GACE Mathematics (6–12) (711)

The real number system encompasses all numbers that can be represented on a number line. Understanding the hierarchical structure of number classifications is fundamental for the GACE Mathematics exam.

Natural numbers are the set of positive integers used for counting: 1, 2, 3, 4, 5, and so on. This set is denoted by ℕ. Natural numbers do not include zero, negative numbers, fractions, or decimals. They are closed under addition and multiplication, meaning adding or multiplying any two natural numbers always produces another natural number.

Whole numbers extend natural numbers by including zero: 0, 1, 2, 3, 4, and so on. This set is denoted by W. Like natural numbers, whole numbers are closed under addition and multiplication but not under subtraction or division.

Integers include all whole numbers and their negative counterparts: ..., -3, -2, -1, 0, 1, 2, 3, .... This set is denoted by ℤ. Integers are closed under addition, subtraction, and multiplication. Division of integers does not always yield an integer result.

Rational numbers are numbers expressible as a ratio of two integers p/q where q ≠ 0. This set is denoted by ℚ. Rational numbers include all integers (since any integer n can be written as n/1), terminating decimals, and repeating decimals. Examples include 3/4, -2/5, 0.75, and 0.333... (which equals 1/3).

Key property: A decimal is rational if and only if it terminates or repeats. The decimal 0.142857142857... is rational because the pattern 142857 repeats indefinitely (this equals 1/7).

Irrational numbers cannot be expressed as a ratio of integers. Their decimal representations neither terminate nor repeat. Common examples include π (pi), e (Euler's number), √2, √3, and the golden ratio φ. The sum or product of a rational and an irrational number is always irrational. However, the sum or product of two irrational numbers may be rational (for example, √2 × √2 = 2).

Real numbers comprise all rational and irrational numbers. This set is denoted by ℝ. Every point on the number line corresponds to exactly one real number, and every real number corresponds to exactly one point on the number line. This is called the completeness property of real numbers.

Complex numbers extend the real number system to include solutions to equations like x² + 1 = 0. A complex number has the form a + bi, where a and b are real numbers and i is the imaginary unit defined by i² = -1.

In the complex number a + bi, the value a is called the real part and b is called the imaginary part. When b = 0, the complex number is simply a real number. When a = 0 and b ≠ 0, the number is called a pure imaginary number.

The powers of i follow a cyclical pattern: i¹ = i, i² = -1, i³ = -i, i⁴ = 1, i⁵ = i, and so on. To find iⁿ for any positive integer n, divide n by 4 and use the remainder: remainder 0 gives 1, remainder 1 gives i, remainder 2 gives -1, remainder 3 gives -i.

For addition: a + b = b + a. For multiplication: a × b = b × a. This property does not hold for subtraction or division.

For addition: (a + b) + c = a + (b + c). For multiplication: (a × b) × c = a × (b × c). This property allows regrouping without changing the result.

Multiplication distributes over addition: a(b + c) = ab + ac. This property connects addition and multiplication and is essential for algebraic manipulation.

The additive identity is 0 (a + 0 = a). The multiplicative identity is 1 (a × 1 = a).

The additive inverse of a is -a (a + (-a) = 0). The multiplicative inverse of a (where a ≠ 0) is 1/a (a × 1/a = 1).

A prime number is a natural number greater than 1 that has exactly two factors: 1 and itself. The first several prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29. Note that 2 is the only even prime number.

A composite number is a natural number greater than 1 that has more than two factors. Every composite number can be expressed as a unique product of prime factors (Fundamental Theorem of Arithmetic). For example, 84 = 2² × 3 × 7.

The number 1 is neither prime nor composite.

Quick divisibility tests help in factoring and simplifying:

Divisible by 2: Last digit is even (0, 2, 4, 6, 8). Divisible by 3: Sum of digits is divisible by 3. Divisible by 4: Last two digits form a number divisible by 4. Divisible by 5: Last digit is 0 or 5. Divisible by 6: Divisible by both 2 and 3. Divisible by 8: Last three digits form a number divisible by 8. Divisible by 9: Sum of digits is divisible by 9. Divisible by 10: Last digit is 0.

The greatest common divisor (GCD) of two or more integers is the largest positive integer that divides each of them. To find the GCD, identify all common prime factors and multiply them together using the lowest power of each.

The least common multiple (LCM) of two or more integers is the smallest positive integer that is divisible by each of them. To find the LCM, identify all prime factors and multiply them together using the highest power of each.

Relationship: For any two positive integers a and b, GCD(a, b) × LCM(a, b) = a × b.