$\sqrt {x^2} = |x|$ : how to relate $\sqrt {x^2}$ to the concept of "distance to $x$ from $0$"?

Question

A basic algbraic fact is that : $ \sqrt {x^2} = |x|$.

There must be a purely algebraic proof of this fact, involving only manipulation of symbols.

But since absolute value has a geometric interpretation , it would be nice to have a geometric interpretation of the number $\sqrt {x^2}$ in terms of " distance of number $x$ from $0$ ", in order to show not only the identity of denotation of the two expressions, but also their conceptual identity.

How can this be achieved?

Is it correct to say : the distance of a number from $0$ is the distance of the point $P=(x, 0)$ from the point $O=(0,0)$, and then to apply the distance formula to point $P$?

How to make this rigorous?

Answer 1

There have been valid comments made, which I incorporated in my answer.

Basics

I shall ignore complex numbers and consider the real case only, because the identity $\sqrt{x^2}=|x|$, as it may be commonly understood, holds for real but not for complex $x$, e.g., $\pm i = \sqrt{i^2}\neq |i| = 1$.

There are several concepts we are dealing with:

• the square of a real number, $(\cdot)^2$,
• the square root of a nonnegative real number, $\sqrt{\cdot}$,
• the absolute value of a real number, $|\cdot|$,
• the distance between numbers or points.

Symbolic manipulations are meaningless, unless we define what they mean. I use the following definitions:

• Given $y\geq 0$, $\sqrt{y}$ is the unique non-negative number $x$, such that $x^2 = y$. Before using this definition, it would be good to demonstrate its validity: the existence and the uniqueness. I will skip that.

• Given a real number $x$, we define $|x|:=\begin{cases} x, & x\geq0,\\ -x, & x<0. \end{cases}$

• The distance between two points is the length of the segment they cut on the line that passes through them. Here I chopped off one Hydra's head, and three new ones grew in its place. I now have to explain how to draw a line through two points, how they cut a segment on it, and how to measure the length of the segment. Fortunately, these "smaller heads" are quite intuitive and might not require further explanation, at least in one, two or three dimensions.

Now let's go through the statements you make.

A basic algbraic fact is that : $ \sqrt {x^2} = |x|$.

This is true for any real number. I am afraid, there is nothing interesting here. If you consider only non-negative reals, $ \sqrt {x^2} = x$ is simply the statement that $\sqrt{\cdot}$ is the inverse function of $(\cdot)^2$. The case of negative reals is reduced to positive reals by observing that $x^2=|x|^2$ and $|x|=||x||$.

But since absolute value has a geometric interpretation ...

Let's make this interpretation explicit: the absolute value of a real number equals the distance between this real number on the real line and $0$. This interpretation is intuitive, but if you start analysing it, it may seem odd: first, we identify numbers with points, second, we identify the distances between pairs of points with numbers (and hence points). One may go crazy questioning the legitimacy of these identification, only to realise that they are as valid as the identification between 5 apples in a fruit bowl and 5 spare lives in Super Mario: the two systems of objects behave identically within a certain scope. The formal term is homomorphism.

Is it correct to say : the distance of a number from $0$ is the distance of the point $P=(x, 0)$ from the point $O=(0,0)$

Yes, this is correct, but this is almost a tautology, which does not say anything! If we drop the zeros that play no role (consider the real line instead of the coordinate plane), the statement becomes

the distance between numbers $x$ and $0$ equals the distance between points $P=x$ and $O=0$

which is an absolutely useless statement, at least in the system of definitions I introduced. It does not even mention the absolute value!

Conclusion

I doubt you can find a geometric interpretation of $\sqrt {x^2} = |x|$ beyond what mentioned above. I will repeat that on the left-hand side you apply a function and then its (almost) inverse. That's all there is. Even the geometric interpretation of $(\cdot)^2$ as the area of a square is being cancelled out.

Generalisations

But it is possible to consider a generalisation of $\sqrt {x^2} = |x|$. The relevant mathematical quantities will have geometrical interpretations, but there will be no geometric interpretation of the identity: it will be just an equality between two non-negative numbers.

The physical "distance" is formalised and generalised by the concept of metric. There are many different metrics we can define on a line, on a plane or in the 3d space. Not all these metrics will have all the properties you might intuitively expect from the distance, but they will have some of these properties, for example, all metrics satisfy the triangle inequality by definition. By the way, a metric can be defined on an arbitrary abstract set, like the set of Power Rangers. In any case, a metric measures how far away two points are from each other.

A special kind of metrics we often deal with are the norm-induced metrics, which are defined in vector spaces. In a vector space, we can add and subtract vectors (or points, if we label an arbitrary point as the origin). A norm is something that tells us how long a vector is. Norm-induced metrics are the ones that measure distance by length. In other words, the distance between two points is defined as the length of the vector connecting them, which is something very natural, but does not have to be the case. Norms are denoted as $\|\cdot\|$, metrics as $d(\cdot,\cdot)$.

A particular variety of norms are norms induced by inner products. We are now generalising the concept of the angle. A dot-product of two unit vectors equals the cosine of the angle between them. And because we can measure angles by measuring their cosines, dot-products basically measure the angle between two vectors. The abstract generalisation of the familiar dot-product is known as the inner product, but in a finite-dimensional space this is simply the $n$-dimensional dot product. Inner products are denoted as $\langle \cdot, \cdot \rangle$.

As a result, we can write $$\sqrt { \langle \mathbf{x}, \mathbf{x} \rangle }= \|\mathbf{x}\|,$$ where have made the following generalisation:

• the square of a real number, $(\cdot)^2$, becomes the dot-product, $\langle \cdot, \cdot \rangle$,
• the square root of a non-negative real number remains as it was, $\sqrt{\cdot}$,
• the absolute value of a real number, $|\cdot|$, becomes a norm, $\|\cdot\|$,
• the distance between numbers or points becomes the norm-induced metric, $d(\cdot,\cdot)$.

For example, consider a vector $\mathbf{x}=(x_1, x_2, x_3)$. On the left-hand side we have

$$\sqrt { \langle \mathbf{x}, \mathbf{x} \rangle } = \sqrt{ x_1^2 + x_2^2 + x_3^2}.$$

You may think we are arriving at something interesting, but no. We have simply obtained a formula for the norm of an $n$-dimensional vector. That's it. We still have no geometrical interpretation of the equality here. Only an interpretation of the quantities on both sides:

The square root of the inner product of a vector $\mathbf{x}$ with itself equals the distance between the point $\mathbf{x}$ and the reference point $\mathbf{0}$ (which, by definition, equals the length of vector $\mathbf{x}).$

Here $\mathbf{x}$ is both a vector and a point.

These generalisations of the distance, the length, and the angle are well-studied and fairly well-known. For example, see this question and other questions linked therein.

Answer 2

This has to do with how we create the concepts of "real number line", "distance", and "absolute value".

The idea behind the real number line is that it idealizes the concept of a ruler. On a ruler, you have markings placed at equal intervals:

When you jump from one marking to the next, you add one. The "add one" operation is specifically created to be equivalent to hopping one unit on the ruler, e.g. from the "1" to the "2" to the "3" on the top edge above. That is, we define $a + 1$ to be the number that would represent the immediately following tick. The generalized addition $a + b$ is then defined in such a way as to permit hopping a uniform fraction of a tick as well as a whole tick.

The real numbers "idealize" this in the sense that they allow for the markings to become infinitely fine (so that we can always have enough resolution to use them to conduct every real-world measurement we may want to) and moreover not to permit any "gaps", where to "have a gap" would mean that we could construct an infinite sequence of real numbers that would look like they "should be" better and better approximating some real number (e.g. 0.3, 0.33, 0.333, 0.3333, ...), and yet for that seemingly "should be there" real number (here, $\frac{1}{3}$) to, in fact, not be there.

That is, we "bake in" to the idea of addition of real numbers the idea of hopping between ruler ticks, and this then permits us to use them in many other situations, e.g. filling up a burette full of water, and thus adding masses of water.

The geometric "shape" of the real number line is not inherent in the individual numbers, which may be any "material" we like to make them from (decimals, Dedekind cuts, continued fractions, whatever method you prefer so long as it makes "just the right [infinite] amount" of them) - it is that part you mention as being a conceptual element. We mentally put the real numbers at equal spatial distances from each other because that is what we want them to have.

"Distance", then, is the qualitative idea of being "closer or further apart", which we can give with the idea of order: if $a < b < c$, then we say $c$ is "farther" from $a$ than $b$ is.

When we talk of measuring distance, we want that measure of distance between two points to be consistent with this, such that the measure of distance agrees with the "ruler-step" method, and that means we should invoke our idea of addition (and its complementary operation, subtraction). And as you know, $0 + x = x$, so it is not hard to see from there that a consistent measure of distance should be $d(0, x) = |x|$. More generally, we see we should have $d(a, b) = |b - a|$.