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# Square Roots

by Ron Kurtus (revised 18 January 2022)

The ** square root** is a number that when multiplied by itself will equal the initial number. For example,

**7**is the square root of

**49**, since

**7**times

**7**equals

**49**. This is an interesting concept and can be difficult to grasp.

The symbol **√** is called the radical sign and represents the square root operation. Many numbers have an integer square root. However, there are many more numbers with irrational square roots.

The result of taking the square root of a number consists of both a positive and negative number. However, when you take the square root of a negative number, the result is called an imaginary number.

Questions you may have include:

- What is the radical sign?
- How do you find the square root of a number?
- What are imaginary numbers?

This lesson will answer those questions.

## Radical sign

The symbol **√** is called the radical sign and represents the square root operation. Thus, **√49** means the square root of **49**, and **√z** means the square root of **z**.

You can also designate a square root by the **1/2** exponent, such as **49 ^{1/2}** or

**z**, but typically the radical sign is used.

^{1/2}## Finding the square root

The result of taking the square root of a number consists of both a positive and negative number. Thus **√49 = ±7** or **+7** and **−7**. This is because both **(+7)*(+7)** and **(−7)*(−7)** equals **49**.

Likewise **√x ^{2} = ±x**

Many numbers have an integer square root. Common integer square roots are:

√1 = ±1√36 = ±6√4 = ±2√49 = ±7√9 = ±3√64 = ± 8√16 = ±4√81 = ±9√25 = ±5√100 = ±10

However, there are many more square roots that are not integers, such as **√2**, **√3**, and **√5**. They are often repeating numbers:

√2 = ±1.414213562373095...or approximately±1.414

√3 = ±1.7320508075688772...or approximately±1.732

√5 = ±2.23606797749979...or approximately±2.236

You can memorize many of the numbers that have an integer square root. Otherwise, you can use a calculator to find the square root or even use Newton's Square Root Approximation method. There are also online square root calculators.

## Imaginary numbers

Note that when you take the square root of a negative number, the result is called an **imaginary** **number**. The square root of **−1** or **√(−1)** is designated as the *imaginary* number **i**. In other words, it is not a real number in our numeric system.

When you take the square root of a negative number such as **√(−49)** you get an answer of **±7i**. The reason is that **−49 = −1*49**.

## Summary

The square root of a number is another number that when multiplied by itself will equal the initial number. The symbol **√** is called the radical sign and represents the square root operation. Many numbers have an integer square root. The result of taking the square root of a number consists of both a positive and negative number. When you take the square root of a negative number, the result is called an imaginary number.

Be radical in your approach

## Resources and references

### Websites

**Roots and Radicals** - Purplemath.com

**Square Root Calculator** - Math.com

**Square Root** - Wikipedia

### Books

(Notice: The *School for Champions* may earn commissions from book purchases)

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## Square Roots