Definition:Root (Analysis)
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Definition
Let $x, y \in \R_{\ge 0}$ be positive real numbers.
Let $n \in \Z$ be an integer such that $n \ne 0$.
Then $y$ is the positive $n$th root of $x$ if and only if:
- $y^n = x$
and we write:
- $y = \sqrt[n] x$
Using the power notation, this can also be written:
- $y = x^{1/n}$
When $n = 2$, we write $y = \sqrt x$ and call $y$ the positive square root of $x$.
When $n = 3$, we write $y = \sqrt [3] x$ and call $y$ the cube root of $x$.
Note the special case where $x = 0 = y$:
- $0 = \sqrt [n] 0$
Also known as
A root is also known as a radical.
The operation of extracting a root of a number is also seen in some sources referred to as evolution.
By a similar naming convention, raising a number to a power is seen the same sources referred to as involution.
Also see
- Existence and Uniqueness of Positive Root of Positive Real Number, which proves the existence and uniqueness of the positive $n$th root
- Results about roots of numbers can be found here.
Sources
- 1951: B. Hague: An Introduction to Vector Analysis (5th ed.) ... (previous) ... (next): Chapter $\text I$: Definitions. Elements of Vector Algebra: $1$. Scalar and Vector Quantities
- 1964: Walter Rudin: Principles of Mathematical Analysis (2nd ed.) ... (previous) ... (next): Chapter $1$: The Real and Complex Number Systems: Real Numbers: $1.37$. Theorem
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 7$: Laws of Exponents: $7.7$
- 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 1$: Real Numbers: $\S 1.9$: Roots
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): Glossary
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.2$: Numbers, Powers, and Logarithms
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): Glossary
- 2004: Ian Stewart: Galois Theory (3rd ed.) ... (previous) ... (next): Historical Introduction: Polynomial Equations
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Entry: root: 2.
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Entry: index (indices) (vii)