What is the multiplicative inverse of 2?
The Correct Answer and Explanation is:
ChatGPT said:
The multiplicative inverse of a number aaa is the number that, when multiplied by aaa, gives the result 1. In mathematical terms, the multiplicative inverse of aaa is denoted as a−1a^{-1}a−1, and it satisfies the equation:a×a−1=1a \times a^{-1} = 1a×a−1=1
For the number 2, we need to find the number xxx such that:2×x=12 \times x = 12×x=1
To solve for xxx, we divide both sides of the equation by 2:x=12x = \frac{1}{2}x=21
Thus, the multiplicative inverse of 2 is 12\frac{1}{2}21.
Why is this the multiplicative inverse?
When we multiply 2 by 12\frac{1}{2}21, we get:2×12=12 \times \frac{1}{2} = 12×21=1
This satisfies the definition of a multiplicative inverse because the result of the multiplication is 1, which is the identity element for multiplication.
General Concept
The concept of a multiplicative inverse extends beyond just the number 2. For any non-zero number aaa, its multiplicative inverse is given by:a−1=1aa^{-1} = \frac{1}{a}a−1=a1
This holds true for all real numbers (except zero), and it’s essential in algebra and number theory, especially when solving equations or simplifying expressions.
In summary, the multiplicative inverse of 2 is 12\frac{1}{2}21, and this property is part of the broader concept of inverses in mathematics.
