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Chapter 0: Problem 21

The multiplicative inverse of a nonzero real number is also called the _____ of the number.

Short Answer

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reciprocal

Step by step solution

01

Define Multiplicative Inverse

The multiplicative inverse of a number is the value that, when multiplied with the original number, gives a product of 1. Mathematically, if we have a number a, its multiplicative inverse is 1/a, such that a * (1/a) = 1.
02

Identify Common Term for Multiplicative Inverse

The multiplicative inverse is commonly referred to using another term in mathematics. Review the standard terminology and definitions that describe this concept.
03

State the Term

The common term for the multiplicative inverse of a nonzero real number is the 'reciprocal.' The reciprocal of a number a is 1/a.

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Key Concepts

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

reciprocal
The term 'reciprocal' might sound complex, but it is quite simple. The reciprocal of a number is essentially the same as its multiplicative inverse. If you have a number, say 5, the reciprocal is obtained by dividing 1 by that number. So, the reciprocal of 5 is \(\frac{1}{5}\). This means when you multiply 5 and \(\frac{1}{5}\) together, you get 1. Reciprocals are a key concept in arithmetic and algebra, especially when dealing with fractions and division.
  • Reciprocal of 2 is \(\frac{1}{2}\)
  • Reciprocal of -3 is \(-\frac{1}{3}\)
  • Reciprocal of \(\frac{4}{5}\) is \(\frac{5}{4}\)
This concept helps simplify complex problems, turning division into multiplication.
real numbers
Real numbers include almost all the numbers you commonly use. They encompass natural numbers, whole numbers, integers, rational numbers, and irrational numbers. In short, all positive and negative numbers, including zero, and fractions and decimals, are real numbers.
Interestingly, all real numbers except zero have a reciprocal.
  • Whole numbers (e.g., 7, -4)
  • Fractions (e.g., \(\frac{3}{8}\), -\(\frac{5}{2}\))
  • Decimals (e.g., 0.25, -0.75)
Understanding real numbers helps in understanding the wider group of numbers and their properties.
product of 1
When discussing multiplicative inverses or reciprocals, the term 鈥榩roduct of 1鈥� is crucial. The product of a number and its reciprocal is always 1. This is why reciprocals are also known as multiplicative inverses. To illustrate, if we multiply any given number \(a\) and its reciprocal \(\(\frac{1}{a}\)\), the result will be 1: \[a \times \frac{1}{a} = 1\] It鈥檚 a straightforward yet powerful concept that can make multiplication and division more manageable. It serves as the backbone for simplifying and solving equations.
nonzero numbers
Nonzero numbers play a vital role in the concept of multiplicative inverses. A nonzero number is any number except zero. These numbers can be positive or negative.
The reason we exclude zero is that zero does not have a reciprocal. Dividing 1 by 0 is undefined in mathematics.
  • Examples of nonzero numbers: 2, -5, 0.7, -0.25
  • Examples of zero: 0
By focusing on nonzero numbers, the concept of reciprocals and multiplicative inverses becomes clearer and more practical for solving real-world math problems.

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Most popular questions from this chapter

The sum of a number and its additive inverse is ____.

The expression \(a^{m / n}\) can be written in radical notation as _________ provided that \(\sqrt[n]{a}\) is a real number.

Suppose that \(x\) represents the larger of two consecutive odd integers. a. Write a polynomial that represents the smaller integer. b. Write a polynomial that represents the sum of the two integers. Then simplify. c. Write a polynomial that represents the product of the two integers. Then simplify. d. Write a polynomial that represents the difference of the squares of the two integers. Then simplify.

On a piano, the note \(\mathrm{A} 440\) (located above Middle \(\mathrm{C}\) ) has a frequency of \(440 \mathrm{~Hz}\). Using the method of equal temperament, a piano tuner can determine the frequency (in \(\mathrm{Hz}\) ) of a note \(n\) keys above or below \(A 440\) by the formula \(f=440 \cdot 2^{n / 12}\) where \(n\) is an integer. a. Use \(n=1,2,\) and 3 to determine the frequencies of the next three notes above \(\mathrm{A} 440 .\) Round to 1 decimal place. b. Determine the frequency of Middle \(\mathrm{C}\) if Middle \(\mathrm{C}\) is located nine notes below A440.

Evaluate each expression for the given values of the variables. \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\) for \(x_{1}=-1.4, x_{2}=2, y_{1}=3.1,\) \(y_{2}=-3.7\)
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