Square Root Of 1: Value, Proof And Both Square Roots Explained

What Is A Square Root?

A square root of a number $n$ is any number that gives $n$ when multiplied by itself. For $n = 1$, both $1 \times 1$ and $(-1) \times (-1)$ give 1, so 1 has two square roots.

The radical sign picks one of them. $\sqrt{1}$ is defined as the non-negative root, called the principal square root, so $\sqrt{1} = 1$ and never $-1$.

In exponent form, the square root is a power of one half: $\sqrt{1} = 1^{\frac{1}{2}} = 1$. That also explains the search "why is square root of $x$ $1/2$": raising to the power $\frac{1}{2}$ undoes squaring, because $\left(x^{\frac{1}{2}}\right)^2 = x^1$.

Where Does The Square Root Of 1 Show Up Beyond Arithmetic, And Why Does It Matter?

The square root of 1 is the side length of the unit square, the square with area 1 that every area measurement is built from. It is also the radius of the unit circle, the circle $x^2 + y^2 = 1$ where every point sits $\sqrt{1} = 1$ unit from the centre.

• Unit vectors: a vector has length 1 exactly when the sum of the squares of its parts is 1, as with $(0.6, 0.8)$ where $0.36 + 0.64 = 1$.

• Calculator experiment: press the square-root key again and again on any positive number and the display creeps toward 1. Starting from 1000, ten presses give 1.0068; starting from 0.001, ten presses give 0.9933.

• Number identity: 1 is the multiplicative identity, so multiplying any radical by $\sqrt{1}$ leaves it unchanged, which is the quiet step behind rationalising denominators.

Why does it matter? The square-root function pulls numbers above 1 down and pushes numbers between 0 and 1 up, and 1 is the fixed point it never moves.

Quick Reference Table

These are the square roots of 0 to 10, with the exact form and the decimal to four places.

The perfect squares in this band are 0, 1, 4 and 9, and they are the only rows with whole-number roots. Notice that 0 and 1 are the only rows where the root equals the number itself.

Is The Square Root Of 1 Positive Or Negative?

The symbol $\sqrt{1}$ is positive: it equals 1. The number 1 itself, though, has one positive and one negative square root, $+1$ and $-1$.

The two statements answer different questions. "What is $\sqrt{1}$?" asks for the principal root, while "Solve $x^2 = 1$" asks for every number that squares to 1, so the answer there is $x = \pm 1$.

The honest difficulty here is notation, not arithmetic. Older textbooks and some calculators phrase "the square root" loosely, which is why this page always says "principal root" when it means the radical symbol.

Is The Square Root Of 1 Rational Or Irrational?

The square root of 1 is rational. Its value, 1, can be written as the fraction $\frac{1}{1}$, and its decimal, 1.0000, terminates.

A whole number has a rational square root exactly when it is a perfect square, and 1 is the smallest positive perfect square. So $\sqrt{1}$ is a rational number, and it is also a natural number and an integer.

How Do You Find The Square Root Of 1?

You can find the square root of 1 straight from the definition, and every standard method confirms it: prime factorization, long division, repeated subtraction, solving $x^2 = 1$ and Newton's iteration all land on 1. Each method below is worth seeing once, because the same steps carry over to larger numbers.

Method 1: Using The Definition

Ask which non-negative number multiplied by itself gives 1.

$$1 \times 1 = 1$$

$$\sqrt{1} = 1$$

Method 2: Prime Factorization

Prime factorization splits a number into primes and takes one prime from each pair. The number 1 is neither prime nor composite, and it has no prime factors at all (its factorization is the empty product).

With nothing to pair, nothing comes out of the root and nothing stays inside it, so $\sqrt{1} = 1$. This is also why 1 is never listed among the prime numbers.

Method 3: Long Division

Write 1 as $1.00\ 00$. The largest whole number whose square is at most 1 is 1.

The remainder is 0 after the first step and every pair brought down is 00, so the decimal digits are all zero. The long division gives $\sqrt{1} = 1.00\ldots = 1$ exactly.

Method 4: Repeated Subtraction Of Odd Numbers

Subtract consecutive odd numbers, starting at 1, until you reach zero.

$$1 - 1 = 0$$

You reach zero in exactly one step, so 1 is a perfect square and its square root is 1. (For 4 it takes two steps, $4 - 1 - 3 = 0$, and for 9 it takes three.)

Method 5: Solving $x^2 = 1$ By Factoring

Move everything to one side and factor the difference of squares.

$$x^2 - 1 = 0$$

$$(x - 1)(x + 1) = 0$$

By the zero product property, $x - 1 = 0$ or $x + 1 = 0$, so $x = 1$ or $x = -1$. The positive solution is the principal square root, $\sqrt{1} = 1$.

Method 6: Newton's Iteration From A Wrong Guess

Newton's (Babylonian) rule for $\sqrt{1}$ is $x_{\text{new}} = \frac{1}{2}\left(x + \frac{1}{x}\right)$. Start from a deliberately poor guess, $x_0 = 3$.

$$x_1 = \frac{1}{2}\left(3 + \frac{1}{3}\right) = \frac{5}{3} \approx 1.6667$$

$$x_2 = \frac{1}{2}\left(\frac{5}{3} + \frac{3}{5}\right) = \frac{17}{15} \approx 1.1333$$

$$x_3 = \frac{257}{255} \approx 1.0078$$

The guesses close in on 1 from above. Here it is just a check, but on a number like 2 the same rule is how calculators find $\sqrt{2}$.

How Do You Find The Square Root Of 1 Over A Number?

The square root of 1 over a number $n$ is 1 over the square root of $n$. In symbols, $\sqrt{\frac{1}{n}} = \frac{\sqrt{1}}{\sqrt{n}} = \frac{1}{\sqrt{n}}$, because the root splits over a fraction.

For $\sqrt{\frac{1}{2}}$, multiply top and bottom by $\sqrt{2}$ to clear the root from the denominator. That step works only because $\frac{\sqrt{2}}{\sqrt{2}} = 1$, the same identity idea from earlier.

What Is The Square Root Of 1 In Complex Numbers?

In the complex number system the square roots of 1 are still just $1$ and $-1$. They are called the square roots of unity, and the principal one is still 1.

Complex numbers only change the picture for negative numbers. The square root of negative 1 is the imaginary unit, $\sqrt{-1} = i$, where $i^2 = -1$; this is not a root of $+1$.

That clears up the autocomplete search "why is square root of 1 i": it isn't. The searcher almost always means $\sqrt{-1}$, and the missing minus sign changes the answer completely.

For contrast, the cube root of 1 has three complex values, $1$ and $\frac{-1 \pm i\sqrt{3}}{2}$, which is why the topic of the cube root of unity gets its own page.

Examples Of Square Root Of 1

Example 1

Evaluate $\sqrt{1} + \sqrt{4} + \sqrt{9}$.

$$\sqrt{1} + \sqrt{4} + \sqrt{9} = 1 + 2 + 3$$

$$= 6$$

Example 2

Find $\sqrt{\frac{1}{9}}$.

Wrong path. Take the root of the 1 only and keep the 9:

$$\sqrt{\frac{1}{9}} = \frac{1}{9}$$

Check it by squaring: $\left(\frac{1}{9}\right)^2 = \frac{1}{81}$, not $\frac{1}{9}$, so the answer fails.

Correct path. Take the root of the numerator and the denominator:

$$\sqrt{\frac{1}{9}} = \frac{\sqrt{1}}{\sqrt{9}} = \frac{1}{3}$$

Check: $\left(\frac{1}{3}\right)^2 = \frac{1}{9}$.

Example 3

Solve $x^2 - 1 = 0$.

$$x^2 = 1$$

$$x = \pm\sqrt{1}$$

$$x = 1 \text{ or } x = -1$$

Both values check, since $1^2 = 1$ and $(-1)^2 = 1$.

Example 4

A square floor tile has an area of 1 square foot. How long is each side?

$$\text{side} = \sqrt{1} = 1 \text{ foot}$$

A length cannot be negative, so the $-1$ root is rejected here.

Example 5

Find $\sqrt{0.01}$.

$$0.01 = \frac{1}{100}$$

$$\sqrt{0.01} = \frac{\sqrt{1}}{\sqrt{100}} = \frac{1}{10} = 0.1$$

The step that usually gets skipped is rewriting the decimal as a fraction first; once 0.01 is read as 1 over 100, the root is immediate.

Example 6

Show that the vector $(0.6, 0.8)$ is a unit vector.

$$\text{length} = \sqrt{0.6^2 + 0.8^2}$$

$$= \sqrt{0.36 + 0.64}$$

$$= \sqrt{1} = 1$$

The length is exactly 1, so it is a unit vector.

What Mistakes Do Students Make With The Square Root Of 1?

The square root of 1 looks too simple to get wrong, yet three errors keep appearing: the sign, the minus under the root, and confusing a root with halving.

Mistake 1: Writing $\sqrt{1} = \pm 1$

Where it slips in: Right after solving $x^2 = 1$, when the $\pm$ from the equation gets copied onto the radical.

Don't do this: Write $\sqrt{1} = \pm 1$ or $\sqrt{1} = -1$.

The correct way: The radical names the principal root, so $\sqrt{1} = 1$. The memorizer who learned "every square root is plus or minus" attaches the sign to the symbol instead of to the equation's solutions.

Mistake 2: Mixing Up $\sqrt{1}$ And $\sqrt{-1}$

Where it slips in: When a minus sign is dropped or misread, often in a hurry.

Don't do this: Say the square root of 1 is $i$.

The correct way: $\sqrt{1} = 1$ and $\sqrt{-1} = i$. The rusher who skips reading the sign under the radical is the one who swaps these, so circle the sign before taking any root.

Mistake 3: Halving Instead Of Rooting

Where it slips in: Early in the square-roots unit, when "square root" and "divide by 2" both feel like making a number smaller.

Don't do this: Write $\sqrt{1} = 0.5$.

The correct way: A square root asks "what times itself gives this?", and $0.5 \times 0.5 = 0.25$, not 1. The first instinct many students have is to halve, and the fix is the squaring check: square your answer and see whether you get the number back.

Where Can Your Child Get Help With Square Roots?

The square root of 1 is where the ideas of principal root, two roots and the identity first meet, and they come back in every radical and quadratic later on. If your child wants live help, an algebra tutor or a middle school math tutor can work through these steps at their pace.

For regular practice, there are math classes online and focused help with algebra sessions.

Conclusion

• The square root of 1 is 1, because $1 \times 1 = 1$, and the radical symbol always means this positive root.

• The number 1 has two square roots, $1$ and $-1$, so $x^2 = 1$ has the solutions $x = \pm 1$.

• $\sqrt{1}$ is rational, and 1 is the smallest positive perfect square.

• For fractions, $\sqrt{\frac{1}{n}} = \frac{1}{\sqrt{n}}$, so $\sqrt{\frac{1}{4}} = \frac{1}{2}$.

• $\sqrt{1} = 1$ but $\sqrt{-1} = i$; reading the sign under the root is the habit that prevents most errors.

Practice These Before Moving On

1. Evaluate $\sqrt{1} \times \sqrt{25}$.

2. Find $\sqrt{\frac{1}{16}}$.

3. Solve $4x^2 = 4$.

4. Find $\sqrt{\frac{1}{2}}$ in simplest radical form and as a decimal to 4 places.

5. A point on the unit circle has $x$-coordinate $0.8$ and positive $y$-coordinate. Find $y$.

Answers:

1. $1 \times 5 = 5$

2. $\frac{1}{4}$

3. $x^2 = 1$, so $x = 1$ or $x = -1$

4. $\frac{\sqrt{2}}{2} \approx 0.7071$

5. $y = \sqrt{1 - 0.64} = \sqrt{0.36} = 0.6$

If question 3 gave you only one answer, go back to the section on positive and negative roots.

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