TExES Mathematics 7-12 (235)

These questions look short, but they reward precise thinking. You will be asked which set is closed under an operation, how a repeating decimal becomes a fraction, where the inverse of a complex number sits in the plane, or how many arrangements satisfy a condition. Many questions are structural rather than computational, so a graphing calculator helps less here than anywhere else on the test. Know the definitions cold and be ready to test a claim against a counterexample.

This chapter covers Competency 001, the real number system: place value and number bases, decimal representations, the algebraic structure of the real numbers and their subsets, the choice of representation (fractions, decimals, percents, roots, exponents and scientific notation), models of operations, deductive justification of algebraic steps, and the way the real numbers solve problems that the rational numbers cannot.

The real numbers are built in layers. Each layer was added to solve problems the previous layer could not.

Natural numbers N = {1, 2, 3, ...} (some texts include 0; TExES questions state the convention when it matters). Whole numbers W = {0, 1, 2, 3, ...}.

Integers Z = {..., -2, -1, 0, 1, 2, ...}, which allow every subtraction.

A number belongs to every set that contains it. The number -7 is an integer, a rational number and a real number, but not a whole or natural number. The number 0.75 is rational because it equals 3/4.

Classifying Real Numbers

Common Trap: A radical sign does not make a number irrational. sqrt(16), cbrt(-27) and sqrt(9/25) are all rational. Simplify before you classify.

Our numeration system is positional with base ten: in 4,305.27 the digit 3 stands for 3 × 10^2 and the 7 stands for 7 × 10^(-2). The same idea works for any base b ≥ 2, using the digits 0 through b - 1 and place values that are powers of b.

To convert from base b to base ten, expand by place value. To convert from base ten to base b, divide repeatedly by b and read the remainders from last to first, or subtract the largest powers of b in turn.

Worked Example: Write 213 (base 5) in base ten, and then write the result in base 7.

Result: 213 (base 5) = 58 (base ten) = 112 (base 7).

Fractional places work the same way. In base 2, 1011.01 = 8 + 0 + 2 + 1 + 0/2 + 1/4 = 11.25. Whether a fraction terminates depends on the base: a fraction in lowest terms terminates in base b exactly when every prime factor of its denominator divides b. So 1/8 terminates in base ten (8 = 2^3), 1/6 does not, and 1/10 terminates in base ten but repeats forever in base two. That last fact is why computers store 0.1 approximately.

Exam Tip: Base questions often test place value disguised as a puzzle, such as "In what base does 3 × 4 = 13?" Write 13 (base b) as b + 3 and solve b + 3 = 12 to get b = 9.

Every rational number has a decimal that terminates or repeats, and every terminating or repeating decimal is rational. To convert a repeating decimal, multiply by powers of ten so that the repeating parts line up, then subtract.

Worked Example: Write 0.2454545... (the block 45 repeats) as a fraction in lowest terms. Let x = 0.2454545...

The test expects you to name the structure of number systems. A set with an operation is a group if four properties hold:

If the operation is also commutative, the group is called abelian. The integers under addition form an abelian group: the sum of integers is an integer, addition is associative and commutative, 0 is the identity, and -a is the inverse of a. The integers under multiplication are not a group, because 2 has no integer multiplicative inverse (1/2 is not an integer).

A field is a set with two operations, addition and multiplication, such that the set is an abelian group under addition, the nonzero elements form an abelian group under multiplication, and multiplication distributes over addition. The rational numbers, the real numbers and the complex numbers are fields. The integers are not a field (no multiplicative inverses), and neither are the natural numbers (no additive inverses or additive identity, depending on the convention).

Structure of Common Number Systems

The real numbers have two extra properties that the rationals lack in part. They are an ordered field (a < b behaves well with addition and with multiplication by positives), and they are complete: every nonempty set of reals that is bounded above has a least upper bound. The set of rationals whose squares are less than 2 is bounded above but has no least upper bound in Q; in R its least upper bound is sqrt(2). Both Q and R are dense: between any two numbers there is another (their average, for example).

Closure questions are a TExES favorite. The question names a set, often an unusual one, and asks which operation always produces a member of the set. To show closure fails, one counterexample is enough. To show closure holds, you need a general argument.

Worked Example: Let S be the set of all numbers of the form a + b·sqrt(2), where a and b are rational. Is S closed under multiplication? Does every nonzero element have a multiplicative inverse in S? Find the inverse of 3 + 2·sqrt(2).

Multiply two general elements: (a + b·sqrt(2))(c + d·sqrt(2)) = ac + ad·sqrt(2) + bc·sqrt(2) + 2bd = (ac + 2bd) + (ad + bc)·sqrt(2).

Because sums and products of rationals are rational, ac + 2bd and ad + bc are rational. So S is closed under multiplication.

For an inverse, rationalize with the conjugate: 1/(3 + 2·sqrt(2)) = (3 - 2·sqrt(2))/((3 + 2·sqrt(2))(3 - 2·sqrt(2))) = (3 - 2·sqrt(2))/(9 - 8) = 3 - 2·sqrt(2).

The same real number can be written many ways, and the best form depends on the job. Fractions keep exact values and make ratios visible. Decimals make comparison and estimation easy. Percents describe parts of 100 and relative change. Radicals and rational exponents describe roots exactly (x^(m/n) = (n-th root of x)^m). Scientific notation, a × 10^n with 1 ≤ |a| < 10, handles very large and very small quantities.

The laws of exponents hold for all real exponents when the base is positive: 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. With negative bases, rational exponents need care: (-8)^(1/3) = -2 is fine, but (-8)^(2/6) is ambiguous, so simplify the exponent first.

Teachers must explain why algorithms work, and the test asks about the models that do it. Area models show multiplication: 23 × 14 is a rectangle split into 20 × 10, 20 × 4, 3 × 10 and 3 × 4, giving 200 + 80 + 30 + 12 = 322. The same picture becomes (x + 3)(x + 4) = x^2 + 7x + 12 in algebra, which is why algebra tiles build on base-ten blocks. Number lines model addition of signed numbers as moves and subtraction as "the distance and direction from b to a." Fraction division by "invert and multiply" can be justified by the common-denominator method: (a/b) ÷ (c/d) = (ad/bd) ÷ (bc/bd) = ad/bc.

Geometric models also show irrational lengths. A right triangle with legs 1 and 1 has hypotenuse sqrt(2), and stacking right triangles (the "spiral of Theodorus") constructs sqrt(3), sqrt(4), sqrt(5) and so on. On a geoboard, a tilted square with vertices (0, 0), (2, 1), (1, 3) and (-1, 2) has side sqrt(5) and area 5, an integer even though the side is irrational.

Every step in solving an equation is justified by a property of the real numbers. The test may show a numbered solution and ask which property justifies a step.

Worked Example: Justify each step in solving 3(x - 4) = 2x + 5.

3x - 12 = 2x + 5 (distributive property).

3x - 12 + 12 = 2x + 5 + 12 (addition property of equality).

3x + 0 = 2x + 17 (additive inverse property, then simplification).

3x = 2x + 17 (additive identity property).

3x - 2x = 17 (addition property of equality, adding -2x).

x = 17 (distributive property to combine 3x - 2x = (3 - 2)x, then multiplicative identity).

Result: x = 17, and each line follows from a field property or a property of equality.

Common Trap: The commutative and associative properties are easy to confuse. Commutative changes order (a + b = b + a). Associative changes grouping ((a + b) + c = a + (b + c)). If the numbers stay in the same order but the parentheses move, the property is associative.

The equation x^2 = 2 has no rational solution. This is one of the classic proofs by contradiction and the test expects you to follow it.

Proof: Suppose sqrt(2) = p/q, where p and q are integers with no common factor and q ≠ 0.

Squaring gives 2 = p^2/q^2, so p^2 = 2q^2. Then p^2 is even.

If p were odd, p^2 would be odd, so p must be even. Write p = 2k.

Then 4k^2 = 2q^2, so q^2 = 2k^2, and q^2 is even. By the same reasoning q is even.

Now p and q are both even, which contradicts the assumption that they have no common factor.

This completes the proof.

The same argument proves that sqrt(n) is irrational whenever n is a positive integer that is not a perfect square. Likewise, the diagonal of a unit square, the circumference of a circle with rational diameter, and the solution of 10^x = 2 are real numbers that are not rational. Measurement creates the need for real numbers: lengths in geometry are not always ratios of whole numbers.

Know the subsets N, W, Z, Q, irrationals and R, and classify a number only after simplifying it.

A fraction in lowest terms terminates in base b exactly when the prime factors of its denominator divide b; every repeating decimal is rational.

The integers form an abelian group under addition; Q, R and C are fields; one counterexample disproves closure.

Choose the representation that fits the task, and carry units through scientific-notation calculations.

Every algebraic step is justified by a named property; the proof that sqrt(2) is irrational shows why the real numbers are needed.