Square Root Symbol (√): Meaning, Uses, and How to Type It

The square root symbol (√), also called the radical sign, asks for the number that, multiplied by itself, gives the value underneath it: √16 = 4 because 4 × 4 = 16. You can type it with Alt + 251 on Windows, Option + V on a Mac, √ in HTML, or \sqrt{x} in LaTeX.

This guide explains the parts of a radical expression, why √16 is 4 and not −4, which roots are irrational, and how to produce the symbol in Word, on your phone, in Excel, Python, and LaTeX.

Copy and Paste the Square Root Symbol

• √ — square root (U+221A)
• ∛ — cube root (U+221B)
• ∜ — fourth root (U+221C)
• √x and √(x + 1) — plain-text forms; use parentheses when the root covers more than one term

The Square Root Symbol Explained

A radical expression has three named parts:

• Radical sign — the √ itself. The horizontal bar that extends over the number is the vinculum; it groups everything underneath.
• Radicand — the number or expression under the bar. In √(x + 1), the radicand is x + 1.
• Index — the small number tucked into the notch of the sign. ∛8 has index 3 and asks for the cube root. When no index is written, the index is 2.

The sign first appeared in print in 1525 in Christoph Rudolff's Coss, probably as a stylized letter r for radix (Latin for "root"). René Descartes added the overbar in 1637.

Meaning and Uses of the Square Root

Principal Root and the ± Sign

Every positive number has two square roots: 16 is both 4² and (−4)². The radical sign, however, always denotes the principal (non-negative) root, so √16 = 4, never −4. When both roots are needed, write ± explicitly:

• √16 = 4 (the symbol alone means the positive root)
• x² = 16 has solutions x = ±√16 = ±4
• √0 = 0, and √x is undefined in the real numbers when x < 0

This distinction is why the quadratic formula carries a ± in front of its square root.

Perfect Squares

A perfect square is a whole number whose square root is also a whole number. The first fifteen are worth memorizing:

Irrational Roots

If the radicand is not a perfect square, its square root is irrational: the decimal never ends and never repeats.

• √2 ≈ 1.41421356
• √3 ≈ 1.73205081
• √5 ≈ 2.23606798
• √10 ≈ 3.16227766

Exact answers keep the radical and simplify it by pulling out perfect-square factors: √50 = √(25 × 2) = 5√2. The irrationality of √2 is the classic example of a proof by contradiction.

Higher Roots

The cube root ∛x is the number whose cube is x, so ∛27 = 3 and ∛(−8) = −2. Unlike square roots, odd-index roots of negative numbers are perfectly real. Roots can also be written as fractional exponents, which is how calculators and programming languages treat them: √x = x^(1/2) and ∛x = x^(1/3). The general rule is ⁿ√(x^m) = x^(m/n).

Where Square Roots Show Up

• Pythagorean theorem — the hypotenuse of a right triangle is c = √(a² + b²)
• Standard deviation — σ = √(Σ(xᵢ − μ)² / N), the square root of the variance
• Distance formula — d = √((x₂ − x₁)² + (y₂ − y₁)²)
• Quadratic formula — x = (−b ± √(b² − 4ac)) / 2a
• Imaginary unit — i = √(−1)

How to Type the Square Root Symbol

Windows: Hold Alt and type 251 on the numeric keypad; the code comes from the old DOS code page and works in nearly every program. In Word, 221A followed by Alt + X does the same thing, and the Equation Editor (Alt + =) accepts \sqrt and draws the full bar over the radicand.

Mac: Press Option + V; it works in any text field. For ∛ or ∜, open the Character Viewer (Control + Command + Space) and search for "root."

iPhone and Android: Neither default keyboard has a root key. Copy the symbol from the block above, or add a text replacement (Settings > General > Keyboard > Text Replacement on iOS) so that typing "sqrt" expands to √.

Excel and Google Sheets: =SQRT(16) returns 4, and =16^(1/2) gives the same result. For a cube root use =27^(1/3) or =POWER(27, 1/3). The symbol itself can be inserted with =UNICHAR(8730).

Python: math.sqrt(2) returns 1.4142135623730951, 2 ** 0.5 gives the same, and cmath.sqrt(-1) returns 1j for negative radicands. NumPy offers np.sqrt and np.cbrt for arrays.

Square Root in LaTeX

The LaTeX command is \sqrt{radicand}, and an optional argument sets the index: \sqrt[3]{x} or, in general, \sqrt[n]{x}. LaTeX draws the vinculum automatically and stretches it to cover whatever you put inside the braces, which is something the plain Unicode character cannot do.

```latex
\documentclass{article}
\usepackage{amsmath}
\begin{document}
\[ \sqrt{2} \approx 1.4142, \qquad \sqrt[3]{27} = 3, \qquad \sqrt[n]{x^m} = x^{m/n} \]
\[ c = \sqrt{a^2 + b^2}, \qquad x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
\[ \sqrt{2 + \sqrt{3}}, \qquad \sqrt{x} + \sqrt{\smash[b]{y}}, \qquad \sqrt{x\vphantom{y}} + \sqrt{y} \]
\[ \surd 2 \]
\end{document}
```

A few finer points:

• Uneven heights. Side by side, \sqrt{x} and \sqrt{y} get different bar heights because y has a descender. \sqrt{\smash[b]{y}} (amsmath) hides the descender so the bars line up; \sqrt{x\vphantom{y}} does the opposite by giving x an invisible y-sized box.
• Nested roots. \sqrt{2 + \sqrt{3}} nests cleanly; amsmath's \leftroot and \uproot fine-tune the position of an index that collides with the outer bar.
• Just the sign. \surd prints √ without a bar, useful in prose or when the vinculum is drawn some other way.
• Unicode vs LaTeX. A pasted √ has no bar and fails under pdfLaTeX without extra font setup. Always use \sqrt{} in math mode.

Related Symbols

Is the Square Root of 16 Equal to 4 or −4?

The symbol √16 means the principal root, which is 4. The equation x² = 16 has two solutions, 4 and −4, and that is written x = ±4. Keeping the two statements separate avoids a very common algebra mistake.

What Is the Square Root of a Negative Number?

There is no real number whose square is negative, so √(−9) is undefined in the real numbers. In the complex numbers, √(−9) = 3i, where i = √(−1).

What Is the Difference Between a Radical and a Square Root?

"Radical" is the general term for any root written with the √ sign, including cube roots and fourth roots. A square root is the specific case where the index is 2, which is the default when no index is shown.

Typesetting Roots the Easy Way

Roots look right only when the bar stretches over the whole radicand and sits at a consistent height, which is exactly what \sqrt{} gives you and a pasted √ does not. Octree, the AI LaTeX editor at useoctree.com, writes the \sqrt{} syntax for you from a plain-language description and previews the compiled output instantly. Browse every root, operator, and relation symbol with its LaTeX command at https://tools.useoctree.com/symbols/math.