Problem 86 - Beginning and Intermediate Algebra

Question

Fill in the table with the opposite (additive inverse), and the reciprocal (multiplicative inverse). Assume that the value of each expression is not 0 $$ \frac{2}{3} $$

Step-by-Step Solution

Answer

Opposite: \(-\frac{2}{3}\), Reciprocal: \(\frac{3}{2}\).

Step 1: Define Opposite

The opposite (additive inverse) of a number is what you add to the number to get zero. For a fraction \( \frac{a}{b} \), the opposite is \( -\frac{a}{b} \). For \( \frac{2}{3} \), the opposite is \( -\frac{2}{3} \).

Step 2: Define Reciprocal

The reciprocal (multiplicative inverse) of a number is what you multiply by the number to get one. For a fraction \( \frac{a}{b} \), the reciprocal is \( \frac{b}{a} \). For \( \frac{2}{3} \), the reciprocal is \( \frac{3}{2} \).

Step 3: Fill in the Table

Using the results from Steps 1 and 2, you can fill in the table. The opposite of \( \frac{2}{3} \) is \( -\frac{2}{3} \). The reciprocal of \( \frac{2}{3} \) is \( \frac{3}{2} \).

Key Concepts

Additive Inverse

An additive inverse is an important concept in mathematics. It helps us understand how to bring numbers back to zero. This process is like balancing a scale. For any number, the additive inverse is the value that, when added to the original number, equals zero. This concept is very handy in solving equations and helps in understanding the properties of numbers.

- For a whole number, say 5, its additive inverse is -5. That’s because adding 5 and -5 gives us 0.- When it comes to fractions like \( \frac{2}{3} \), the additive inverse is \( -\frac{2}{3} \). Therefore, \( \frac{2}{3} + (-\frac{2}{3}) = 0 \).

Additive inverses highlight the idea of getting back to a starting point, as zero represents a neutral position in mathematics. Knowing how to find the additive inverse is particularly useful when you need to solve equations or simplify expressions.

Multiplicative Inverse

The multiplicative inverse might sound complex, but it's a simple concept: it's the number you multiply by to get one. This is crucial in algebra and ensures we can "cancel out" values in equations, bringing us back to a neutral one.

- For a regular number, like 5, the multiplicative inverse is \( \frac{1}{5} \) since \( 5 \times \frac{1}{5} = 1 \).- When dealing with fractions, such as \( \frac{2}{3} \), its multiplicative inverse is \( \frac{3}{2} \). Here, multiplying \( \frac{2}{3} \times \frac{3}{2} \) equals 1.

Understanding this inverse has many practical applications. It helps solve equations, especially when fractions are involved. By multiplying both sides of an equation by the reciprocal of a fraction, we can isolate the variable and simplify our solution.

Fractions

Fractions might seem tricky, but they're just a way of representing parts of a whole. A fraction consists of two numbers: the numerator and the denominator. The numerator is the top part, representing how many parts we have, while the denominator, at the bottom, shows the total parts that make up a whole.

- A fraction like \( \frac{2}{3} \) means we have 2 out of 3 equal parts.- Understanding fractions involves recognizing that they can represent numbers less than one as well as more than one.

Fractions are also essential in understanding other math concepts like additive and multiplicative inverses. They teach us how to divide numbers into equal parts. Learning how to work with fractions, including performing operations like addition, subtraction, multiplication, and division, is fundamental for mastering math.