CSET Mathematics Exam (211, 212, 213)

Subtest I does not allow a calculator. Every question in this domain is built to be worked by hand, so expect clean numbers, exact answers such as 3 sqrt(2) or 4/11, and items that test whether you understand why a property holds rather than how fast you can compute. The constructed-response assignment is often a short proof or a justified computation, such as an induction argument or a Euclidean algorithm calculation with a linear combination.

This chapter covers Subtest I requirement 0001, The Real and Complex Number Systems (SMR 1.1). It reviews the properties of the real numbers and their subsets, equivalent forms of numbers (fractions, decimals, exponents, radicals), building expressions for real situations, proportional reasoning, units and dimensional analysis, and the arithmetic and geometry of complex numbers.

The number sets nest inside one another. The natural numbers N = {1, 2, 3, ...} sit inside the whole numbers {0, 1, 2, ...}, which sit inside the integers Z = {..., -2, -1, 0, 1, 2, ...}. The rational numbers Q are all numbers of the form a/b with a and b integers and b ≠ 0. The real numbers R are all points on the number line; the reals that are not rational are the irrational numbers, such as sqrt(2), pi, and e. The complex numbers C contain R as the numbers a + bi with b = 0.

A number is rational exactly when its decimal expansion terminates or eventually repeats. A terminating decimal such as 0.375 equals 375/1000 = 3/8. A repeating decimal can always be converted to a fraction with the subtraction trick shown in the worked example that follows. A decimal that neither terminates nor repeats, such as 0.101001000100001..., is irrational.

Worked Example: Write 0.363636... (the block 36 repeating) as a fraction in lowest terms. Let x = 0.363636....

The repeating block has two digits, so multiply by 100: 100x = 36.363636....

For a decimal with a non-repeating start, such as 0.1666..., multiply to move the decimal point past the non-repeating part and then past one full block: 100x = 16.666... and 10x = 1.666..., so 90x = 15 and x = 1/6.

The real numbers satisfy the field axioms under addition and multiplication. For all real a, b, c:

Field Axioms for the Real Numbers

The reals also satisfy the order axioms (making R an ordered field, covered in Chapter 2.1) and the completeness axiom: every nonempty set of reals that is bounded above has a least upper bound. Completeness is what separates R from Q. The set of rationals whose square is less than 2 is bounded above in Q but has no least upper bound in Q, because its supremum would have to be sqrt(2).

Both Q and the irrationals are dense: between any two distinct real numbers there is a rational number and an irrational number. For example, between a and b the average (a + b)/2 lies strictly between them, and if a and b are rational so is their average.

Which subsets are closed under which operations is a favorite exam question.

Closure of Common Subsets

For a > 0 and rational exponents, the laws of exponents are a^m · a^n = a^(m+n), a^m / a^n = a^(m-n), (a^m)^n = a^(mn), (ab)^n = a^n b^n, a^0 = 1, and a^(-n) = 1/a^n. A rational exponent combines a power and a root: a^(m/n) = (the nth root of a)^m = the nth root of a^m.

Multiply numerator and denominator by the conjugate 3 + sqrt(3).

The denominator becomes (3: sqrt(3))(3 + sqrt(3)) = 9 - 3 = 6.

The numerator becomes 6(3 + sqrt(3)).

Writing an expression or equation for a situation means naming variables, stating their units, and translating each relationship. A quantity y is proportional to x when y = kx for a constant k (the constant of proportionality). The graph of a proportional relationship is a line through the origin, and the ratio y/x is constant. The relationship y = kx + b with b ≠ 0 is linear but not proportional. Inverse variation, y = k/x, has a constant product xy.

Worked Example: On a map, 2 inches represent 5 miles. How many miles do 7.2 inches represent, and what is the constant of proportionality?

Let d be miles and m be map inches. Then d = km.

From 5 = k · 2, k = 2.5 miles per inch. Substitute m = 7.2: d = 2.5 × 7.2 = 18.

Result: 7.2 inches represent 18 miles, with k = 2.5 miles per inch.

Dimensional analysis treats units as algebraic factors. Multiply by conversion factors equal to 1 (such as 5280 feet / 1 mile) arranged so unwanted units cancel. Checking that the units of an answer make sense is a fast way to catch a setup error.

Worked Example: Convert 60 miles per hour to feet per second.

Write the rate with units: (60 miles / 1 hour).

Multiply by (5280 feet / 1 mile) so miles cancel.

Multiply by (1 hour / 3600 seconds) so hours cancel.

Compute: 60 × 5280 / 3600 = 316,800 / 3600 = 88.

Result: 60 miles per hour = 88 feet per second.

Quantitative reasoning also means choosing an appropriate level of precision and unit. Squared and cubed units convert with squared and cubed factors: 1 square yard = 9 square feet, and 1 cubic meter = 1,000,000 cubic centimeters because (100 cm)^3 = 10^6 cm^3.

Common Trap: Converting area or volume with the linear factor. A room of 30 square yards is 270 square feet, not 90.

The imaginary unit i satisfies i^2 = -1. A complex number is z = a + bi with a, b real; a is the real part and b is the imaginary part. Two complex numbers are equal exactly when their real parts are equal and their imaginary parts are equal.

Operations follow the usual algebra, replacing i^2 with -1:

Multiply numerator and denominator by 1 + 2i, the conjugate of the denominator.

Properties of the conjugate are used constantly in proofs (including the Conjugate Root Theorem in Chapter 2.2):

Conjugate and Modulus Facts

The complex number a + bi is plotted as the point (a, b), with the real axis horizontal and the imaginary axis vertical. This picture turns algebra into geometry:

|z| is the distance from z to the origin, and |z - w| is the distance between z and w.

Addition is vector addition (the parallelogram rule).

The conjugate z̄ is the reflection of z across the real axis; -z is the reflection through the origin.

Multiplying by i rotates a point 90° counterclockwise about the origin: i(a + bi) = -b + ai, so (a, b) goes to (-b, a).

In general, multiplying two complex numbers multiplies their moduli and adds their angles (arguments). Polar form and De Moivre's theorem are developed in the trigonometry chapter for Subtest III.

Worked Example: Describe the set of complex numbers z with |z - (2 - i)| = 3.

|z - w| is the distance between z and w in the complex plane.

So the condition says the distance from z to the point 2 - i, that is (2, -1), is 3.

A real number is rational exactly when its decimal terminates or repeats; convert repeating decimals with the multiply-and-subtract method.

R is a complete ordered field; Q is an ordered field that is not complete. Rational plus irrational is always irrational, but two irrationals can combine to give a rational.

Proportional means y = kx (line through the origin); dimensional analysis multiplies by factors equal to 1, squared for area and cubed for volume.

For complex numbers, divide using the conjugate, use z · z̄ = |z|^2, and read |z - w| as distance; multiplying by i rotates 90° counterclockwise.