1. Number Concepts

This is Part 1 of a 6-part refresher series covering every domain of the TExES Mathematics 7-12 (235) exam. It's built for candidates who have already studied the material and just need a fast, structured recall before test day, not a first-time introduction. Today: real numbers, complex numbers, and number theory (Competencies 001-003).

• Place rational, irrational, and complex numbers correctly within the nested structure of the number system.
• Apply field properties, order axioms, and absolute value as distance on the number line.
• Perform arithmetic with complex numbers, including conjugates and modulus, and explain why the complex numbers are not an ordered field.
• Apply the Fundamental Theorem of Algebra to count and locate polynomial roots.
• Use core number theory tools: primes, GCF/LCM, divisibility rules, and modular arithmetic.

• The Real Number System
• The Complex Number System
• Number Theory
• Exercises

The Real Number System

The real numbers split into rationals and irrationals: \( \mathbb{R} = \mathbb{Q} \cup \mathbb{Q}^{c} \). A rational number is any terminating or repeating decimal, equivalently any ratio of integers. An irrational number never terminates and never repeats, like \( \sqrt{2} \), \( \pi \), or \( e \). These sets nest inside each other in a strict chain:

\[ \mathbb{N} \subset \mathbb{W} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \]

(naturals, whole numbers, integers, rationals, reals). One detail the exam likes to test: the TExES definition of the natural numbers is \( \{1, 2, 3, \dots\} \), so 0 is a whole number but not a natural number.

The real numbers form a field: they're closed under addition and multiplication, both operations are commutative and associative, multiplication distributes over addition, 0 and 1 are the additive and multiplicative identities, and every real number except 0 has a multiplicative inverse. They also satisfy the order axioms: trichotomy (for any \( a \) and \( b \), exactly one of \( a<b \), \( a=b \), \( a>b \) holds) and transitivity. Absolute value \( |a-b| \) measures the distance between \( a \) and \( b \) on the number line, which is why \( |x - 3| < 5 \) reads as "x is within 5 units of 3."

A classic point of confusion: \( 0.\overline{9} = 1 \) exactly, not "close to" 1. The cleanest proof is algebraic: let \( x = 0.\overline{9} \), so \( 10x = 9.\overline{9} \), and subtracting gives \( 9x = 9 \), so \( x = 1 \).

The Complex Number System

The imaginary unit is defined by \( i = \sqrt{-1} \), so \( i^2 = -1 \). A complex number \( a + bi \) has real part \( a \) and imaginary part \( b \). Powers of \( i \) cycle with period 4:

\[ i^1 = i, \quad i^2 = -1, \quad i^3 = -i, \quad i^4 = 1 \]

Add and subtract complex numbers by combining like terms. Multiply by distributing (FOIL) and replacing \( i^2 \) with \( -1 \). To divide, multiply numerator and denominator by the conjugate of the denominator, since the conjugate of \( a + bi \) is \( a - bi \), and their product is always a real number:

\[ (a+bi)(a-bi) = a^2 + b^2 \]

The complex numbers form a field, but not an ordered field: an expression like \( 3+2i > 1+5i \) has no mathematical meaning, since there is no total order on \( \mathbb{C} \) that respects the field operations. What complex numbers do have is a modulus, the distance from the origin in the complex plane: \( |a + bi| = \sqrt{a^2 + b^2} \).

The Fundamental Theorem of Algebra guarantees that a degree-\( n \) polynomial has exactly \( n \) complex roots, counting multiplicity. If the polynomial's coefficients are real, any non-real roots always come in conjugate pairs, so if \( 3+2i \) is a root, \( 3-2i \) must be one too.

Number Theory

A prime number has exactly two positive divisors, 1 and itself. By that definition, 1 is neither prime nor composite, and 2 is the only even prime. A useful shortcut connects the greatest common factor and least common multiple of two numbers:

\[ \gcd(a,b) \times \operatorname{lcm}(a,b) = a \times b \]

so you only ever need to compute one of them directly. Divisibility rules worth having memorized: 2 (even last digit), 3 (digit sum divisible by 3), 4 (last two digits divisible by 4), 5 (ends in 0 or 5), 6 (divisible by both 2 and 3), 9 (digit sum divisible by 9), and 10 (ends in 0).

Modular arithmetic writes \( a \equiv b \pmod{n} \) to mean \( n \) divides evenly into \( a - b \), which is the formal machinery behind clock arithmetic and remainders. Finally, the Fundamental Theorem of Arithmetic states that every integer greater than 1 has a unique prime factorization, up to the order the factors are written in.

Exercises

1. Which of the following is an irrational number? A) \( 0.\overline{45} \)   B) \( \sqrt{16} \)   C) \( \pi/2 \)   D) \( -7/3 \)
2. Simplify \( (3 + 2i)(1 - 4i) \). A) \( 3 - 8i^2 \)   B) \( 11 - 10i \)   C) \( 3 - 10i - 8i^2 \)   D) \( 11 + 10i \)
3. What is \( \gcd(48, 60) \times \operatorname{lcm}(48, 60) \)? A) 48   B) 60   C) 2,880   D) 240
4. A teacher wants students to justify why \( 0.\overline{9} = 1 \). Which argument is mathematically valid? A) It's just a rounding convention. B) Let \( x = 0.\overline{9} \), then \( 10x = 9.\overline{9} \), so \( 10x - x = 9 \), giving \( 9x=9 \), so \( x=1 \). C) \( 0.\overline{9} \) is the largest number less than 1. D) Since 1 has fewer digits, they can't be equal.
5. Which statement about complex numbers is true? A) Every polynomial equation with real coefficients has at least one real root. B) Complex numbers can be ordered the same way real numbers are. C) If \( 3+2i \) is a root of a polynomial with real coefficients, \( 3-2i \) must also be a root. D) \( i^4 = -1 \)

• 1. C. Pi is irrational, and dividing an irrational number by a nonzero rational number keeps it irrational. A is a repeating decimal (rational), B simplifies to 4 (rational), and D is a ratio of integers (rational).
• 2. B. \( 3(1) + 3(-4i) + 2i(1) + 2i(-4i) = 3 - 12i + 2i - 8i^2 = 3 - 10i - 8(-1) = 11 - 10i \).
• 3. C. \( \gcd \times \operatorname{lcm} \) always equals \( a \times b \), so \( 48 \times 60 = 2{,}880 \).
• 4. B. This is the standard algebraic proof. A, C, and D all reflect common student misconceptions about infinite decimals.
• 5. C. This is the Complex Conjugate Root Theorem. A is false (\( x^2+1=0 \) has no real root); B is false, there is no total order on \( \mathbb{C} \); D is false, \( i^4 = 1 \).

Continue to Part 2: Patterns and Algebra →, covering functions, quadratics, and everything through exponential, logarithmic, trigonometric, and calculus basics, the largest domain on the exam.