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Chapter 1: Problem 107

Find the reciprocal, or multiplicative inverse, if it exists. $$ 0 $$

Short Answer

Expert verified
The reciprocal of 0 does not exist.

Step by step solution

01

Define the Reciprocal

The reciprocal or multiplicative inverse of a number is a value that, when multiplied by the original number, results in 1. For a number \( x \), its reciprocal is \( \frac{1}{x} \).
02

Identify the Given Number

The given number is 0. We need to find the reciprocal of 0, if it exists.
03

Reciprocal of Zero

To find the reciprocal of 0, we set up the expression \( \frac{1}{0} \). Division by zero is undefined in mathematics because there's no number that we can multiply by 0 to get 1. Therefore, the reciprocal of 0 does not exist.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

multiplicative inverse
A multiplicative inverse, also called a reciprocal, is a number which when multiplied by the original number results in 1. For a number such as \( x \), its multiplicative inverse is \( \frac{1}{x} \). This means that \( x \times \frac{1}{x} = 1 \). This concept is crucial for understanding how division and multiplication are connected. It shows us how to 'undo' multiplication through division. For example, the multiplicative inverse of 2 is \( \frac{1}{2} \), because \( 2 \times \frac{1}{2} = 1 \). In simple terms, it's like flipping the number around.
undefined in mathematics
We sometimes encounter situations where mathematical operations do not produce a meaningful result. Such cases are termed as 'undefined' in mathematics. One common example is dividing by zero. Since there is no number that can be multiplied by 0 to get 1, the reciprocal of zero is undefined. Undefined results indicate the boundaries and limitations of mathematical rules. Whenever you come across an undefined operation, it usually tells you that the operation doesn't have a place in conventional arithmetic.
division by zero
Division by zero is a prohibited mathematical operation. When we try to divide a number by zero, we are essentially asking what number multiplied by 0 would give us the original number. But no number fits this requirement, because anything multiplied by 0 is always 0. This makes the division by zero undefined. It's like asking an impossible question in math. Think of it as trying to share something into zero groups, which doesn't make any sense. This is why any expression involving division by zero is considered undefined.

In summary:
  • Division by zero has no meaningful result.
  • The reciprocal of zero doesn't exist.
  • Such operations help highlight the constraints of mathematical logic.
This concept reminds us to be cautious and aware of these limitations when solving problems.

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Most popular questions from this chapter

Calculate using the rules for order of operations. If an expression is undefined, state this. $$ 19-\left(4+2 \cdot 3^{2}\right) $$

To the student and the instructor: Synthesis exercises are designed to challenge students to extend the concepts or skills studied in each section. Many synthesis exercises require the assimilation of skills and concepts from several sections. On a quiz, Mia answers \(6 \in \mathbb{Z}\) while Giovanni writes \(\\{6\\} \in \mathbb{Z} .\) Giovanni's answer does not receive full credit while Mia's does. Why?

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