TExES Mathematics 4-8 (115)

This chapter covers the sets of numbers from the natural numbers through the complex numbers, how each set extends the one before it, the properties of operations (closure, identity, inverse, commutative, associative, distributive), the density of the rational and real numbers, rational and irrational decimals, and arithmetic with complex numbers. It serves Competency 001, which includes properties of number sets through the complex numbers and the reasons number systems are extended.

Each number set used in school mathematics contains the one before it. The following table lists them with their usual symbols.

The Number Sets

The nesting is N ⊂ W ⊂ Z ⊂ Q ⊂ R ⊂ C. The irrational numbers are the real numbers that are not rational, so the rationals and irrationals together make up the reals and share no members. Every real number is also a complex number with imaginary part 0, so 7 = 7 + 0i.

A rational number is any number that can be written as a/b where a and b are integers and b ≠ 0. The key words are "can be written." The number √49 is rational because it equals 7 = 7/1. The number 0.25 is rational because it equals 1/4. The number −3 is rational because it equals −3/1.

Common Trap: Many students and some adults think π equals 22/7 or 3.14. Both are rational approximations. π is irrational, so no fraction of integers equals it exactly. On the exam, a choice that treats 22/7 as the exact value of π is wrong.

Competency 001 asks you to explain why we extend number systems. Each extension makes an operation always possible that was not always possible before.

A good way to state the pattern is: we extend a number system when an equation we care about has no solution in the current system, and we extend it so that the old rules still work. For example, the rules for adding and multiplying integers were chosen so that the distributive property still holds. That is the real reason a negative times a negative is positive: (−1)(−1) must equal 1 so that (−1)(1 + (−1)) = (−1)(0) = 0 works out.

Exam Tip: When a question asks which equation requires a particular extension, solve it and look at the answer. x + 9 = 4 needs integers (x = −5); 9x = 4 needs rationals (x = 4/9); x^2 = 5 needs irrationals (x = ±√5); x^2 = −9 needs complex numbers (x = ±3i).

The following table summarizes the field properties for addition and multiplication of real numbers.

Properties of Real Number Operations

Subtraction and division are not commutative or associative: 8 − 3 ≠ 3 − 8, and (24 ÷ 4) ÷ 2 = 3 while 24 ÷ (4 ÷ 2) = 12. Students who apply "you can switch the order" to subtraction make errors such as computing 3 − 8 as 5.

Teachers name properties to justify steps. Mental math with 4 × 37 × 25 uses the commutative and associative properties to regroup as (4 × 25) × 37 = 100 × 37 = 3,700. Computing 8 × 47 as 8 × 40 + 8 × 7 = 320 + 56 = 376 uses the distributive property.

A set is closed under an operation if performing the operation on any two members of the set always gives a member of the set. To show closure fails, one counterexample is enough. To show closure holds, you need a general argument.

Closure of Common Sets

The irrationals fail every closure test. For addition, √2 + (−√2) = 0. For multiplication, √3 × √3 = 3. For division, π ÷ π = 1.

The exam may also ask about less familiar sets. The odd integers are closed under multiplication (odd × odd is odd) but not addition (3 + 5 = 8). The even integers are closed under both addition and multiplication. The set {−1, 0, 1} is closed under multiplication but not addition (1 + 1 = 2). The set of numbers of the form a + b√2, where a and b are rational, is closed under addition and multiplication, because (a + b√2)(c + d√2) = (ac + 2bd) + (ad + bc)√2, which has the same form.

Worked Example: Determine whether the set S = {numbers of the form 3k + 1, where k is an integer} is closed under addition and under multiplication.

S contains ..., −2, 1, 4, 7, 10, .... Test addition with a pair: 1 + 4 = 5. Is 5 of the form 3k + 1? 5 = 3(1) + 2, so no.

One counterexample is enough: S is not closed under addition.

Test multiplication generally: (3m + 1)(3n + 1) = 9mn + 3m + 3n + 1 = 3(3mn + m + n) + 1.

Because 3mn + m + n is an integer, the product has the form 3k + 1.

Result: S is closed under multiplication but not under addition.

A rational number in lowest terms a/b has a terminating decimal exactly when the only prime factors of b are 2 and 5. Otherwise its decimal repeats. So 7/40 = 0.175 terminates (40 = 2^3 × 5), while 5/12 = 0.41666... repeats (12 has a factor of 3).

Any repeating decimal can be converted to a fraction with algebra.

Worked Example: Write 1.2454545... (the digits 45 repeat) as a fraction in lowest terms.

Let x = 1.2454545.... The repeating block has two digits and starts after one nonrepeating decimal digit.

Result: 1.2454545... = 137/110.

A quick shortcut for pure repeating decimals: the repeating block over the same number of 9s. So 0.363636... = 36/99 = 4/11, and 0.123123... = 123/999 = 41/333. The decimal 0.999... = 9/9 = 1, a fact that surprises many students.

To prove that √2 is irrational, use indirect proof. Suppose √2 = a/b in lowest terms. Then 2 = a^2/b^2, so a^2 = 2b^2, which makes a^2 even, so a is even. Write a = 2c. Then 4c^2 = 2b^2, so b^2 = 2c^2, which makes b even. Now a and b are both even, which contradicts "lowest terms." So √2 is irrational. The same argument shows √p is irrational for any prime p, and √n is irrational for any positive integer n that is not a perfect square.

Rational and irrational numbers interact predictably. A nonzero rational times an irrational is irrational (3√2), and a rational plus an irrational is irrational (5 + π). Two irrationals can combine to give either type: √2 × √8 = √16 = 4 is rational, while √2 × √3 = √6 is irrational.

A set of numbers is dense if between any two different members there is always another member. The rationals are dense: between a and b you can always find their average (a + b)/2, which is rational when a and b are. The reals and the irrationals are also dense. The integers and whole numbers are not dense, since there is no integer between 3 and 4.

Density means there is no "next" rational number after 1/2. Whatever number you name, there is another one closer to 1/2. Students who think of fractions like whole numbers often say there are no fractions between 3/7 and 4/7. Renaming as 6/14 and 8/14 shows 7/14 between them, and renaming as 30/70 and 40/70 shows nine fractions between them.

Worked Example: Find two different rational numbers between 5/8 and 7/11, and confirm they lie between.

Check with decimals: 5/8 = 0.625, 7/11 ≈ 0.63636, 111/176 ≈ 0.63068, 12/19 ≈ 0.63158.

Both values are greater than 0.625 and less than 0.63636.

Result: 111/176 and 12/19 both lie between 5/8 and 7/11.

Exam Tip: The density property is part of Competency 001 and shows up as a conceptual question, often framed around a student who claims there is nothing between two fractions or decimals. The best answer usually renames the numbers with a common, larger denominator or more decimal places.

The imaginary unit i is defined by i^2 = −1. It appears on the Definitions and Formulas page, but the arithmetic rules do not, so practice them. A complex number has the form a + bi, where a is the real part and b is the imaginary part. Two complex numbers are equal only when both parts are equal.

Operations follow the usual rules of algebra, with i^2 replaced by −1.

Add or subtract by combining like parts: (3 + 5i) − (7 − 2i) = −4 + 7i.

Multiply by distributing: (3 − 2i)(4 + 5i) = 12 + 15i − 8i − 10i^2 = 12 + 7i + 10 = 22 + 7i.

Divide by multiplying the numerator and denominator by the conjugate of the denominator.

The modulus (absolute value) is |a + bi| = √(a^2 + b^2), the distance from 0 in the complex plane.

Powers of i cycle every four: i^1 = i, i^2 = −1, i^3 = −i, i^4 = 1, then the cycle repeats. To find i^n, divide n by 4 and use the remainder. For i^47, 47 = 4 × 11 + 3, so i^47 = i^3 = −i.

Worked Example: Write (5 + i)/(2 − 3i) in the form a + bi.

Multiply the numerator and denominator by the conjugate 2 + 3i.

Complex roots of real polynomials come in conjugate pairs. If 2 + i is a root of a polynomial with real coefficients, then 2 − i is also a root. The quadratic with those roots is (x − (2 + i))(x − (2 − i)) = x^2 − 4x + 5. This connection between complex numbers and quadratics is developed further in Section 2.

The real numbers are ordered: for any two different reals, one is greater. The complex numbers are not ordered in a way that respects multiplication. Asking whether 3i > 2 has no meaning. What you can compare is the modulus: |3i| = 3 and |2| = 2, so 3i is farther from 0 than 2 is. On the exam, a choice that says one non-real complex number is "greater than" another is incorrect.

Classroom Scenario: A seventh grader argues that 0.999... must be less than 1 "because it never quite gets there." The student is thinking of 0.999... as a process that is always unfinished rather than as a fixed number. The best next move is to give the student two arguments to test. First, 1/3 = 0.333..., and three times 1/3 is 1, while three times 0.333... is 0.999.... Second, if 0.999... were less than 1, there would have to be a number between them (density), but no decimal can fit between them. The teacher can then ask the student to explain which argument convinces them and why, which builds reasoning instead of simply asserting the answer.

Memory Aid: For the cycle of i, remember "i, −1, −i, 1" as "eye, minus one, minus eye, one," and remember that any multiple of 4 as the exponent gives 1.

The number sets are nested: N ⊂ W ⊂ Z ⊂ Q ⊂ R ⊂ C, and rationals and irrationals together form the reals.

Number systems are extended so that operations or equations that were impossible become possible, while keeping the old properties true.

To disprove closure, give one counterexample; to prove it, write a general argument, as with (3m + 1)(3n + 1).

A fraction in lowest terms terminates exactly when its denominator has no prime factors other than 2 and 5; repeating decimals convert to fractions with algebra.

The rationals, irrationals, and reals are dense; the integers are not.