Fraction Notation and Operations

Factors and Prime Factorizations

Understanding factors and prime factorization is essential for working with fractions and simplifying algebraic expressions.

• Prime Number: A prime number is a natural number greater than 1 that has exactly two different factors: 1 and itself.

• Composite Number: A composite number is a natural number greater than 1 that is not prime; it has more than two factors.

• Prime Factorization: The prime factorization of a composite number is a way of expressing the number as a product of prime numbers.

Examples:

• List the factors of 18: 1, 2, 3, 6, 9, 18

• Prime factorization of 18:

• Classify numbers: - 29: Prime - 4: Composite - 1: Neither prime nor composite

Practice:

• List the factors of 40: 1, 2, 4, 5, 8, 10, 20, 40

• Prime factorization of 40:

• Classify numbers: - 13: Prime - 16: Composite - 28: Composite

Multiplication, Division, and Simplification of Fractions

Fractions are multiplied, divided, and simplified using specific rules that help maintain equivalence and simplify calculations.

Multiplication of Fractions

• For any two fractions and :

• Multiply the numerators together and the denominators together.

Examples:

Practice:

Reciprocals (Multiplicative Inverses)

• Two numbers whose product is 1 are called reciprocals or multiplicative inverses.

• Example: The reciprocal of is because .

Division of Fractions

• For any two fractions and :

• To divide by a fraction, multiply by its reciprocal.

Examples:

Addition and Subtraction of Fractions

Adding and subtracting fractions requires a common denominator. If denominators are the same, add or subtract the numerators. If not, find a common denominator first.

Addition of Fractions

• For any two fractions and (same denominator):

Subtraction of Fractions

• For any two fractions and (same denominator):

• If denominators are different, find a common denominator using the identity property of 1.

Identity Property of 1

• For any number :

• This property allows us to multiply by a form of 1 (such as ) to create equivalent fractions with a common denominator.

Examples:

• (find common denominator: 60) ,

Practice:

• (common denominator: 12)

• (common denominator: 40) ,

Additional info: The above notes expand on the definitions, provide step-by-step examples, and clarify the process of finding common denominators, as is standard in beginning algebra textbooks.