Concept
Radical Sign vs. Radicand: What Each Part of √ Actually Means
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The Two Main Parts
Radical Sign
The tick-shaped symbol itself. It indicates a root operation is being performed.
Radicand
The number or expression placed under the radical sign that is being evaluated.
Together they form a radical expression. For example, in √49: the radical sign is √ and the radicand is 49. Since 7 × 7 = 49, the expression equals 7.
The Full Anatomy of a Radical
A complete radical expression can have up to four parts:
| Part | Name | Description |
|---|---|---|
| √ | Radical sign | The tick-shaped symbol (U+221A) |
| n | Radicand | The number or expression under the radical |
| — | Vinculum | The horizontal bar over the radicand (in typeset math) |
| 2, 3… | Index | Small number in the notch showing which root (2 = square, 3 = cube). Index 2 is usually omitted. |
The Radicand in Detail
The radicand can be:
- A positive integer: √16 = 4
- A fraction: √(1/4) = 1/2
- A decimal: √0.25 = 0.5
- An algebraic expression: √(x² + y²)
- Zero: √0 = 0
The radicand cannot be negative in real-number mathematics. A negative radicand produces an imaginary result involving the unit i.
The Vinculum: An Often-Forgotten Part
In printed or typeset mathematics, the radical sign is always drawn with a vinculum — the horizontal bar that extends over the radicand. The vinculum acts as a grouping symbol, making the scope of the radical unambiguous:
- √(a + b) means “root of the sum of a and b”
- √a + b means “root of a, then add b”
In plain text (like this webpage), the vinculum cannot be drawn, so parentheses are used instead.
The Index: Choosing Which Root
The small number written in the notch of the radical sign specifies which root to take:
| Expression | Index | Meaning | Example |
|---|---|---|---|
| √n | 2 (omitted) | Square root | √9 = 3 |
| ∛n | 3 | Cube root | ∛27 = 3 |
| ∜n | 4 | Fourth root | ∜16 = 2 |
| Un | n | nth root | n√32 = 2 (n=5) |
Frequently Asked Questions
What is the radicand in √(3x + 1)?
The radicand is the entire expression 3x + 1. Everything under the radical sign is the radicand. In typeset math, the vinculum (overline) shows exactly where the radicand ends.
Is the radical sign the same as the square root symbol?
Yes. The radical sign (√), the square root symbol, and the radix sign all refer to the same character (U+221A). The term “radical sign” is technically correct in all contexts, while “square root symbol” is used when the index is specifically 2.
What does radicand mean etymologically?
Radicand comes from the Latin radicandus, meaning “that which is to be rooted.” The root word is radix (root), which also gives us the words radical and radish.
Can the radicand be negative?
Not in real numbers. √(−4) is undefined in real mathematics because no real number squared equals a negative. In complex numbers, √(−4) = 2i, where i is the imaginary unit.