College Algebra Foundations: Numbers, Operations, Exponents, Radicals, Logarithms, and Sequences

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: $a + b = b + a$

• Multiplication: $a \times b = b \times a$

Associative Properties

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

• Addition: $(a + b) + c = a + (b + c)$

• Multiplication: $(a \times b) \times c = a \times (b \times c)$

Identity Properties

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

• Additive Identity: $a + 0 = a$

• Multiplicative Identity: $a \times 1 = a$

Order of Operations

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

• Example: $30 \div 5 \times (4 - 2)^2 = 30 \div 5 \times 4 = 6 \times 4 = 24$

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: $40 = 2 \times 2 \times 2 \times 5$

Fractions

Definition of Fractions

A fraction represents a part of a whole, written as $\frac{a}{b}$ where $a$ is the numerator and $b$ is the denominator.

• Proper Fraction: Numerator is less than denominator (e.g., $\frac{3}{4}$).

• Improper Fraction: Numerator is greater than or equal to denominator (e.g., $\frac{7}{4}$).

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: $\frac{2}{5} + \frac{3}{10} = \frac{4}{10} + \frac{3}{10} = \frac{7}{10}$

Multiplying and Dividing Fractions

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

• Multiplication: $\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}$

• Division: $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$

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: $1\frac{3}{4} = \frac{7}{4}$

Exponents

Definition and Properties

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

• Product Rule: $a^m \times a^n = a^{m+n}$

• Quotient Rule: $\frac{a^m}{a^n} = a^{m-n}$

• Power Rule: $(a^m)^n = a^{mn}$

• Zero Rule: $a^0 = 1$ (for $a \neq 0$)

• Negative Exponent: $a^{-n} = \frac{1}{a^n}$

Examples and Applications

• Example: $2^3 = 2 \times 2 \times 2 = 8$

• Example: $\frac{2^5}{2^3} = 2^{5-3} = 2^2 = 4$

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: $\sqrt{a} \times \sqrt{b} = \sqrt{ab}$

• Quotient Rule: $\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$

Simplifying Radicals

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

• Example: $\sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}$

Logarithms

Definition and Properties

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

• Definition: $\log_b(x) = n$ means $b^n = x$

• Product Rule: $\log_b(xy) = \log_b(x) + \log_b(y)$

• Quotient Rule: $\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)$

• Power Rule: $\log_b(x^n) = n\log_b(x)$

• Change of Base Rule: $\log_b(x) = \frac{\log_k(x)}{\log_k(b)}$

Examples

• Example: $\log_{10}(100) = 2$ because $10^2 = 100$

• Example: $\log_2(8) = 3$ because $2^3 = 8$

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 $\frac{8}{12} = \frac{2}{3}$

Proportions

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

• Example: $\frac{a}{b} = \frac{c}{d}$

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: $5,000 = 5 \times 10^3$

• Example: $0.0042 = 4.2 \times 10^{-3}$

Sequences

Arithmetic Sequences

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

• General Term: $a_n = a_1 + (n-1)d$

• Sum of n Terms: $S_n = \frac{n}{2}(a_1 + a_n)$

• Example: For $a_1 = 3$, $d = 2$, $a_5 = 3 + (5-1)\times2 = 11$