Root 3 – Value, derivation and examples

Significance of root 3 

Root 3 is the positive real number that, when multiplied by itself, yields the number 3. It is denoted mathematically as √3. Widely used in mathematics, root 3 is an irrational number, which means that it cannot be represented in the form of a fraction. The number of decimals in the fractional form of root three is infinite.

What are square roots 

The square root of a number is the value that gives the original value when multiplied by itself. It is the inverse operation of squares.  

The value of root 3

Root 3 is an irrational number. This makes it difficult to be represented in a fractional manner. With its decimals stretching up to infinity, root 3 is mostly represented in its shortened form as 1.732 (rounded up to three decimal places) to facilitate calculations. This value can be both positive and negative. But in the article, we will only look at the positive value of root 3.

Here, we’re going to have a look at the method of finding out the value of root 3; also, we’re going to learn how to find its position on a number line.

Determining the value of root 3 using the long-division method:-

The long division method is one of the most convenient and easiest ways to determine the square root of any given number. It is also one of the most trusted ways of determining the root values of numbers.

Step 1:

We begin by placing the bar over 3 to determine its square root. Add four pairs of zeros to determine the root value up to 4 decimal places. Now, we group the digits after the decimal point in pairs as 3. 00 00 00 00

Step 2:

Next, we choose a perfect square whose value is less than3. The desired digit, in this case, is ‘1’, and its square is ‘1’.

Step 3:

We write 1 in the place of dividend and quotient. The number ‘1’ is written below ‘3’, which is subtracted from ‘3’, and the difference, i.e., 2 is written.

Step 4:

We bring down the first pair of zeroes in the dividend and place a decimal point in the quotient.

Step 5:

Next, we add 1 to the divisor. The resultant sum will be 2. Now we choose a number succeeding 2 to get a 2 digit number such that when that two-digit number is multiplied by the number chosen, the resultant number is less than 200. Through trial and error method, we’ll find the desired digit to be ‘7’ (as 27 x 7 = 189, which is less than 200). So now we’re left with ‘11’ as the remainder, and the next divisor is 34. The dividend now becomes 1100 with the next pair of zeroes being taken down.

Step 6:

The above steps are continued, and we’ll see that the final answer obtained in the place of the quotient is 1.732, which is our value of root three up to three decimal places.

Representing root three on a number line

Step 1:

We begin by drawing a number line having three points, O, A and D.

Step 2:

Next, we denote point 0 as “O” and point 1 as “A” such that OA is equal to 1 unit.

Step 3:

Now, at point A draw a perpendicular of length AB=1 unit. Make sure that its length is equal to the distance from O to A in the number line.

Step 4:

Now join the points O and A, so that OAB is a right-angled triangle. Then by using the Pythagorean theorem,

OB²=OA²+AB²

=>  OB²= 1² + 1²

=> OB = √2

Step 5:

Now, draw a perpendicular at point B of length BC= 1 unit and join the points O and C. We’re left with another right-angled triangle.

Step 6:

Again by using Pythagoras’ theorem, we get the value of OC to be √3.

Step 7:

Then using a compass, we take the radius as OC and draw an arc so that it cuts the number line at D. The length of OD will give us the square root of 3.

Examples –

1. Solve for x : 7x² = 3x² +12

7x²- 3x² = 12

4x² = 12

x² = 3 x= √3

Therefore, x = ±1.732 (truncated)

2. Find the value of √12

√12 can be simplified as = √2x2x3

= √(2² X3)

Taking the bases out of the radical sign, we get = 2√3

We know, √3 = 1.732 So, 2√3 = 2 x 1.732

= 3.464

Therefore, the value of √12 = 3.464