The arithmetic value which is used for representing the quantity and used in making calculations are defined as Numbers. A symbol like “4,5,6” which represents a number is known as a numeral. Without numbers, we can’t do counting of things, dates, time, money, etc., these numbers are also used for measurement and used for labelling.

The properties of numbers make them helpful in performing arithmetic operations on them. These numbers can be written in numeric forms and also in words.

For example, 3 is written as three in words, 35 is written as thirty-five in words, etc. Students can write the numbers from 1 to 100 in words to learn more.

There are different types of numbers, which we can learn. They are whole and natural numbers, odd and even numbers, rational and irrational numbers, etc.

What is a Number System?

A Number System is a method of showing numbers by writing, which is a mathematical way of representing the numbers of a given set, by mathematically using the numbers or symbols. The writing system for denoting numbers logically using digits or symbols is defined as the Number System.

We can use the digits from 0 to 9 to form all the numbers. With these digits, anyone can create infinite numbers.

For example, 156, 3907, 3456, 1298, 784859, etc.

What is a Square Root?

Square roots of a number are defined as a number which on multiplication by itself gives the original number. Suppose a is the square root of b, then it is represented as

a = √b

We can express the same equation as a² = b. Here, ’√’ this symbol we used to represent the root of numbers is termed as are.

The positive number when it is to be multiplied by itself represents the square of the number. The square root of the square of any positive number gives the original number.

For example, the square of 4 is 16, 4² = 16, and the square root of 16, √16 = ±4. Since 4 is a perfect square, hence it is easy to find the square root of such numbers, but for an imperfect square, it’s really tricky.

To represent a number ‘a’ as a square root using this symbol can be written as: ‘√a‘, where a is the number.

The number here under the radical symbol is called the radicand. For example, the square root of 4 is also represented as a radical of 4. Both represent the same value.

Formula to find the square root is: a = √b

Properties of Square Roots

It is defined as a one-to-one function that takes a positive number as an input and returns the square root of the given input number.

f(x) = √x

For example, here if x = 9, then the function returns the output value as 3.

Properties of the square root are as follows:

• If a number is a perfect square number, then there definitely exists a perfect square root.
• If a number ends with an even number of zeros (0’s), then we can have a square root.
• The two square root values can be multiplied. For example, √3 can be multiplied by √2, then the result will be √6.
• When two same square roots are multiplied, then the result must be a radical number. It shows that the result is a non-square root number. For example, when √7 is multiplied by √7, the result obtained is 7.
• The square root of negative numbers is undefined. Hence the perfect square cannot be negative.
• Some of the numbers end with 2, 3, 7, or 8 (in the unit digit), then the perfect square root does not exist.
• Some of the numbers end with 1, 4, 5, 6, or 9 in the unit digit, then the number will have a square root.

It is easy to find the square root of a number that is a perfect square.

Perfect squares are those positive numbers that can be written as the multiplication of a number by itself, or you can say that a perfect square is a number which is the value of power 2 of any integer.

Number that can be expressed as the product of two equal integers.

For example, 16 is a perfect square because it is the product of two equal integers, 4 × 4 = 16. However, 24 is not a perfect square because it cannot be expressed as the product of two equal integers. (8 × 3 = 24).

Number which is obtained by squaring a whole number is termed as a perfect square. If we assume N is a perfect square of a whole number y, this can be written as

N = Product of y and y = y².

So, the perfect square formula can be expressed as:

N = Y²

Let’s Use the formula with values.

If y = 9, and N = y².

This means, N = 9² = 81.

Here, 81 is a perfect square of 9 because it is the square of a whole number.

So real square roots of 81 is +9, -9

With the help of square roots, we can identify whether a number is a perfect square or not, if we calculate the square root of the given number.

If the square root is a whole number, then the given number will be a perfect square, and if the square root value is not a whole number, then the given number is not a perfect square.

For example, to check whether 24 is a perfect square or not, we will calculate its square root. √24 = 4.898979. As we can see, 4.898979 is not a whole number, so, 24 is not a perfect square.

Let’s take another example of

The number 49. √49 = ±7. We can see that 7 is a whole number, therefore, 49 is a perfect square.

Why do we use plus or minus in square root?

If we want both the positive and the negative square root of a radicand then we put the symbol ± (read as plus minus) in front of the root. 

The numbers that are not a perfect square are members of the irrational numbers. This means that numbers or square root can’t be written as the quotient of two integers.

Related Article

• Why minus into minus equals to plus?
• If the square root of a number plus two is the same number, then find the number?
• What is the Square Root of a Negative Number?
• a plus b minus c Whole Square
• Why do we use numbers?
• Square Root

Sample Problems

Question 1: What are the two square roots of 100?

Solution:

Here 100 is the perfect square of 10, so this can have two roots one negative and one positive

or we can say real square root of 100 is ±10

or              10² = 10 × 10 = 100

             (-10)² = – 10 × – 10 = 100

Hence, the two square roots of 100 are +10 and -10.

Question 2: What are the square roots of 12?

Solution: 

Square root of 12 

Here 12 is not a perfect square so this number doesn’t have two square roots we can’t write it as √12 = ±3.464

Therefore √12 = 3.464 is an irrational number, the numbers that are not a perfect square are members of the irrational numbers. This means that numbers or square roots can’t be written as the quotient of two integers.

Question 3: What are the two square roots of 144? 

Solution: 

square root of 144

Here square root of 144 is perfect square of 12, i.e a whole number this has two square roots +12, -12

Therefore √144 = ± 12

Square root- FAQs

Why do we use the plus or minus sign in square roots?

The plus or minus sign in square roots indicates that there are two possible solutions to the equation, one positive and one negative. This is because the square of both a positive number and its negative counterpart yields the same result.

When do we use the plus or minus sign in square roots?

The plus or minus sign is used when solving equations involving quadratic functions or when finding the roots of equations. It is particularly relevant when taking the square root of both sides of an equation to solve for a variable.

What does the plus or minus sign represent in square roots?

The plus sign represents the principal square root, which is the positive square root of a number. The minus sign represents the negative square root, which is the opposite of the principal square root.

Why are there two solutions in square roots?

Since squaring both a positive number and its negative counterpart results in the same value, equations involving square roots can have two valid solutions—one positive and one negative.

Are there situations where we only consider one solution in square roots?

Yes, in some contexts, such as when dealing with physical quantities that cannot be negative (e.g., length, time), only the principal square root (the positive solution) is considered. However, in mathematics, both solutions are typically considered valid unless specified otherwise.

How do we determine whether to use the positive or negative solution in square roots?

The context of the problem or equation often dictates which solution is appropriate. For example, when solving for a distance or a magnitude, the positive solution is typically chosen. However, when considering situations with opposites or when dealing with equations that involve both positive and negative values, both solutions may be relevant.