Review of Algebra

Real Numbers

Real numbers include all numbers that can be found on the number line, encompassing positive, negative, zero, fractions, and decimals.

• Definition: Any number on the number line, including rational and irrational numbers.

• Examples: -3, 0, 2.5, 1/2, √2

Integers

Integers are whole numbers that can be positive, negative, or zero. They do not include fractions or decimals.

• Examples: ..., -3, -2, -1, 0, 1, 2, 3, ...

Commutative Properties

The commutative property states that the order of addition or multiplication does not affect the result.

• Addition:

• Multiplication:

Associative Properties

The associative property states that when adding or multiplying three or more numbers, the grouping does not affect the result.

• Addition:

• Multiplication:

Identity Properties

The identity property refers to the existence of an identity element for addition (0) and multiplication (1).

• Additive Identity:

• Multiplicative Identity:

Order of Operations

Mathematical expressions must be evaluated in a specific order: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction (PEMDAS).

• Example:

Prime Numbers

Prime numbers are natural numbers greater than 1 that have only two factors: 1 and themselves.

• Examples between 60 and 70: 61, 67

Prime Factorization

Every composite number can be written as a product of prime numbers.

• Example:

Fractions

Definition of Fractions

A fraction represents a part of a whole, written as where is the numerator and is the denominator.

• Proper Fraction: Numerator is less than denominator (e.g., ).

• Improper Fraction: Numerator is greater than or equal to denominator (e.g., ).

Adding and Subtracting Fractions

To add or subtract fractions, a common denominator is required.

1. Find the least common denominator (LCD).

2. Rewrite each fraction with the LCD.

3. Add or subtract the numerators, keep the denominator.

• Example:

Multiplying and Dividing Fractions

To multiply fractions, multiply numerators and denominators. To divide, multiply by the reciprocal of the divisor.

• Multiplication:

• Division:

Mixed and Improper Fractions

Mixed fractions combine a whole number and a fraction. Improper fractions have numerators greater than denominators.

• Converting Mixed to Improper: Multiply the whole number by the denominator, add the numerator, place over denominator.

• Example:

Exponents

Definition and Properties

An exponent shows how many times to use a number in multiplication.

• Product Rule:

• Quotient Rule:

• Power Rule:

• Zero Rule: (for )

• Negative Exponent:

Examples and Applications

• Example:

• Example:

Radicals

Definition and Properties

A radical expression involves roots, such as square roots or cube roots.

• Principal Root: The non-negative root for even indices.

• Product Rule:

• Quotient Rule:

Simplifying Radicals

Write the factorization of the number under the radical and simplify by taking out perfect squares.

• Example:

Logarithms

Definition and Properties

A logarithm is the power to which a base must be raised to yield a given number.

• Definition: means

• Product Rule:

• Quotient Rule:

• Power Rule:

• Change of Base Rule:

Examples

• Example: because

• Example: because

Decimals

Operations with Decimals

Decimals are added and subtracted by aligning decimal points. Multiplication and division require counting decimal places and adjusting the result accordingly.

• Multiplication: Multiply as whole numbers, then count total decimal places in the factors for the product.

• Division: Move the decimal in both numbers to make the divisor a whole number, then divide as usual.

Ratios and Proportions

Ratios

A ratio compares two quantities, showing the relative size of one to the other.

• Example: The ratio of 8 to 12 is

Proportions

A proportion is an equation stating that two ratios are equal.

• Example:

Scientific Notation

Definition and Conversion

Scientific notation expresses numbers as a product of a number between 1 and 10 and a power of 10.

• Example:

• Example:

Sequences

Arithmetic Sequences

An arithmetic sequence is a list of numbers with a constant difference between consecutive terms.

• General Term:

• Sum of n Terms:

• Example: For , ,