5.4 - Finding the Square of a Number

The square of a number is the result of multiplying that number by itself. It is a fundamental concept in mathematics, particularly when dealing with algebra, geometry, and number theory.

To find the square of a number, follow these simple steps:

• Step 1: Identify the number you need to square.
• Step 2: Multiply the number by itself.

The formula for finding the square of a number is:

Square of a number = Number × Number

In mathematical notation, if $x$ is the number, then the square of $x$ is written as $x^2$.

Example 1: Find the square of 5.

Solution: The square of 5 is calculated as $5 × 5 = 25$. Therefore, $5^2 = 25$.

Example 2: Find the square of 12.

Solution: The square of 12 is calculated as $12 × 12 = 144$. Therefore, $12^2 = 144$.

In the case of negative numbers, squaring a negative number results in a positive number.

Example 3: Find the square of -7.

Solution: The square of -7 is calculated as $(-7) × (-7) = 49$. Therefore, $(-7)^2 = 49$.

Properties of Square Numbers:

• Squares of all integers (whether positive or negative) are non-negative numbers.
• The square of any number is always greater than or equal to zero.
• Squares of even numbers are always divisible by 4.
• Squares of odd numbers are always 1 more than a multiple of 4.

Example 4: Check if the square of a number is divisible by 4.

Solution: Let's square the even number 6. We have $6 × 6 = 36$, and since 36 is divisible by 4, the square of 6 is divisible by 4.

Square of a Fraction:

To find the square of a fraction, square both the numerator and the denominator.

For example, to find the square of the fraction $\frac{2}{3}$, calculate:

Square of $\frac{2}{3} = \frac{2^2}{3^2} = \frac{4}{9}$.

Square of a Decimal:

To find the square of a decimal, treat the decimal as a fraction and square it, or multiply the decimal by itself directly.

For example, to find the square of 0.6:

$0.6 × 0.6 = 0.36$.

Square of a Negative Decimal:

Just like with negative integers, the square of a negative decimal is positive. For example, the square of -0.4 is:

$(-0.4) × (-0.4) = 0.16$.

Importance of Squares in Geometry:

The concept of squaring is also important in geometry. For example, in the calculation of the area of a square, if the side length of the square is $a$, then the area of the square is $a^2$.

Real-Life Applications of Squares:

Squares are used in various real-life situations, such as:

• Calculating areas (like the area of a square plot of land).
• In physics, when calculating acceleration or the square of velocity in formulas.
• In finance, for calculating certain financial models or investments that involve squared terms.