Why Is The Square Root of 1 Equal To 1?

On the surface, it appears as though I am examining the most mundane mathematical fact here. But as it turns out, a journey through the seemingly mundane often leads us to the heart of profound mathematical truths.

No, seriously. You would be surprised how far this simple question will lead us into the labyrinth of mathematical reasoning. Are you game? Let us begin.

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What is the Square Root of 1? — The Casual Answer

To any casual observer, the answer to this question is: 1. But to any mathematically inclined person, this “obvious” answer is a siren call that begs a deeper exploration of the mathematical reasoning involved.

That’s exactly what I plan to do. Let me start with the most basic question I can come up with:

“What do we mean by the square root of 1?”

The square root of a number, say ‘x’, is a value that, when multiplied by itself, gives ‘x’. In short, we are looking for a number that, when multiplied by itself, gives us 1. At this point, the casual observer would be quick to remark:

“Aha! That’s obviously 1!”

Hold your horses! Here’s the thing: mathematically speaking, there is also another number, which when multiplied by itself, gives us 1. It is −1.

1 * 1 = 1

−1 * −1 = 1

So, now we face a challenge. Is the square root of 1 equal to 1 or −1? Enter the domain of the “principal square root”.

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What is the Square Root of 1? — The Serious Answer

According to mathematical convention, when we refer to “the square root” of a number, we mean the “principal square root”. Wolfram MathWorld (Eric Weisstein) defines the principle square root as follows:

The principal square root is the unique non-negative square root of a non-negative real number.

Consequently, although −1 does indeed square to 1, by convention, when we talk about “the square root of 1”, we refer to the principal square root, which is 1. But then again, any sceptic might question:

“Why are we biased toward positive numbers in the context of square roots?”

Why is the Square Root of 1 Equal to 1?

The reason we define the principal square root is not so much bias as it is a method we use to main consistency and avoid confusion. You see, mathematics thrives on consistency and precision.

In the world of equations, having two possible values for the square root of a number could lead to “complex” situations.

The line of reasoning that I am following here has its “roots” (pun totally intended) in the theory of complex numbers.

You see, in the world of complex numbers, every non-zero number has exactly two square roots — one positive and one negative.

The astute reader might note a challenging conundrum here which involves the square root of a negative number and ‘i’.

But that is a topic for a future essay, as it is significantly more technical than the topic we are dealing with in this essay.

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Conclusion

In the world of mathematics, nothing is ever truly as simple as it seems. Even something as fundamental as 1 has its share of mysteries and surprises.

This unassuming number, the essential ‘unit’ (or building block) of all other numbers, when subjected to one of the fundamental mathematical operations, unfolds an intricate labyrinth of how mathematics deals with complexities even in simplicity.

In my (unpopular?) opinion, as far as mathematics is concerned, simplicity and complexity are but two sides of the same coin. This is evident from the fact that even though both 1 and −1 square to 1, we define 1 as the principal square root and −1 as the “other” real root.

If there is one thing that topics like these have taught me, it is to continually keep asking “why?” even in the face of the most mundane mathematical facts. This process often leads to the most profound answers!

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Further reading that might interest you:

•  Why Is Negative Times Negative Really Positive?
• Why Is A Number Raised To The Power Zero = One?
• Why Exactly Is Zero Factorial Equal To One?

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