# What does the L2 or Euclidean norm mean?

Here’s a quick tutorial on the L2 or Euclidean norm.

First of all, the terminology is not clear. So let’s start with that.

### Many equivalent names

All these names mean the same thing:
Euclidean norm == Euclidean length == L2 norm == L2 distance == $l^2$ norm

Although they are often used interchangable, we will use the phrase “L2 norm” here.

### Many equivalent symbols

Now also note that the symbol for the L2 norm is not always the same.

Let’s say we have a vector, $\vec{a} = [3,1,4,3,1]$.

The L2 norm is sometimes represented like this, $||\vec{a}||$

Or sometimes this, $||\vec{a}||_2$

Other times the L2 norm is represented like this, $|\vec{a}|$

Or even this, $|\vec{a}|_2$

To help distinguish from the absolute value sign, we will use the $||\vec{a}||$ symbol.

### Equation

Now that we have the names and terminology out of the way, let’s look at the typical equations.
$||\vec{a}|| = \sqrt{\sum_i^n (a_i)^2} = \sqrt{(a_1)^2 + (a_2)^2 + \dots + (a_n)^2}$
where $n$ is the number of elements in $\vec{a}$ (in this case $n=5$).

In words, the L2 norm is defined as, 1) square all the elements in the vector together; 2) sum these squared values; and, 3) take the square root of this sum.

### A quick example

Let’s use our simple example from earlier, $\vec{a} = [3,1,4,3,1]$.

We compute the L2 norm of the vector $\vec{a}$ as,
$||\vec{a}|| = \sqrt{(3^2 + 1^2 + 4^2 + 3^2 + 1^2)} = \sqrt{9 + 1 + 16 + 9 + 1} = \sqrt{36} = 6$

And there you go!

So in summary, 1) the terminology is a bit confusing since as there are equivalent names, and 2) the symbols are overloaded. Finally, 3) we did a small example computing the L2 norm of a vector by hand.

If you are hungry for a code example, I wrote a small MATLAB example (computing L2 distance) here.