Integers are evenly spread and get bigger as you move to the right.
- The beginning or middle point is zero.
- Around 0, both positive and negative integers are the same.
- Opposite integers, like +4 and -4, are both the same distance from 0.
Seeing integers on a line helps you understand how they work in math, especially when learning to add and subtract negative numbers.
Integers vs Whole Numbers
Whole numbers go from zero to positive infinity and include all of the numbers in between. There are no negative numbers in them. It is common to write the set of whole numbers as:
W = {0, 1, 2, 3, 4, …}
On the other hand, integers add negative numbers to this idea, along with zero and positive whole numbers. To sum up, the set of integers is:
ℤ = {…, -3, -2, -1, 0, 1, 2, 3, …}
Not having any negative numbers makes the main difference. Not all integers are whole numbers, but all whole numbers are integers.
In cases where negative numbers are necessary, such as when recording temperature drops, bank withdrawals, or losses, this difference is essential. If you have -₹500 in your bank account, that is an integer, but not a whole number. So, integers are a better way to keep track of both positive and negative numbers.
Integers vs Natural Numbers
The counting numbers we use every day to count and organize things are usually known as natural numbers:
N = {1, 2, 3, 4, …}
Some definitions add 0 to the list of natural numbers, but the most common one starts with 1. On the other hand, integers are made up of all natural numbers, zero, and their negative forms. The set of integers is:
ℤ = {…, -3, -2, -1, 0, 1, 2, 3, …}
On the other hand, integers can show both gain and loss, as well as increases and decreases, whereas natural numbers can only show positive counts.
You’d use natural numbers to count the people in a room. But you’d need integers to figure out the net profit or loss. So, even though natural numbers are a type of integer, they can’t represent values like -7 or 0 (based on the meaning). This makes integers more useful in math and real life.
Integers vs Rational Numbers
What is an integer? Integers are whole numbers that don’t have any fractional or decimal parts. They can be positive, negative, or zero. Values like -4, 0, and 7 are among them.
Rational numbers, on the other hand, are any numbers that can be written as a ratio (p/q), where p and q are both integers and q ≠ 0. This includes decimals (0.75), fractions (1/2), and decimals that repeat (0.333…).
You can write any integer ‘a’ as a/1, which is what a rational number is, so all integers are rational numbers. There are, however, some rational numbers that are not integers. For example, 1/3 is rational but not an integer.
When you compare integers and rational numbers, the main difference is that rational numbers include parts of whole numbers. As a whole, rational numbers are a bigger group that includes integers, fractions, and decimals. This makes it possible to show the amount and measurement in more depth.
Rules of Integers in Math
Following the proper sign rules for each action when working with integers is essential. You can use these rules for addition, subtraction, multiplication, and division. They help you figure out the correct answer based on whether the numbers are positive or negative.
Addition Rules:
Same signs: Add the absolute values and keep the common sign.
- Example: (-5) + (-3) = -8
Different signs: Subtract the smaller absolute value from the larger one. Use the sign of the larger number.
- Example: (-6) + 4 = -2
Subtraction Rules:
Convert subtraction into addition of the opposite (keep-change-change rule).
- Example: 7 – (-2) becomes 7 + 2 = 9
- Example: (-3) – 5 becomes (-3) + (-5) = -8
Multiplication Rules:
When multiplying integers, the result depends entirely on whether the signs are the same or different.
- Same signs: Positive result
- Example: (-3) × (-4) = 12
- Different signs: Negative result
- Example: (5) × (-2) = -10
Division Rules:
For division, just like multiplication, the sign of the result depends on the signs of the numbers involved.
- Same signs: Positive result
- Example: (-8) ÷ (-2) = 4
- Different signs: Negative result
- Example: (6) ÷ (-3) = -2
Knowing these rules about integers makes math easier and keeps you from making mistakes with signs in algebra questions.
Examples of Integers in Real Life
It’s not just mathematics where integers are used; they also have real-world applications. Examining some real-world applications of integers:
- Temperature: -15°C in Antarctica vs. +30°C in Delhi
- Finance: Balance of -₹2,000 shows debt; +₹3,500 shows savings.
- Stock Market: Profit: +7%, Loss: -12%
- Elevation:
- Dead Sea (-430 meters below sea level)
- Mount Everest (+8,848 meters above sea level)
- Scoring Systems: A tennis player wins by +3 games or loses by -2.
These applications of integers demonstrate how important they are for making decisions, researching, and showing facts. For additional math resources and interactive examples, you can explore Chrysalis High for detailed explanations and practice exercises.
FAQs:
- What exactly is an integer in simple words?
An integer is a whole number that can be positive, negative, or zero, but never a fraction or decimal.
- Is zero really considered an integer?
Yes. Zero is an integer and acts as the neutral element between positive and negative numbers.
- Are negative numbers also integers?
Yes. Negative whole numbers such as -1, -50, and -1000 are all integers.
- Can decimals or fractions ever be integers?
No. Numbers like 2.5, 1/2, or -3.75 are not integers because integers must be whole numbers.
- What’s the difference between whole numbers and integers?
Whole numbers include only zero and positive numbers, while integers also include negative ones.
