Question

which set of numbers contains both an additive inverse and a multiplicative inverse?
integers
whole numbers
natural numbers
rational numbers

Explanation:

1. Additive inverse: For any number $x$, a number $-x$ such that $x + (-x) = 0$.
2. Multiplicative inverse: For any non-zero number $x$, a number $\frac{1}{x}$ such that $x \times \frac{1}{x} = 1$.

• Natural numbers (positive counting numbers: 1,2,3...) lack additive inverses (negative numbers are not in the set).
• Whole numbers (0,1,2,3...) lack additive inverses for positive values, and 0 has no multiplicative inverse.
• Integers (...-2,-1,0,1,2...) have additive inverses, but non-zero integers like 2 have a multiplicative inverse $\frac{1}{2}$ which is not an integer, and 0 has no multiplicative inverse.
• Rational numbers (all numbers that can be written as $\frac{a}{b}$, where $a,b$ are integers, $b

eq0$): For any rational number $\frac{a}{b}$, its additive inverse is $-\frac{a}{b}$ (also rational), and its multiplicative inverse is $\frac{b}{a}$ (also rational, as long as $a
eq0$).

Answer:

D. Rational numbers