Let’s Explore Square Root of 2.

Square root of 2! Yes, it can be calculated like other non-perfect squares. When we heard this square root of 2, we feel a little awkward as it is different from perfect squares like 4, or 9, or 16, or 25. The square root of any perfect square is very easy to calculate.

Let’s begin a small session of learning the square root of 2.

Let us learn the basics of the square root of 2. If we try to find out the value of the square root of 2, we see a lot of irrational numbers come after the decimal point.

Now, do you think all the numbers can use in our calculation? No, it is not feasible to write all those numbers in our calculations. So, how to resolve this problem?

It’s simple! Just write the square root of 2 as √2.

But many times, writing √2 is not helpful and it’s numerical is required to consider in our calculation, Here, we should know the numeric value of the square root of 2.

Fig.1 Square Root of 5

The square root of 2 is basically a number, if two same number multiplies and gives the result of 2. It means,

√**Z** x √**Z** =**Z**

It is written by √2.

So, as per basic understanding, we can say,

√2x√2 =2

There is a lot of research on the square root of 2 to indicate the maximum numbers after decimal point.

In many research, number becomes in billions!

We have shown, the numerical value of the square root of 2 up to 15 decimal places,

**√5 = 1.414213562373095**

There are a lot of applications in mathematics, where we use, √2. We can understand the example, by solving a simple equation.

Let’s us calculate the value of x.

- 11x
^{2}=2x^{2}+18 - Or, 9x
^{2}=18 - Or, x
^{2 }=18/9=2 - Or, x = √2

We used to write,

- 1+√2
- 9+√2
- 3-√2

Or sometimes, we multiply,

- 3x√2=3√2
- √2x√2x√2=2√2
- 11x3x√2=33√2 and so on.

Hence, it is very common to use √2 symbol wherever square root of 2 comes. This value is not a numerical value. So, how do we specify the exact value of √2?

For example, if your friend ask you to provide 2√2kg of wheat, how do you measure?

Yes! It can be possible if you know the numerical value of square root of 2. Let’s find the numerical value.

There are three most common methods to find out the square root of 2.

- Long division method
- Average method
- Equation method

Whenever we try to find out a square too of any number other than perfect square, we opt for this method. This method is a little long but easy. This method is also called a long division method as well. Let’s learn all the steps,

**Step#1:Write the number 2 as 2.00000000, remember we are not doing any change of the value of 2, as both numerical values are the same.**

**Step#2: Order the zeros, like, 2.00 00 00 00 as both are the same and it will help in the division method.**

**Step#3: Now, try to find out the perfect square just below 2. If we analyze, 0,1 are numbers among below 2, and 1 is the perfect square.**

As we know, 1 = 1×1 = 1^{2}

**Step#4: Find out the square root of the perfect square, below 1. Here, it is 1 and the square root of 1 is 1.**

**Step#5: Make a division table and write the number 1 in quotient place as well as in the divisor place.**

**Step#6: Make the division considering,**

- Divisor 1
- Quotient 1
- Dividend 2

So, by the division method, 1 multiplied by 1 gives 1. Subtract 1 from 2, there will be remainder 1.

**Step#7: As 1 is the remainder, we must carry down 2 zeros after 1. So, it will become 100. Next, carry down two zeros and write it down after 1. Simultaneously decimal point will come after 1 in the quotient.**

**Step#8: Now, we will add 1 in the divisor and make it 2.**

**Step#9: We have to select a number next to 2, in such a way that if we multiply the new combined number with the new number, then the value should be equal to100 or less than that.**

Now,

- If we take 3, the combination number will be 23. So, 23×3=69.
- If we take 4, the combination number will become 24. So, 24×4=96.
- If we take 5, the combination number will become 25. So, 25×5=125, which is more than 100.

Hence, the number 4 is acceptable and we get 96. Now, we have to subtract 96 from 100, and we get the remainder 4. So, we get 4 after the decimal.

**Step#10: As 4 is the remainder, we should carry down 2 zeros after 4. So, it will become 400.**

**Step#11: Now, we get 24, So, 24+4=28. We have to select another number next to 28, in such a way that if we multiply the new combined number with the new number, then the value should be equal to 400 or less than that.**

Now,

- If we take 1, the combination number will be 281. So, 441×1=441.
- If we take 2, the combination number will become 282. So, 282×2=564.
- If we take 2, the combination number will become 564, which is more than 400.

Hence, if we take 1, the combination number will become 281. So, 281×1= 281, which is less than 400. 1 will be added in the second decimal point.

Hence, the number 1 is acceptable and we get 281. Now, we have to subtract 281 from 400, and we get the remainder 119.

Fig. 6 Square root 5 reminder 271

**Step#12: As 119 is the remainder, we should carry down 2 zeros after 119. So, it will become 11900.**

**Step#13: Now, we get 281, So, 281+1=282. We have to select another number next to 282, in such a way that if we multiply the new combined number with the new number, then the value should be equal to 11900 or less than that.**

Now,

- If we take 3, the combination number will be 2823. So, 2823×3=8469.
- If we take 4, the combination number will become 2824. So, 2824×4=11296.
- If we take 5, the combination number will become 2825. So, 2825×5=14125.
- If we take 5, the combination number will become 14125, this is more than 11900.

Hence, if we take 4, the combination number will become 2824. So, 2824×4=11296, which is less than 11900. 4 will be added in the third decimal point.

Hence, the number 4 is acceptable and we get 11296. Now, we have to subtract 11296 from 11900, and we get the remainder 704.

**Step#14: As 704 is the remainder, we should carry down 2 zeros after 704. So, it will become 70400.**

**Step#15: Now, we get 2824, So, 2824+4=2828. We have to select another number next to 2828, in such a way that if we multiply the new combined number with the new number, then the value should be equal to 70400 or less than that.**

This process will go on, however, we have taken up to three decimal point.

In ancient times itself, the average method was used to find the square root of 2. This is very simple, and we get the value of the square root of 2 based on averages in a few steps. Let us see all steps,

**Step#1: We have to check the perfect squares just below 2 and above 2. In the case of number 2, 1 is the perfect square below 2, and 4 is the perfect square above 2.**

1 2 4

**Step#2: Find out the square root of the perfect square of both the numbers.**

1^{2} & 2^{2}

So, square roots are 1 & 2.

**Step#3: Square root of 2 will be between the square root of 1, i.e. 1 and the square root of 4, i.e. 2.**

1<2<4

1^{2}<√2<2^{2}

^{or, }1<Square root of 2<2.

**Step#4: Divide the number 2 by any of one number, 1 or 2.**

Let us divide 2 by 2,

So, 2/2=1.0

**Step#5: Hence, we get a value of 1.0, and we will consider divisor 2 which was divided. Now, we will calculate the average value of them,**

Average value = (2+1.0)/2=3.0/2=1.5

**Step#6: In case, if you required further precise value, the process to be continued. The above 1.5 vale is almost closer, and we can select the same.**

In the method, you have to remember a simple equation. It’s not very complex, but you remember, you can calculate the square root of any number in a few seconds.

Equation

Square root 5 equation

√(x±y) = √x ±y/(2√x)

Here, x or y is the perfect square. It is better to remember all the basic perfect square table,

Perfect square to remember

**Step#1: Split the digit into two numbers, in such a way that x can be written as a perfect square. **

We are going to calculate, the square root of 2. Hence, 2 can be written as 1+1, since we know from the perfect square table, 1 is a perfect square, that means x=1 and y=1.

**Step#2: Input the value of x & y in the equation.**

√(x±y) = √x ±y/(2√x)

or, √(1+1) = √1 +1/(2√1)

or, √2 = 1 +1/(2×1)

or, √2 = 1 +0.5 =1.5 (approx value)

Hence, we have learned the square root of 1 along with different methods. Any doubt or if you any suggestions, please let us know.

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