Base
ArithmeticIn a power expression, the base is the number that is multiplied by itself the number of times indicated by the exponent.
Formula
\text{base}^{\text{exponent}} = \text{power}
Definition
The base is the number being multiplied in a power expression. It is the big number on the bottom, and the exponent tells you how many times to multiply it by itself.
Example
In $5^3$, the base is $5$. You multiply $5 \times 5 \times 5 = 125$.
Key Insight
The base is what you start with. The exponent controls how many copies you multiply together.
Definition
In $b^n$, $b$ is the base and $n$ is the exponent. The base can be any real number (positive, negative, zero, or fractional). Bases and exponents together define the full power expression. The number system we use (base-$10$) is named for the base of its positional notation.
Example
$(-2)^3 = (-2)(-2)(-2) = -8$ (negative base, odd exponent: negative result). $(-2)^4 = 16$ (negative base, even exponent: positive result). $(1/3)^2 = 1/9$.
Key Insight
The sign rule for negative bases: negative base to an even exponent is positive; to an odd exponent is negative. This is because pairs of negatives cancel.
Definition
In the general theory, the base $b$ of the exponential $b^r$ for real $r$ requires $b > 0$ to be well-defined in $\mathbb{R}$ (since $b^{1/2} = \sqrt{b}$ requires $b \ge 0$ and negative bases cause $b^r$ to be complex for non-integer $r$). In complex analysis, $z^w = e^{w \operatorname{Log} z}$ where $\operatorname{Log}$ is the principal logarithm, a multi-valued function.
Example
The choice of base in logarithms determines the unit: base $2$ gives bits, base $10$ gives bels, base $e$ gives nats. The change-of-base formula: $\log_b(x) = \ln(x)/\ln(b)$ converts between any two bases.
Key Insight
Base $e$ is the "natural" base because it is the base for which the derivative of $b^x$ at $x=0$ equals $1$. No other base has this property, making $e$ the unique fixed point of calculus's fundamental operation.