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Chapter 7: Problem 413

For the number \(\frac{2}{5}\) find the (a) additive inverse (b) multiplicative inverse.

Short Answer

Expert verified
(a) Additive inverse: \(-\frac{2}{5}\), (b) Multiplicative inverse: \(\frac{5}{2}\).

Step by step solution

01

Understanding Inverses

To solve this problem, one needs to understand the concepts of additive and multiplicative inverses. The additive inverse of a number is what you add to it to get zero. The multiplicative inverse of a number is what you multiply by it to get one.
02

Finding the Additive Inverse

The additive inverse of \(\frac{2}{5}\) is the number that, when added to \(\frac{2}{5}\), equals zero. This number is \(-\frac{2}{5}\). Therefore, the additive inverse is \(-\frac{2}{5}\).
03

Finding the Multiplicative Inverse

The multiplicative inverse of \(\frac{2}{5}\) is the number that, when multiplied by \(\frac{2}{5}\), equals one. This number is the reciprocal of \(\frac{2}{5}\), which means flipping the numerator and the denominator. Therefore, the multiplicative inverse is \(\frac{5}{2}\).

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Key Concepts

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

Understanding the Additive Inverse
To find an additive inverse, think about what number you need to add to your original number to make the sum zero. For any number, its additive inverse is simply the negative of that number.

For example, the additive inverse of \(\frac{2}{5}\) is \(-\frac{2}{5}\).

When you add \(\frac{2}{5}\) and \(-\frac{2}{5}\) together, the result is zero: \(\frac{2}{5} + -\frac{2}{5} = 0\).

This is a crucial concept in arithmetic and algebra because it helps balance equations and solve for unknowns.

Remember:
  • The additive inverse of a positive number is its negative counterpart.
  • The additive inverse of a negative number is its positive counterpart.
  • Zero is its own additive inverse because \(0 + 0 = 0\).
Understanding the Multiplicative Inverse
The multiplicative inverse is also known as the reciprocal. To find the multiplicative inverse of a number, you need to identify the number that, when multiplied by your original number, gives the product as one.

For a fraction \(\frac{a}{b}\), its multiplicative inverse is \(\frac{b}{a}\).

For example, the multiplicative inverse (or reciprocal) of \(\frac{2}{5}\) is \(\frac{5}{2}\). When you multiply these together, you get one: \(\frac{2}{5} \times \frac{5}{2} = 1\).

Some key points to remember:
  • The multiplicative inverse of a fraction \(\frac{2}{5}\) is found by flipping the numerator and the denominator.
  • The multiplicative inverse of a whole number (like 3) is its reciprocal \(\frac{1}{3}\).
  • The number one is its own multiplicative inverse because \(1 \times 1 = 1\).
Understanding Reciprocals
A reciprocal is another name for the multiplicative inverse of a number. It is the number you multiply by your original number to get a product of one.

For instance, \(\frac{2}{5}\) and \(\frac{5}{2}\) are reciprocals because \(\frac{2}{5} \times \frac{5}{2} = 1\).

To find the reciprocal of a fraction, simply swap the numerator and the denominator.

Reciprocals are essential in solving equations that involve fractions and division.

Key points to consider:
  • Multiplying a fraction by its reciprocal always gives one.
  • The reciprocal of zero does not exist because division by zero is undefined.
  • To simplify division of fractions, multiply by the reciprocal of the divisor.

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Most popular questions from this chapter

In the following exercises, solve, and state your answer in mixed units. Every day last week, Pedro recorded the amount of time he spent reading. He read for \(50,25,83,45,32,60,\) and 135 minutes. How much time, in hours and minutes, did Pedro spend reading?

In the following exercises, simplify. $$ \frac{1}{6} \cdot 17 \cdot 12 $$

In the following exercises, simplify the given expression. $$ \frac{0}{8} $$

In the following exercises, identify whether each example is using the identity of addition or multiplication. $$ -35(1)=-35 $$

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