Operations on Numbers: Foundations for College Algebra

Operations on Numbers

Introduction

Understanding basic operations on numbers is essential for success in College Algebra. These operations include addition, subtraction, multiplication, and division, and are applied to both numbers and variables. This section introduces key vocabulary and rules for working with positive and negative numbers, fractions, and algebraic expressions.

Vocabulary and Key Concepts

• Numerator: The top part of a fraction, representing the quantity being divided.

• Denominator: The bottom part of a fraction, representing the quantity by which the numerator is divided.

• Expression: A term or combination of terms, such as $3x + 2$.

• Least Common Multiple (LCM): The smallest number that two or more numbers divide into evenly. For example, the LCM of 2 and 5 is 10.

• Reciprocal: The reciprocal of a number $a$ is $\frac{1}{a}$.

Sign Rules for Multiplying Positive and Negative Numbers

• Multiplying two positive numbers yields a positive result.

• Multiplying two negative numbers yields a positive result.

• Multiplying a positive and a negative number yields a negative result.

Symbols in Arithmetic and Algebra

• Arithmetic uses $\times$ and $\div$ for multiplication and division.

• Algebra uses $\cdot$ or parentheses for multiplication, e.g., $a \cdot b$ or $(a)(b)$.

• The negative sign for a fraction can be placed in front of the fraction, in the numerator, or in the denominator: $-\frac{a}{b} = \frac{-a}{b} = \frac{a}{-b}$.

Adding Positive and Negative Fractions

When fractions have the same denominator, add their numerators and keep the denominator unchanged.

• Example: $\frac{2}{5} + \frac{3}{5} = \frac{2+3}{5} = \frac{5}{5} = 1$

• If denominators differ, find the least common multiple and rewrite each fraction with the common denominator before adding.

Subtracting Fractions with Double Negatives

Subtracting a negative fraction is equivalent to adding its positive counterpart.

• Example: $\frac{4}{7} - (-\frac{2}{7}) = \frac{4}{7} + \frac{2}{7} = \frac{6}{7}$

• A double negative becomes a positive.

Multiplying Positive and Negative Fractions

To multiply fractions, multiply their numerators and denominators separately.

• Example: $\frac{2}{3} \times \frac{-4}{5} = \frac{2 \times -4}{3 \times 5} = \frac{-8}{15}$

• Multiplying two negative fractions yields a positive result.

Dividing Positive and Negative Fractions

Dividing by a fraction is the same as multiplying by its reciprocal.

• Example: $\frac{3}{4} \div \frac{2}{5} = \frac{3}{4} \times \frac{5}{2} = \frac{15}{8}$

• "Flip and multiply": Invert the divisor and multiply.

Order of Operations with Positive and Negative Numbers

Follow the order of operations, often remembered by the acronym PEMDAS:

• P: Parentheses

• E: Exponents

• M: Multiplication

• D: Division

• A: Addition

• S: Subtraction

Example: $2 + 3 \times (4 - 1)^2 = 2 + 3 \times 3^2 = 2 + 3 \times 9 = 2 + 27 = 29$

Evaluating Expressions with Integers

To evaluate an algebraic expression, substitute given values for variables, often using parentheses to avoid errors.

• Example: If $x = -2$, evaluate $3x^2 + 5$:

• $3(-2)^2 + 5 = 3 \times 4 + 5 = 12 + 5 = 17$

• Always use parentheses when substituting negative numbers.

Summary Table: Fraction Operations

Operation	Rule	Example
Addition	Add numerators, keep denominator (if same denominator)	$\frac{1}{4} + \frac{3}{4} = \frac{4}{4} = 1$
Subtraction	Subtract numerators, keep denominator (if same denominator)	$\frac{5}{6} - \frac{2}{6} = \frac{3}{6} = \frac{1}{2}$
Multiplication	Multiply numerators and denominators	$\frac{2}{3} \times \frac{4}{5} = \frac{8}{15}$
Division	Multiply by reciprocal of divisor	$\frac{3}{4} \div \frac{2}{5} = \frac{3}{4} \times \frac{5}{2} = \frac{15}{8}$

Additional info: This section provides foundational skills for College Algebra, including operations with fractions, integers, and variables, as well as the order of operations. Mastery of these concepts is essential for solving equations, working with functions, and understanding more advanced algebraic topics.