Are you trying to solve a quadratic equation? Maybe you need to calculate the length of one side of a right triangle. For these types of equations and more, the Python square root function, ** sqrt()**, can help you quickly and accurately calculate your solutions.

**By the end of this article, you’ll learn:**

- What a square root is
- How to use the Python square root function,
`sqrt()`

- When
`sqrt()`

can be useful in the real world

Let’s dive in!

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## Square Roots in Mathematics

In algebra, a **square**, *x*, is the result of a number, *n*, multiplied by itself: *x = n²*

You can calculate squares using Python:

```
>>> n = 5
>>> x = n ** 2
>>> x
25
```

The Python `**`

operator is used for calculating the power of a number. In this case, 5 squared, or 5 to the power of 2, is 25.

The square root, then, is the number *n*, which when multiplied by itself yields the square, *x*.

In this example, *n*, the square root, is 5.

25 is an example of a **perfect square**. Perfect squares are the squares of integer values:

```
>>> 1 ** 2
1
>>> 2 ** 2
4
>>> 3 ** 2
9
```

You might have memorized some of these perfect squares when you learned your multiplication tables in an elementary algebra class.

If you’re given a small perfect square, it may be straightforward enough to calculate or memorize its square root. But for most other squares, this calculation can get a bit more tedious. Often, an estimation is good enough when you don’t have a calculator.

Fortunately, as a Python developer, you do have a calculator, namely the Python interpreter!

## The Python Square Root Function

Python’s `math`

module, in the standard library, can help you work on math-related problems in code. It contains many useful functions, such as `remainder()`

and `factorial()`

. It also includes the Python square root function, `sqrt()`

.

You’ll begin by importing `math`

:

```
>>> import math
```

That’s all it takes! You can now use `math.sqrt()`

to calculate square roots.

`sqrt()`

has a straightforward interface.

It takes one parameter, `x`

, which (as you saw before) stands for the square for which you are trying to calculate the square root. In the example from earlier, this would be `25`

.

The return value of `sqrt()`

is the square root of `x`

, as a floating point number. In the example, this would be `5.0`

.

Let’s take a look at some examples of how to (and how not to) use `sqrt()`

.

### The Square Root of a Positive Number

One type of argument you can pass to `sqrt()`

is a positive number. This includes both `int`

and `float`

types.

For example, you can solve for the square root of `49`

using `sqrt()`

:

```
>>> math.sqrt(49)
7.0
```

The return value is `7.0`

(the square root of `49`

) as a floating point number.

Along with integers, you can also pass `float`

values:

```
>>> math.sqrt(70.5)
8.396427811873332
```

You can verify the accuracy of this square root by calculating its inverse:

```
>>> 8.396427811873332 ** 2
70.5
```

### The Square Root of Zero

Even `0`

is a valid square to pass to the Python square root function:

```
>>> math.sqrt(0)
0.0
```

While you probably won’t need to calculate the square root of zero often, you may be passing a variable to `sqrt()`

whose value you don’t actually know. So, it’s good to know that it can handle zero in those cases.

### The Square Root of Negative Numbers

The square of any real number cannot be negative. This is because a negative product is only possible if one factor is positive and the other is negative. A square, by definition, is the product of a number and itself, so it’s impossible to have a negative real square:

```
>>> math.sqrt(-25)
Traceback (most recent call last):
File "<stdin>", line 1, in <module>
ValueError: math domain error
```

If you attempt to pass a negative number to `sqrt()`

, then you’ll get a `ValueError`

because negative numbers are not in the domain of possible real squares. Instead, the square root of a negative number would need to be complex, which is outside the scope of the Python square root function.

## Square Roots in the Real World

To see a real-world application of the Python square root function, let’s turn to the sport of tennis.

Imagine that Rafael Nadal, one of the fastest players in the world, has just hit a forehand from the back corner, where the baseline meets the sideline of the tennis court:

Now, assume his opponent has countered with a drop shot (one that would place the ball short with little forward momentum) to the opposite corner, where the other sideline meets the net:

How far must Nadal run to reach the ball?

You can determine from regulation tennis court dimensions that the baseline is 27 feet long, and the sideline (on one side of the net) is 39 feet long. So, essentially, this boils down to solving for the hypotenuse of a right triangle:

Using a valuable equation from geometry, the Pythagorean theorem, we know that *a² + b² = c²*, where *a* and *b* are the legs of the right triangle and *c* is the hypotenuse.

Therefore, we can calculate the distance Nadal must run by rearranging the equation to solve for *c*:

You can solve this equation using the Python square root function:

```
>>> a = 27
>>> b = 39
>>> math.sqrt(a ** 2 + b ** 2)
47.43416490252569
```

So, Nadal must run about 47.4 feet (14.5 meters) in order to reach the ball and save the point.

## Conclusion

Congratulations! You now know all about the Python square root function.

You’ve covered:

- A brief introduction to square roots
- The ins and outs of the Python square root function,
`sqrt()`

- A practical application of
`sqrt()`

using a real-world example

Knowing how to use `sqrt()`

is only half the battle. Understanding when to use it is the other. Now, you know both, so go and apply your newfound mastery of the Python square root function!