Square Root of 3 (√3)

Square Root of 3

Here we will define, analyze, simplify, and calculate the square root of 3. We start off with the definition and then answer some common questions about the square root of 3. Then, we will show you different ways of calculating the square root of 3 with and without a computer or calculator. We have a lot of information to share, so let's get started!

Square root of 3 definition
The square root of 3 in mathematical form is written with the radical sign like this √3. We call this the square root of 3 in radical form. The square root of 3 is a quantity (q) that when multiplied by itself will equal 3.

√3 = q × q = q²

Is 3 a perfect square?
3 is a perfect square if the square root of 3 equals a whole number. As we have calculated further down on this page, the square root of 3 is not a whole number.

3 is not a perfect square.

Is the square root of 3 rational or irrational?
The square root of 3 is a rational number if 3 is a perfect square. It is an irrational number if it is not a perfect square. Since 3 is not a perfect square, it is an irrational number. This means that the answer to "the square root of 3?" will have an infinite number of decimals. The decimals will not terminate and you cannot make it into an exact fraction.

√3 is an irrational number

Can the square root of 3 be simplified?
You can simplify 3 if you can make 3 inside the radical smaller. We call this process "to simplify a surd". The square root of 3 cannot be simplified.

√3 is already in its simplest radical form.

How to calculate the square root of 3 with a calculator
The easiest and most boring way to calculate the square root of 3 is to use your calculator! Simply type in 3 followed by √x to get the answer. We did that with our calculator and got the following answer with 9 decimal numbers:

√3 ≈ 1.732050808

How to calculate the square root of 3 with a computer
If you are using a computer that has Excel or Numbers, then you can enter SQRT(3) in a cell to get the square root of 3. Below is the result we got with 13 decimals. We call this the square root of 3 in decimal form.

SQRT(3) ≈ 1.7320508075689

What is the square root of 3 rounded?
The square root of 3 rounded to the nearest tenth, means that you want one digit after the decimal point. The square root of 3 rounded to the nearest hundredth, means that you want two digits after the decimal point. The square root of 3 rounded to the nearest thousandth, means that you want three digits after the decimal point.

10th: √3 ≈ 1.7

100th: √3 ≈ 1.73

1000th: √3 ≈ 1.732

What is the square root of 3 as a fraction?
Like we said above, since the square root of 3 is an irrational number, we cannot make it into an exact fraction. However, we can make it into an approximate fraction using the square root of 3 rounded to the nearest hundredth.

√3
≈ 1.73/1
≈ 173/100
≈ 1 73/100

What is the square root of 3 written with an exponent?
All square roots can be converted to a number (base) with a fractional exponent. The square root of 3 is no exception. Here is the rule and the answer to "the square root of 3 converted to a base with an exponent?":

√b = b^½

√3 = 3^½

How to find the square root of 3 by long division method
Here we will show you how to calculate the square root of 3 using the long division method with one decimal place accuracy. This is the lost art of how they calculated the square root of 3 by hand before modern technology was invented.

Step 1)
Set up 3 in pairs of two digits from right to left and attach one set of 00 because we want one decimal:

Step 2)
Starting with the first set: the largest perfect square less than or equal to 3 is 1, and the square root of 1 is 1. Therefore, put 1 on top and 1 at the bottom like this:

1
3	00
1

Step 3)
Calculate 3 minus 1 and put the difference below. Then move down the next set of numbers.

1
3	00
1
2	00

Step 4)
Double the number in green on top: 1 × 2 = 2. Then, use 2 and the bottom number to make this problem:

2? × ? ≤ 200

The question marks are "blank" and the same "blank". With trial and error, we found the largest number "blank" can be is 7. Now, enter 7 on top:

1	7
3	00
1
2	00

That's it! The answer is on top. The square root of 3 with one digit decimal accuracy is 1.7.

Square Root of a Number
Please enter another number in the box below to get the square root of the number and other detailed information like you got for 3 on this page.

Notes
Remember that negative times negative equals positive. Thus, the square root of 3 does not only have the positive answer that we have explained above, but also the negative counterpart.

We often refer to perfect square roots on this page. You may want to use the list of perfect squares for reference.

Square Root of 4
Here is the next number on our list that we have equally detailed square root information about.