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Chapter 3: Problem 15

(GRAPH CANNOT COPY) Find the negative reciprocal of each number. a. 6 b. \(-\frac{7}{8}\) c. \(-1\)

Short Answer

Expert verified
a. \(-\frac{1}{6}\), b. \(\frac{8}{7}\), c. 1

Step by step solution

01

Understanding Reciprocals

A reciprocal of a number is calculated by taking the number 1 and dividing it by that given number. For example, the reciprocal of 6 is \( \frac{1}{6} \).
02

Understanding Negative Reciprocals

Unlike a simple reciprocal, a negative reciprocal involves taking the reciprocal of a number and then multiplying it by -1. So, if the reciprocal of a number is \( \frac{1}{x} \), the negative reciprocal would be \( -\frac{1}{x} \).
03

Calculate Negative Reciprocal of a

For 6, the reciprocal is \( \frac{1}{6} \), so the negative reciprocal is \( -\frac{1}{6} \).
04

Calculate Negative Reciprocal of b

For \(-\frac{7}{8}\), the reciprocal is \( -\frac{8}{7} \). By multiplying it by -1, the negative reciprocal becomes \( \frac{8}{7} \).
05

Calculate Negative Reciprocal of c

For \(-1\), the reciprocal is \(-1\) (since \( \frac{1}{-1} = -1 \)). The negative reciprocal, consequently, is \( 1 \).

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Key Concepts

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

Reciprocals
Reciprocals can sound tricky, but they are actually quite simple! A reciprocal of a number is what you multiply the original number with to get a product of 1. Basically, if you have a number, say 6, you flip it upside down to find its reciprocal.
So, 6 becomes \( \frac{1}{6} \). Here鈥檚 the key point:
  • For any whole number, like 6, the reciprocal is 1 divided by that number, which makes it a fraction.
  • Fractions are a bit trickier; you switch the numerator and denominator. For example, the reciprocal of \( \frac{3}{4} \) is \( \frac{4}{3} \).
  • The reciprocal of 1 is still 1. If you go to 0, watch out! There is no reciprocal because you can鈥檛 divide by zero.
Knowing the reciprocal helps in dividing fractions and in many algebraic equations. It really is the flip-side of the original number!
Multiplying Fractions by -1
When we talk about negative reciprocals, we need to understand what happens when multiplying fractions by -1. Multiplying a fraction by -1 changes its sign. Here鈥檚 how it works:
  • If the fraction is positive, it becomes negative, and vice versa. For example, multiplying \( \frac{3}{4} \) by -1 gives \( -\frac{3}{4} \).
  • If you have a negative fraction, like \( -\frac{7}{8} \), multiplying by -1 flips it to positive, giving you \( \frac{7}{8} \).
  • This step is crucial when finding negative reciprocals because after finding a fraction鈥檚 reciprocal, multiplying by -1 is what gives it the negative spin!
This simple step ensures we complete the process of finding the negative reciprocal correctly. It鈥檚 all about keeping the rule of changing the sign!
Reciprocal Calculation Steps
Calculation of negative reciprocals is a sequence of clear steps that help unlock this mathematical concept. Let鈥檚 look at how exactly this is done:
  • Start with the original number, like 6. First, find its reciprocal: divide 1 by that number. For 6, the reciprocal is \( \frac{1}{6} \). Now, make it negative, resulting in \( -\frac{1}{6} \).
  • For fractions like \(-\frac{7}{8}\), flip the fraction to get \(-\frac{8}{7}\). Then, multiply by -1 to shift the signs. The result is \( \frac{8}{7} \).
  • With 鈭�1, the reciprocal is still 鈭�1 because \( \frac{1}{-1} = -1 \). When you put it through the negative spin by multiplying by -1, it becomes 1.
Breaking down the steps this way aids in understanding. The pattern to follow is clear: reciprocal first, then negative! This orderly approach makes and keeps calculations straightforward. Whether for cross-referencing or solving equations, following these steps ensures you never get lost in the math jungle!

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Most popular questions from this chapter

Find the slope and the y-intercept of the graph of each equation and graph it. See Examples 4 and 5. $$ 2 x+y=-6 $$

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