Applications of Integers

Integers are an essential part of mathematics and have great importance in everyday life. In Class 7, understanding integers and their applications is important to build a strong foundation for more advanced math topics. Integers are whole numbers that can be positive, negative or zero. They do not have any fractional or decimal part.

What are integers?

In simple terms, integers can be understood as all the numbers on the number line, including zero, without any fractions or decimals. A number line with integers typically looks like this:

      ... -3, -2, -1, 0, 1, 2, 3 ...

Positive integers are numbers greater than zero. They represent quantities or values that have a certain amount, such as 1, 2, 3...

Negative integers are numbers less than zero. They represent a decrease, loss or debt, such as -1, -2, -3...

Zero is neutral and acts as a divider between positive and negative integers.

Illustrations of integers: The number line

The number line is a visual representation of integers where each integer has a specific position. The number line extends to infinity in both directions.

Consider the following line representation:

Using this line we can easily compare integers. For example, 2 is greater than -3 because it appears to the right of -3 on the number line.

Applications of integers in real life

Integer numbers play an important role in everyday life. Let's take a look at some common applications:

Thermometer

Thermometers use integers to represent temperature. Positive values represent temperatures above zero (e.g., 25°C), while negative values represent temperatures below zero (e.g., -5°C).

Banking and finance

The integers in the bank balance represent deposits and withdrawals. A positive balance means a credit (for example, +500 dollars), while a negative balance means a debt (for example, -200 dollars).

Game

In sports, integers are used to represent scores and rankings. For example, in golf, a positive score means under par, while a negative score, such as -3, indicates that the player is three strokes below par.

Height and depth

Heights above sea level are represented as positive integers, while depths below sea level are represented by negative integers. For example, Mount Everest might be +8,848 meters, and the Dead Sea might be -430 meters.

Arithmetic operations with integers

Arithmetic operations such as addition, subtraction, multiplication, and division can be performed using integers.

Add

Adding integers involves combining numbers:

        4 + (-2) = 2

On the number line, this is like taking four steps forward and two steps back to get to 2.

Subtraction

Subtracting integers is virtually the same as adding opposites:

        5 - (-3) = 8

Here subtracting -3 is the same as adding positive 3 So, the result is 8.

Multiplication

When multiplying integers, if the signs of the two numbers are the same then the product is positive and if the signs are different then the product is negative:

        (-6) * 5 = -30
        7 * (-3) = -21
        (-4) * (-2) = 8

Division

Like multiplication, division of integers also follows sign rules. The quotient is positive if both signs are the same and negative if they are different:

        (-15) / 3 = -5
        40 / (-8) = -5
        (-28) / (-4) = 7

Visualization of operations on integers

Let's look at these functions graphically:

In the graphic above:

• The red line shows the increase from -4 to 0, which can be seen by adding 4.
• The blue line represents the increase from 0 to 2, which is a sum of 2.
• The green line shows the movement from 2 to 4, which is an additional multiple of 2.

Integers in programming

Integers are not only useful in mathematical calculations, but are also fundamental in programming and computer science. In these fields, integers are used to calculate and control loops, express conditions, and represent data in memory.

Here's a simple programming example using integers to count iterations:

        for(int i=0; i<5; i++) {
            console.log("Iteration: " + i);
        ,

In this code, i is an integer indicating the number of iterations from 0 to 4 (5 iterations in total).

Solving real-world problems with integers

Real-world problems involving changes in values can often be solved using integers. Here are some examples:

Example 1: Temperature change

Suppose the temperature of a city changes from -2°C in the morning to 5°C in the afternoon. Calculate the temperature change.

        Temperature change = Afternoon temperature - Morning temperature
                          = 5 - (-2)
                          = 5 + 2
                          = 7° Celsius

The temperature increased by 7°C.

Example 2: Financial transaction

If a person has a bank balance of +200 dollars and makes the following transactions: withdraws $150, deposits $100, and withdraws $75. What will be the final balance?

        Opening balance = +200
        Withdrawal = -150
        Deposit amount = +100
        Clearance = -75

        Final balance = 200 – 150 + 100 – 75
                     = 75 + 100 - 75
                     = 100 dollars

The final bank balance is $100.

Example 3: Height difference

Calculate the difference in altitude if an aircraft descends from 10,000 m to 2,500 m.

        Height difference = Initial height – Final height
                            = 10,000 - 2,500
                            = 7,500 meters

The aircraft descended to 7,500 metres.

Understanding absolute values

The absolute value of an integer is its distance from zero on the number line, regardless of its sign. Absolute values are represented by vertical bars, i.e. |x|

Consider the following example:

        |-5| = 5
        |8| = 8

The absolute value of -5 is 5, and the absolute value of 8 is simply 8.

Summary

Integers are fundamental in mathematics and practical applications. From basic arithmetic operations to solving real-world problems, the versatility of integers makes them an integral part of education and beyond. By understanding how integers work and recognizing their applications in daily life, students can greatly enhance their problem-solving skills.

Through visual models, real-life scenarios, and arithmetic calculations, students can effectively understand these concepts, and lay the foundation for more complex mathematical theories.

Whether you're balancing accounts, planning a trip's altitude, or calculating changes in temperature, integers provide an organized, numerical approach to problem-solving in a variety of contexts.

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