#### What Is Dilation in Math? Definition, Examples & How-to

Read on to find easy-to-follow definitions, real-life examples, and a simple guide to dilations in math.

Jul 10, 2024 | Queen Creek

This beginner-friendly overview of square numbers in math is for everyone, from seven- to seventy-year-olds, looking to learn new mathematical concepts or refresh their memory. Read on to find simple definitions, and fun applications and exercises.

**A square number** is the result of multiplying a number by itself.

For instance, if you take the number 3 and multiply it by 3, you get 9.

9 is a square number because it’s the product of multiplying 3 by itself.

In math, we write this as 3² (we read it as “three to the second power” or “three squared”). We call the “**3**” in 3² the **base** and we call “**²**” the **exponent**.

Let’s look at some examples to understand square numbers better:

- 2² which can also be expressed as 2 x 2 = 4
- 4² which can also be expressed as 4 x 4 = 16
- 5² which can also be expressed as 5 x 5 = 25

**Refresh Your Memory: ****What is a Factor in Math?**

In addition to “square numbers,” you might have also heard about “square roots.”

Square numbers and square roots are like two sides of the same coin — they represent opposite actions.

How?

Since square roots are a whole new topic, let’s look at a brief example:

As we said, a square number is a result of multiplying a number by itself. For example, 3 multiplied by 3 equals 9 (3² = 9), so the number 9 is a square number.

A square root is the opposite of a square number. **A square root **is the number you multiply by itself to get another number. We show it with this symbol: √ (radical symbol or surd). For example, the square root of 9 is 3 (√9 = 3), and the square root of 25 is 5 (√25 = 5).

If you thought square numbers must have something to do with squares, you would have been correct!

Square numbers are called “squared” because they make the shape of a square.

Squares have sides of equal length. To find the area of a square, you just need to multiply one side by itself, or “square” it.

Squares are like puzzle pieces in math. They fit into various math classes and fields.

**Pre-Algebra and Algebra**: In algebra, we use square numbers in equations. Let’s say that we have the expression
. In algebra, this means “x squared” or “x multiplied by itself.” For example, if
, then
would be
or
which equals
.

**Geometry**: In geometry we use square numbers to measure the area inside of a square based on its side lengths. Let’s look at a square with sides of length that are 4 centimeters long for this example. To find the area of this square, we square the length of one of its sides. In this case, the length of one side is 4 centimeters. So, the area of the square would be 16 square centimeters.

**Number Theory**: In number theory, we use square numbers to study patterns and relationships between numbers. For example, the sequence 1, 4, 9, 16, 25, 36 is a list of square numbers. How do we know this? Each number in the sequence is made by multiplying a natural number by itself.

Here’s proof:

- 1 = 1 × 1
- 4 = 2 × 2
- 9 = 3 × 3
- 16 = 4 × 4
- 25 = 5 × 5
- 36 = 6 × 6

Check out** ****this video**** **demonstrating a cool trick for squaring numbers ending in 0 or 5.

Let’s see how numbers from 1 to 12 are squared.

- 1² = 1 which can also be expressed 1 × 1 = 1
- 2² = 4 which can also be expressed 2 × 2 = 4
- 3² = 9 which can also be expressed 3 × 3 = 9
- 4² = 16 which can also be expressed 4 × 4 = 16
- 5² = 25 which can also be expressed 5 × 5 = 25
- 6² = 36 which can also be expressed 6 × 6 = 36
- 7² = 49 which can also be expressed 7 × 7 = 49
- 8² = 64 which can also be expressed 8 × 8 = 64
- 9² = 81 which can also be expressed 9 x 9 = 81
- 10² = 100 which can also be expressed 10 x 10 = 100
- 11² = 121 which can also be expressed 11 × 11 = 121
- 12² = 144 which can also be expressed 12 × 12 = 144

Check out** ****this short guide** to squaring any number (no matter how big!) using mental math and Number Sense.

We can easily use (and practice!) square numbers outside the classroom, especially when we want to calculate the surfaces.

Here are a couple of examples of how you can put square numbers to practice, plus how we use them to plan and build the spaces you are using every day:

Take your toy bricks with studs and a studded baseplate to place them on. The rule of the game is: count the number of studs on a piece and put that many bricks one next to another on the baseplate.

- Start with the smallest brick which has 1 stud. Since the brick only has 1 stud, leave it alone on your baseplate.
- Next, take the brick with 2 studs. Since we have 2 studs, we’ll put 2 bricks on the baseplate, one next to another.
- Now let’s take a brick with 3 studs. Since we have 3 studs, we’ll put 3 bricks next to one another on the baseplate.

And so on!

As you start to count the number of studs in each group of bricks, you will get square numbers of studs on single bricks.

- 1
^{2 }(1 brick only) equals 1 stud - 2
^{2 }(or 2 studs x 2 studs) equals 4 studs - 3
^{2 }(or 3 studs x 3 studs) equals 9 studs

Kids over the age of 12 might like this one!

If you are familiar with virtual house-building games, you’ll know that these games often provide you with “plots” overlaid with square patterns

Next time you outline your square living room or bedroom, count the number of squares and you’ll get a square number. Simple as that!

From planning playgrounds to rooms and buildings, you can see square numbers in action in any square-shaped space.

Take the measuring tape and measure one side of your room. Multiply that number by itself and you’ll get the size of your floor if the room is perfectly square. For example, if each side of your room is 10ft wide, the surface of your square floor would be 10x10ft = 100ft^{2} (read as “square feet”).

Is 100 a square number?

Yes! Yes, it is.

Ask your parents, engineers, or architects in your family to see how they use square numbers to plan and build the spaces you use every day.

Your time to shine! Let’s review what we’ve learned with these simple exercises.

**Exercise 1**: Square Number Multiplication

Let’s calculate these:

7^{2}=___________

9^{2}=___________

5^{2}=___________

11^{2}=___________

**Exercise 2**: Missing Square Numbers

Fill in the missing square numbers in the sequence:

1, __, 9, __, 25, __, 49, 64, __, 100

**Exercise 3:** Square Number Word Problems

Try to solve these 3 word problems:

- If a square garden has an area of 64 square feet, what is the length of each side? (Hint: What number can we multiply by itself to get 64?)
- Sarah wants to build a square picture frame with an area of 100 square inches. What should be the length of each side? (Hint: What number can we multiply by itself to get 100?)
- A square rug has an area of 49 square meters. What is the length of one side of the rug? (Hint: What number can we multiply by itself to get 49?)

Completed the exercises?

Scroll to the end to check your answers.

**A Neat Trick: ****Learn How to Square Any Number**

Find answers to common queries regarding the properties and applications of square numbers.

Students usually encounter the basic square numbers when learning addition in early elementary school. They then formally begin to learn about square numbers later in elementary school, typically around grades 4 to 6.

Mathnasium works with elementary school students of all ages and skill levels to help them master math, including square numbers.

Yes, zero is a square number because 0 x 0 = 0.

Not all positive numbers can be square numbers. For example, 7 is a positive number, but it’s not a square number because you can’t make 7 by multiplying a number by itself.

Yes, you can square a negative number. Squaring a negative number also means multiplying the number by itself. When you square a negative number, the result is always positive because multiplying a negative number by a negative number always makes a positive number.

Let’s look at these examples:

-2 squared is (-2) x (-2) = 4

-3 squared is (-3) x (-3) = 9

-4 squared is (-4) x (-4) = 16

** **

Mathnasium’s specially trained tutors work with students of all skill levels to help them learn and master any K-12 math topic, including squared numbers.

Our tutors assess each student’s skills to create personalized learning plans that will put them on the best path to math mastery.

Find a Mathnasium Learning Center near you, schedule an assessment, and enroll today!

Completed your square number exercises? Check your answers:

**Exercise 1**: Square Number Multiplication

**5 ^{2 }= 25**

**7 ^{2 }= 49**

**9 ^{2 }= 81**

**11 ^{2} = 121**

** **

**Exercise 2**: Missing Square Numbers

1, _**4**_, 9, _**16**_, 25, _**36**_, 49, 64, _**81**_, 100

**Exercise 3:** Square Number Word Problems

- If a square garden has an area of 64 square feet, what is the length of each side? (Hint: What number can we multiply by itself to get 64?)
**The length of each side is 8 ft, because 64 = 8**^{2}. - Sarah wants to build a square picture frame with an area of 100 square inches. What should be the length of each side? (Hint: What number can we multiply by itself to get 100?)
**The length of each side should be 10 inches, because 100 = 10**^{2}. - A square rug has an area of 49 square meters. What is the length of one side of the rug? (Hint: What number can we multiply by itself to get 49?)
**The length of one side of the rug is 7 meters, because 49=7**^{2}.