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Chapter 1: Problem 4

Give the reciprocal of each number. $$ -5 / 13 $$

Short Answer

Expert verified
The reciprocal of \(-\frac{5}{13}\) is \(-\frac{13}{5}\).

Step by step solution

01

Identify the Original Fraction

The original number given in the problem is a fraction: \(-\frac{5}{13}\). To find the reciprocal, we need to understand this as a fraction, which it already is.
02

Understanding the Concept of a Reciprocal

The reciprocal of a number is what you multiply that number by to get 1. For a fraction, the reciprocal is found by swapping the numerator and the denominator.
03

Swap the Numerator and Denominator

To find the reciprocal of \(-\frac{5}{13}\), swap the denominator and numerator to get \(-\frac{13}{5}\).
04

Sign of the Reciprocal

The negative sign is part of the numerator in the original fraction. When swapping the numerator and the denominator in a reciprocal, the negative sign remains with the numerator. Hence, the reciprocal is \(-\frac{13}{5}\).

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Key Concepts

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

Fraction
A fraction represents a part of a whole or a division of a quantity. For instance, a fraction like \(\frac{5}{13}\) indicates that 5 parts out of a total of 13 are considered. Here, 5 is the numerator and 13 is the denominator. They form a ratio, effectively displaying how many parts we have versus the whole quantity.
  • The numerator (top number) states how many parts are being taken into account.
  • The denominator (bottom number) clarifies the total number of equal parts the whole can be divided into.
Fractions can also be negative, like \(-\frac{5}{13}\). This signifies that the value is less than zero, indicating a deficiency or a direction opposite to the positive. This concept is crucial not only in arithmetic calculations but also in understanding proportions and ratios in everyday contexts.
Reciprocal
The reciprocal of a fraction is quite an interesting concept. Simply put, it is what you need to multiply a number by in order to get 1. In the case of fractions, the reciprocal is found by flipping the numerator and denominator.
  • If you have \(\frac{a}{b}\), then its reciprocal is \(\frac{b}{a}\).
  • This means the positions of the numerator and the denominator swap.
So, flipping \(-\frac{5}{13}\) would give us \(-\frac{13}{5}\). By understanding reciprocals, you gain insight into division and multiplication, especially when dividing fractions, as multiplying by a reciprocal is equivalent to dividing a number.
Negative Sign in Fractions
When it comes to fractions, the presence of a negative sign can change the whole meaning. It indicates that the fraction is less than zero. This negative sign has an important role even when working with reciprocals.
  • In \(-\frac{5}{13}\), the negative is associated with the numerator.
  • When you find the reciprocal, this negative sign does not go away. It stays with the numerator.
Therefore, the reciprocal of \(-\frac{5}{13}\) becomes \(-\frac{13}{5}\), maintaining the negative aspect of the original fraction. This concept of keeping track of the negative sign helps ensure accuracy in your calculations, especially in situations involving both positive and negative values.

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