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WEST-B

Practice QuestionsStudy Guide

10.1 Number Classification, Properties of Operations & Place Value

Key Takeaways

  • Real numbers partition into rational numbers (expressible as a/b with b ≠ 0, yielding terminating or repeating decimals) and irrational numbers (non-terminating, non-repeating decimals such as √2, √3, and π).
  • Base-ten place value extends infinitely in both directions; scientific notation expresses numbers as a × 10^b where 1 ≤ a < 10 and b is an integer.
  • The fundamental arithmetic properties (commutative, associative, distributive, identity, and inverse) govern algebraic manipulation; commutative and associative properties hold strictly for addition and multiplication, not subtraction or division.
  • Signed number operations require strict sign governance: subtracting a signed number is equivalent to adding its opposite (additive inverse), while products and quotients of numbers with identical signs yield positive results and differing signs yield negative results.
  • Divisibility rules streamline prime factorization; the Greatest Common Factor (GCF) takes the lowest power of shared prime factors, whereas the Least Common Multiple (LCM) takes the highest power of all occurring prime factors.
Last updated: August 2026

Number Classification, Properties of Operations & Place Value

Quick Answer: On the WEST-B Mathematics subtest (Objective 0013), candidates must demonstrate conceptual mastery and operational fluency with real numbers. The real number system $\mathbb{R}$ comprises rational numbers $\mathbb{Q}$ (integers, terminating decimals, and repeating decimals expressible as $\frac{a}{b}$) and irrational numbers $\mathbb{I}$ (non-terminating, non-repeating values like $\sqrt{2}$ and $\pi$). Arithmetic operations are governed by core field properties: commutative ($a+b=b+a$), associative ($(a+b)+c=a+(b+c)$), distributive ($a(b+c)=ab+ac$), identity ($a+0=a, a \times 1=a$), and inverse ($a+(-a)=0, a \times \frac{1}{a}=1$). Number theory principles—including divisibility tests, prime factorization, Greatest Common Factor (GCF), and Least Common Multiple (LCM)—form the computational backbone for fraction arithmetic and algebraic reasoning.

1. The Real Number System & Classification Hierarchy

Every number encountered on the WEST-B Mathematics subtest belongs to the set of Real Numbers ($\mathbb{R}$). The real numbers are partitioned into nested subsets based on their mathematical definitions and decimal representations.

+---------------------------------------------------------------------------------------------------+
|                                 REAL NUMBER SYSTEM HIERARCHY (ℝ)                                  |
|                                                                                                   |
|   +-------------------------------------------------------------------+   +-------------------+   |
|   |                       RATIONAL NUMBERS (ℚ)                         |   |    IRRATIONAL     |   |
|   |   Expressible as a/b (a, b ∈ ℤ, b ≠ 0). Terminating or repeating. |   |   NUMBERS (𝕀)     |   |
|   |                                                                   |   |                   |   |
|   |   +-----------------------------------------------------------+   |   | Non-terminating,  |   |
|   |   |                       INTEGERS (ℤ)                        |   |   | non-repeating     |   |
|   |   |   {... -3, -2, -1, 0, 1, 2, 3 ...}                        |   |   | decimals.         |   |
|   |   |                                                           |   |   |                   |   |
|   |   |   +---------------------------------------------------+   |   |   Examples:       |   |
|   |   |   |                   WHOLE NUMBERS (𝕎)               |   |   |   √2 ≈ 1.4142...  |   |
|   |   |   |   {0, 1, 2, 3, 4, 5 ...}                          |   |   |   √3 ≈ 1.7320...  |   |
|   |   |   |                                                   |   |   |   π  ≈ 3.14159... |   |
|   |   |   |   +-------------------------------------------+   |   |   |   e  ≈ 2.71828... |   |
|   |   |   |   |            NATURAL NUMBERS (ℕ)            |   |   |   |   √7, ∛5          |   |
|   |   |   |   |   {1, 2, 3, 4, 5 ...} (Counting numbers)  |   |   |   |                   |   |
|   |   |   |   +-------------------------------------------+   |   |   |                   |   |
|   |   |   +---------------------------------------------------+   |   |                   |   |
|   |   +-----------------------------------------------------------+   |   |                   |   |
|   |                                                                   |   |                   |   |
|   |   Fractions & Decimals: -3/4, 0.75, 0.333..., -5.12, 14/3         |   |                   |   |
|   +-------------------------------------------------------------------+   +-------------------+   |
+---------------------------------------------------------------------------------------------------+

Definitions of Subsets

  1. Natural Numbers (Counting Numbers, $\mathbb{N}$): ${1, 2, 3, 4, 5, \dots}$. These are positive integers used for counting discrete objects. Note that zero is not a natural number.
  2. Whole Numbers ($\mathbb{W}$): ${0, 1, 2, 3, 4, 5, \dots}$. Formed by including zero with the set of natural numbers ($\mathbb{W} = \mathbb{N} \cup {0}$).
  3. Integers ($\mathbb{Z}$): ${\dots, -3, -2, -1, 0, 1, 2, 3, \dots}$. Formed by combining whole numbers and their negative opposites (additive inverses). Integers contain no fractional or fractional-decimal components.
  4. Rational Numbers ($\mathbb{Q}$): Any number that can be expressed as the ratio of two integers in the form $\frac{a}{b}$, where $a, b \in \mathbb{Z}$ and $b \neq 0$.
    • Terminating Decimals: Decimals that end after a finite number of digits (e.g., $0.625 = \frac{5}{8}$, $-2.5 = -\frac{5}{2}$, $0.004 = \frac{1}{250}$).
    • Repeating (Periodic) Decimals: Decimals that continue indefinitely in a recurring cycle of one or more digits (e.g., $0.\overline{3} = \frac{1}{3}$, $0.\overline{142857} = \frac{1}{7}$, $0.8\overline{3} = \frac{5}{6}$).
  5. Irrational Numbers ($\mathbb{I}$): Real numbers that cannot be expressed as a ratio of two integers. Their decimal expansions are non-terminating and non-repeating.
    • Square roots of non-perfect squares: $\sqrt{2}, \sqrt{3}, \sqrt{5}, \sqrt{10}, \sqrt{50}$. (Note: $\sqrt{49} = 7$, which is rational!)
    • Transcendental constants: $\pi \approx 3.14159265\dots$, $e \approx 2.7182818\dots$
    • Sums or products of non-zero rationals and irrationals: $3 + \sqrt{2}$, $2\pi$, $\frac{\sqrt{5}}{2}$ are all irrational.

2. Base-Ten Place Value, Scientific Notation & Rounding

Our standard numeration system is a positional base-ten system where the value of each digit depends upon its position relative to the decimal point. Each position represents a power of 10.

+---------------------------------------------------------------------------------------------------+
|                                    BASE-TEN PLACE VALUE CHART                                     |
|                                                                                                   |
|  Thousands | Hundreds | Tens | Ones | . | Tenths | Hundredths | Thousandths | Ten-Thousandths     |
|    10³     |   10²    |  10¹ | 10⁰  | . |  10⁻¹  |    10⁻²    |    10⁻³     |      10⁻⁴           |
|   1,000    |   100    |  10  |  1   | . |  0.1   |    0.01    |    0.001    |     0.0001          |
+---------------------------------------------------------------------------------------------------+

Expanded Form

Expanded notation expresses a number as the sum of each digit multiplied by its corresponding place value:

4,508.37=(4×1,000)+(5×100)+(0×10)+(8×1)+(3×0.1)+(7×0.01)4,508.37 = (4 \times 1,000) + (5 \times 100) + (0 \times 10) + (8 \times 1) + (3 \times 0.1) + (7 \times 0.01)

=(4×103)+(5×102)+(8×100)+(3×10−1)+(7×10−2)= (4 \times 10^3) + (5 \times 10^2) + (8 \times 10^0) + (3 \times 10^{-1}) + (7 \times 10^{-2})

Scientific Notation

Scientific notation standardizes very large or very small real numbers into the product form:

a×10bwhere1≤∣a∣<10andb∈Za \times 10^b \quad \text{where} \quad 1 \le |a| < 10 \quad \text{and} \quad b \in \mathbb{Z}
  • Large Numbers ($b > 0$): Move the decimal point to the left until one non-zero digit remains to the left of the decimal. The number of places moved is the positive exponent $b$.
    • Example: $4,850,000 = 4.85 \times 10^6$ (moved 6 places left).
  • Small Numbers ($b < 0$): Move the decimal point to the right until one non-zero digit remains to the left. The number of places moved is the negative exponent $b$.
    • Example: $0.0000732 = 7.32 \times 10^{-5}$ (moved 5 places right).

Multiplication and Division in Scientific Notation

  1. Multiplication: Multiply the coefficients $a_1 \times a_2$ and add the exponents of 10 ($10^{b_1 + b_2}$). Then adjust coefficient if not in $[1, 10)$: (3.0×104)×(5.0×107)=(3.0×5.0)×104+7=15.0×1011=1.5×1012(3.0 \times 10^4) \times (5.0 \times 10^7) = (3.0 \times 5.0) \times 10^{4+7} = 15.0 \times 10^{11} = 1.5 \times 10^{12}
  2. Division: Divide coefficients $\frac{a_1}{a_2}$ and subtract exponents of 10 ($10^{b_1 - b_2}$): 1.2×1034.0×108=(1.24.0)×103−8=0.3×10−5=3.0×10−6\frac{1.2 \times 10^3}{4.0 \times 10^8} = \left(\frac{1.2}{4.0}\right) \times 10^{3-8} = 0.3 \times 10^{-5} = 3.0 \times 10^{-6}

Standard Rounding Protocol

To round a number to a specified place value:

  1. Identify the target digit at the specified place value.
  2. Inspect the digit immediately to its right (the determining digit):
    • If the determining digit is 5 or greater (5, 6, 7, 8, 9), increase the target digit by 1 and replace subsequent digits with zeros (or drop them if to the right of the decimal point).
    • If the determining digit is less than 5 (0, 1, 2, 3, 4), leave the target digit unchanged and replace subsequent digits with zeros (or drop them).
  • Example: Round $34.8647$ to the nearest hundredth. The hundredths digit is $6$; the digit to its right is $4$ ($< 5$). Thus, $34.8647 \approx 34.86$.

3. Fundamental Arithmetic Properties of Operations

Field properties establish the structural rules that allow expressions to be simplified, factored, and solved algebraically.

+---------------------------------------------------------------------------------------------------+
|                                 CORE FIELD PROPERTIES OF ARITHMETIC                               |
|                                                                                                   |
|   +-----------------------+-------------------------------+-----------------------------------+   |
|   | PROPERTY              | ADDITION FORMULATION          | MULTIPLICATION FORMULATION        |   |
|   +-----------------------+-------------------------------+-----------------------------------+   |
|   | Commutative           | a + b = b + a                 | a × b = b × a                     |   |
|   | (Order changes)       | 7 + 12 = 12 + 7               | 4 × 9 = 9 × 4                     |   |
|   +-----------------------+-------------------------------+-----------------------------------+   |
|   | Associative           | (a + b) + c = a + (b + c)     | (a × b) × c = a × (b × c)         |   |
|   | (Grouping changes)    | (3 + 8) + 2 = 3 + (8 + 2)     | (5 × 2) × 7 = 5 × (2 × 7)         |   |
|   +-----------------------+-------------------------------+-----------------------------------+   |
|   | Distributive          | a × (b + c) = (a × b) + (a × c)                                   |   |
|   | (Multiplication over +)| 6 × (10 + 4) = (6 × 10) + (6 × 4) = 60 + 24 = 84                  |   |
|   +-----------------------+-------------------------------+-----------------------------------+   |
|   | Identity              | a + 0 = a                     | a × 1 = a                         |   |
|   | (Value preserved)     | (0 is Additive Identity)      | (1 is Multiplicative Identity)    |   |
|   +-----------------------+-------------------------------+-----------------------------------+   |
|   | Inverse               | a + (-a) = 0                  | a × (1/a) = 1  (for a ≠ 0)        |   |
|   | (Yields identity)     | (-a is Additive Inverse)      | (1/a is Reciprocal)               |   |
|   +-----------------------+-------------------------------+-----------------------------------+   |
|   | Zero Product          | N/A                           | a × 0 = 0                         |   |
|   | Property              |                               | If ab = 0, then a = 0 or b = 0    |   |
|   +-----------------------+-------------------------------+-----------------------------------+   |
+---------------------------------------------------------------------------------------------------+

Critical Distinctions & Non-Properties

  • Subtraction and Division are NEITHER Commutative NOR Associative:
    • $10 - 4 = 6$, but $4 - 10 = -6$ (Not commutative).
    • $(12 - 4) - 2 = 8 - 2 = 6$, but $12 - (4 - 2) = 12 - 2 = 10$ (Not associative).
    • $24 \div 6 = 4$, but $6 \div 24 = 0.25$ (Not commutative).
    • $(24 \div 6) \div 2 = 4 \div 2 = 2$, but $24 \div (6 \div 2) = 24 \div 3 = 8$ (Not associative).

4. Signed Number Arithmetic & Absolute Value

Absolute Value ($|x|$)

Geometrically, the absolute value of a real number $x$ represents its non-directed distance from zero on the number line:

∣x∣={xif x≥0−xif x<0|x| = \begin{cases} x & \text{if } x \ge 0 \\ -x & \text{if } x < 0 \end{cases}
  • Distance is always non-negative: $|7| = 7$, $|-7| = 7$, $|0| = 0$.
  • Watch the Sign Outside: $-|-12| = -(12) = -12$. The negative sign outside the absolute value bars applies after evaluating the magnitude inside.

Rules for Adding Signed Numbers

  1. Same Signs: Add their absolute values and attach the common sign.
    • $(+7) + (+5) = +12$
    • $(-8) + (-6) = -(8 + 6) = -14$
  2. Different Signs: Subtract the smaller absolute value from the larger absolute value. The result carries the sign of the number with the larger absolute value.
    • $(-15) + (+9) \implies 15 - 9 = 6 \implies -6$ (since $|-15| > |+9|$)
    • $(+18) + (-7) \implies 18 - 7 = 11 \implies +11$

Rules for Subtracting Signed Numbers (Adding the Opposite)

Subtraction is universally defined as adding the additive inverse: $a - b = a + (-b)$.

  • Change the subtraction sign to addition and reverse the sign of the subtrahend (second number).
    • $8 - 14 = 8 + (-14) = -6$
    • $-9 - 7 = -9 + (-7) = -16$
    • $5 - (-8) = 5 + (+8) = 13$
    • $-12 - (-15) = -12 + (+15) = +3$

Rules for Multiplying and Dividing Signed Numbers

  • Like Signs Yield a Positive Result:
    • $(+) \times (+) = +$ and $(-) \times (-) = +$
    • $(+) \div (+) = +$ and $(-) \div (-) = +$
    • Examples: $(-6) \times (-7) = +42$; $\frac{-54}{-9} = +6$.
  • Unlike Signs Yield a Negative Result:
    • $(+) \times (-) = -$ and $(-) \times (+) = -$
    • $(+) \div (-) = -$ and $(-) \div (+) = -$
    • Examples: $(-8) \times (+5) = -40$; $\frac{48}{-6} = -8$.
  • Parity of Negative Factors: In a product of multiple non-zero factors, if the count of negative factors is even, the product is positive; if the count is odd, the product is negative.
    • $(-2) \times (-3) \times (-4) = -24$ (3 negative factors $\implies$ negative).
    • $(-2) \times (-3) \times (-4) \times (-5) = +120$ (4 negative factors $\implies$ positive).

5. Prime Numbers, Divisibility Rules, GCF & LCM

Primes vs. Composites

  • Prime Number: A natural number strictly greater than 1 that has exactly two distinct positive divisors: 1 and itself. First ten primes: $2, 3, 5, 7, 11, 13, 17, 19, 23, 29$.
    • Note: 2 is the only even prime number and the smallest prime number. 0 and 1 are neither prime nor composite.
  • Composite Number: A natural number greater than 1 that has more than two positive divisors (e.g., $4, 6, 8, 9, 10, 12, \dots$).

Rapid Divisibility Tests

+---------------------------------------------------------------------------------------------------+
|                                    DIVISIBILITY TEST MATRIX                                       |
|                                                                                                   |
|   +-----------+-----------------------------------------------+-------------------------------+   |
|   | DIVISOR   | CONDITION / TEST RULE                         | WORKED EXAMPLE                |   |
|   +-----------+-----------------------------------------------+-------------------------------+   |
|   | 2         | Last digit is even (0, 2, 4, 6, 8)            | 3,578 ends in 8 (Even) -> Yes |   |
|   | 3         | Sum of all digits is divisible by 3           | 4,731: 4+7+3+1=15 (÷3) -> Yes |   |
|   | 4         | Last two digits form a number divisible by 4  | 9,524: 24 ÷ 4 = 6 -> Yes      |   |
|   | 5         | Last digit is 0 or 5                          | 1,285 ends in 5 -> Yes        |   |
|   | 6         | Divisible by BOTH 2 and 3                     | 732: Even (÷2) & 7+3+2=12 (÷3)|   |
|   | 8         | Last three digits form a number divisible by 8| 5,168: 168 ÷ 8 = 21 -> Yes    |   |
|   | 9         | Sum of all digits is divisible by 9           | 6,822: 6+8+2+2=18 (÷9) -> Yes |   |
|   | 10        | Last digit is 0                               | 4,930 ends in 0 -> Yes        |   |
|   +-----------+-----------------------------------------------+-------------------------------+   |
+---------------------------------------------------------------------------------------------------+

Prime Factorization

The Fundamental Theorem of Arithmetic states that every integer greater than 1 is either prime or can be uniquely factored into a product of prime numbers (up to the order of factors).

  • Example: Find the prime factorization of $360$.
    • $360 = 36 \times 10 = (6 \times 6) \times (2 \times 5) = (2 \times 3 \times 2 \times 3) \times (2 \times 5) = 2^3 \times 3^2 \times 5^1$.

Greatest Common Factor (GCF) vs. Least Common Multiple (LCM)

  • Greatest Common Factor (GCF / GCD): The largest integer that divides evenly into two or more given integers without a remainder.
    • Prime Factorization Rule: Take the lowest exponent of each common prime factor.
  • Least Common Multiple (LCM): The smallest positive integer that is a multiple of two or more given integers.
    • Prime Factorization Rule: Take the highest exponent of all occurring prime factors.
Fundamental Identity: GCF(a,b)×LCM(a,b)=∣a×b∣\text{Fundamental Identity: } \text{GCF}(a, b) \times \text{LCM}(a, b) = |a \times b|

Worked Example: GCF and LCM of 72 and 120

  1. Prime factorizations:
    • $72 = 2^3 \times 3^2$
    • $120 = 2^3 \times 3^1 \times 5^1$
  2. Find GCF:
    • Shared prime factors: $2$ and $3$.
    • Lowest powers: $2^{\min(3,3)} = 2^3 = 8$; $3^{\min(2,1)} = 3^1 = 3$.
    • $\text{GCF}(72, 120) = 2^3 \times 3^1 = 8 \times 3 = \mathbf{24}$.
  3. Find LCM:
    • All prime factors: $2, 3, 5$.
    • Highest powers: $2^{\max(3,3)} = 2^3 = 8$; $3^{\max(2,1)} = 3^2 = 9$; $5^{\max(0,1)} = 5^1 = 5$.
    • $\text{LCM}(72, 120) = 2^3 \times 3^2 \times 5^1 = 8 \times 9 \times 5 = \mathbf{360}$.
  4. Check identity:
    • $\text{GCF} \times \text{LCM} = 24 \times 360 = 8,640$.
    • $a \times b = 72 \times 120 = 8,640$. (Exact match!)

6. Diagnostic Error Analysis & Educator Strategies

Candidate Error / MisconceptionMathematical Reality & Diagnostic ExplanationCorrect Pedagogical Intervention
Thinking $\sqrt{49}$ or $\sqrt{100}$ is irrational because it contains a radical symbol.A radical $\sqrt{x}$ is rational whenever $x$ is a perfect square ($49 = 7^2 \implies \sqrt{49} = 7 \in \mathbb{N} \subset \mathbb{Q}$). Only non-perfect square roots are irrational.Instruct candidates to simplify radical expressions completely before classifying them into number sets.
Believing $0$ is positive or that $0$ is a natural counting number.Zero is a whole number, integer, and rational number ($0 = 0/1$), but it is neutral (neither positive nor negative) and not in $\mathbb{N}$.Use a number line visual showing 0 as the origin dividing positive and negative integers.
Applying commutative property to division: $12 \div 3 = 3 \div 12$.Division and subtraction are non-commutative ($12/3 = 4 \neq 3/12 = 0.25$). Commutativity applies strictly to addition and multiplication.Emphasize inverse operations: division is multiplication by a reciprocal ($12 \times \frac{1}{3} \neq 3 \times \frac{1}{12}$).
Confusing $-4^2$ with $(-4)^2$.In $-4^2$, the exponent binds strictly to the base 4 before the negation: $-4^2 = -(4^2) = -16$. In $(-4)^2$, the base is $-4$: $(-4)(-4) = +16$.Teach parentheses as grouping the negative sign with the base. Without parentheses, exponentiation precedes negation.
Multiplying GCF prime factors with highest powers instead of lowest.GCF seeks the common divisor (restricted to the smallest shared power), whereas LCM requires the full multiple (requiring the highest powers).Use Venn diagram models of prime factor sets to show GCF as the intersection and LCM as the union.
Loading diagram...
Number Theory: GCF and LCM Prime Factorization Venn Diagram
Test Your Knowledge

Which of the following statements correctly classifies the given real numbers?

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Test Your Knowledge

Which of the following equations correctly illustrates the Distributive Property of Multiplication over Addition?

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Test Your Knowledge

A middle school science teacher is preparing identical laboratory kits. She has 72 test tubes and 120 pipettes. She wants to create the maximum number of identical kits with no items left over. How many total items (test tubes plus pipettes) will be placed in EACH kit?

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Test Your Knowledge

Evaluate the following numerical expression: -|-14 + 6| - (-3) × (-4) + (-36) ÷ (-9)

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10.2 Fraction and Decimal Operations, Conversions & Order of Operations

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