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Chapter 1: Problem 126

Why is it that 0 has no multiplicative inverse?

Short Answer

Expert verified
0 has no multiplicative inverse because, by definition, a multiplicative inverse of a number \(x\) is a number that, when multiplied by \(x\), gives the result as 1. However, multiplying anything by 0 always results in 0, not 1. Therefore, 0 cannot have a multiplicative inverse.

Step by step solution

01

Understanding Multiplicative Inverse

A multiplicative inverse of a number \(x\) is a number such that when \(x\) is multiplied by this number, the result is 1. We denote the multiplicative inverse of \(x\) as \(x^{-1}\). So, basically, \(x^{-1} * x = 1\).
02

Why 0 has no Multiplicative Inverse

If 0 had a multiplicative inverse, let's say \(a\), then when we multiply 0 by \(a\), it should result in 1 (since that's the definition of multiplicative inverse). However, any number multiplied by 0 is always 0, not 1. Hence, it's impossible to find a number \(a\) that when multiplied by 0 results in 1. Therefore, 0 does not have a multiplicative inverse.

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