Problem 60 - Algebra 2

Question

Find all values of \(\pm \frac{a}{b}\) given each replacement set. \(a=\\{1,2\\} ; b=\\{1,2,7,14\\}\)

Step-by-Step Solution

Answer

The values are \( \pm 1, \pm 2, \pm \frac{1}{2}, \pm \frac{1}{7}, \pm \frac{1}{14}, \pm \frac{2}{7} \).

Step 1: Understand the Expression

We need to find all values of \( \pm \frac{a}{b} \) using the given sets \( a = \{1, 2\} \) and \( b = \{1, 2, 7, 14\} \). This means we will form fractions with each \( a \) and divide by each \( b \), then consider both the positive and negative of each of these fractions.

Step 2: List Fractions for Each a

Start by making fractions using each value of \( a \):For \( a = 1 \):- \( \frac{1}{1}, \frac{1}{2}, \frac{1}{7}, \frac{1}{14} \)For \( a = 2 \):- \( \frac{2}{1}, \frac{2}{2}, \frac{2}{7}, \frac{2}{14} \)

Step 3: Consider Positive and Negative Values

For each fraction obtained, determine both the positive and the negative values:- From \( \frac{1}{1} \): \( \pm 1 \)- From \( \frac{1}{2} \): \( \pm \frac{1}{2} \)- From \( \frac{1}{7} \): \( \pm \frac{1}{7} \)- From \( \frac{1}{14} \): \( \pm \frac{1}{14} \)- From \( \frac{2}{1} \): \( \pm 2 \)- From \( \frac{2}{2} \): \( \pm 1 \)- From \( \frac{2}{7} \): \( \pm \frac{2}{7} \)- From \( \frac{2}{14} \): \( \pm \frac{1}{7} \) (simplified)

Step 4: Combine and Eliminate Duplicates

Combine all unique values found from the fractions:- Unique positive fractions: 1, \( \frac{1}{2}, \frac{1}{7}, \frac{1}{14}, 2, \frac{2}{7} \)- Consider both positive and negative: \( \pm 1, \pm \frac{1}{2}, \pm \frac{1}{7}, \pm \frac{1}{14}, \pm 2, \pm \frac{2}{7} \)

Key Concepts

Rational Numbers

A rational number is any number that can be expressed as the quotient or fraction \( \frac{a}{b} \) of two integers, where the numerator \( a \) and the denominator \( b \) have no common factors other than 1, and the denominator \( b \) is not zero. This is what categorizes them as rational—they can be written as a ratio.

• The numerator ( is the top part of the fraction, representing parts of the whole.
• The denominator ( is the bottom part of the fraction, indicating into how many parts the whole is divided.

Rational numbers can be positive or negative, depending on the signs of the numerator and denominator.
If both the numerator and denominator have the same sign, the fraction will be positive. If they have opposite signs, the fraction will be negative. This makes it possible for rational numbers to span all sections of the number line, offering flexibility in number representation.

Positive and Negative Fractions

Fractions can be either positive or negative, which highly depends on the sign of the numerator and the denominator.

• When both the numerator and denominator are positive, or both are negative, the fraction is positive.
• If one is positive and the other is negative, the fraction is negative.

For example, let’s examine \( \frac{1}{2} \):

• The fraction is positive when both 1 (numerator) and 2 (denominator) are positive: \( \frac{1}{2} = +\frac{1}{2} \).
• The fraction becomes negative if we change the sign of either the numerator or the denominator: \( -\frac{1}{2} = \frac{-1}{2} \) or \( \frac{1}{-2} \).

Understanding the signs of fractions helps in identifying positive and negative values during calculations, such as the case in our original exercise where each fraction of \( a \) over \( b \) can either be positive or negative.

Number Sets

Number sets are crucial in understanding the classification of numbers and how they fit into different categories in mathematics. In this context, we consider the sets used in the exercise to form fractions.

• The set of numbers \( a \) was given as \{1, 2\}, and \( b \) as \{1, 2, 7, 14\}.
• The idea is to pair each number from set \( a \) with each number from set \( b \) to form a fraction.

The concept of sets helps us organize numbers into collections, making it easier to perform operations and derive all possible outcomes.
Knowing number sets allows straightforward permutations of numbers, making our task predictable and systematic. This understanding is essential as it helps manage scenarios like dealing with several fractions, as shown in the fractions found when different \( a \) and \( b \) are combined. It simplifies the process of forming rational numbers in various exercises and real-life situations.