What you'll learn
Number and Quantity forms the foundation of US Common Core Math at the high school level, encompassing the real number system, complex numbers, quantities and their units, and vector and matrix representations. This domain appears across multiple question types on Common Core assessments, from direct computation problems to multi-step applications requiring fluency with different number systems and dimensional analysis.
Key terms and definitions
Real numbers — the set of all rational and irrational numbers that can be represented on a number line, including integers, fractions, terminating decimals, and non-terminating decimals.
Imaginary unit (i) — defined as i = √(-1), where i² = -1, forming the basis for the complex number system.
Complex number — a number of the form a + bi where a is the real part and b is the imaginary part, with both a and b being real numbers.
Rational exponent — an exponent expressed as a fraction m/n, where a^(m/n) = ⁿ√(a^m) = (ⁿ√a)^m for appropriate values of a.
Dimensional analysis — a method of converting between units by multiplying by conversion factors in fraction form, ensuring units cancel appropriately.
Vector — a quantity possessing both magnitude and direction, often represented as an ordered pair or matrix in two dimensions.
Matrix — a rectangular array of numbers arranged in rows and columns, used to represent data, transformations, or systems of equations.
Closure property — a set is closed under an operation if performing that operation on elements of the set always produces another element in the same set.
Core concepts
The Real Number System
The real number system extends from natural numbers through increasingly comprehensive sets. Understanding the properties and relationships between these subsets appears frequently on Common Core assessments.
Number set hierarchy:
- Natural numbers (ℕ): {1, 2, 3, 4, ...}
- Whole numbers (W): {0, 1, 2, 3, ...}
- Integers (ℤ): {..., -2, -1, 0, 1, 2, ...}
- Rational numbers (ℚ): numbers expressible as p/q where p and q are integers and q ≠ 0
- Irrational numbers: real numbers that cannot be expressed as a ratio of integers (π, √2, e)
- Real numbers (ℝ): the union of rational and irrational numbers
Rational versus irrational identification requires recognizing that rational numbers have decimal representations that either terminate (0.75 = 3/4) or repeat (0.333... = 1/3), while irrational numbers have non-repeating, non-terminating decimal expansions.
Properties of operations apply across the real numbers:
- Commutative: a + b = b + a and ab = ba
- Associative: (a + b) + c = a + (b + c) and (ab)c = a(bc)
- Distributive: a(b + c) = ab + ac
- Identity elements: 0 for addition, 1 for multiplication
- Inverse elements: -a for addition, 1/a for multiplication (a ≠ 0)
Exam questions test whether specific sets are closed under given operations. The integers are closed under addition, subtraction, and multiplication but not division (5 ÷ 2 = 2.5, not an integer). The irrational numbers are not closed under addition (√2 + (-√2) = 0, a rational number).
Exponents and Radicals
Rational exponents connect exponential and radical notation. The expression a^(m/n) means the nth root of a raised to the mth power, where n is the index of the root.
Key exponent properties:
- a^m · a^n = a^(m+n)
- (a^m)^n = a^(mn)
- a^m / a^n = a^(m-n)
- (ab)^n = a^n · b^n
- a^0 = 1 (a ≠ 0)
- a^(-n) = 1/a^n
Converting between forms: 16^(3/4) = (⁴√16)³ = 2³ = 8, or equivalently, 16^(3/4) = ⁴√(16³) = ⁴√4096 = 8.
Simplifying radicals involves factoring out perfect powers: √72 = √(36 · 2) = 6√2. For cube roots and higher: ³√54 = ³√(27 · 2) = 3³√2.
Rationalizing denominators eliminates radicals from the denominator:
- For simple radicals: 5/√3 = (5√3)/(√3 · √3) = 5√3/3
- For binomial denominators: 1/(2 + √3) = (2 - √3)/[(2 + √3)(2 - √3)] = (2 - √3)/(4 - 3) = 2 - √3
Complex Numbers
The complex number system extends the real numbers to include solutions to equations like x² + 1 = 0. Every complex number takes the form a + bi where a and b are real numbers.
Operations with complex numbers:
Addition and subtraction: Combine real parts and imaginary parts separately.
- (3 + 4i) + (2 - 5i) = (3 + 2) + (4 - 5)i = 5 - i
- (6 + 2i) - (1 + 7i) = (6 - 1) + (2 - 7)i = 5 - 5i
Multiplication: Use the distributive property and substitute i² = -1.
- (3 + 2i)(4 - i) = 12 - 3i + 8i - 2i² = 12 + 5i - 2(-1) = 14 + 5i
Division: Multiply numerator and denominator by the complex conjugate of the denominator.
- The conjugate of a + bi is a - bi
- (3 + 4i)/(2 - i) = [(3 + 4i)(2 + i)]/[(2 - i)(2 + i)] = (6 + 3i + 8i + 4i²)/(4 - i²) = (6 + 11i - 4)/(4 + 1) = (2 + 11i)/5 = 2/5 + (11/5)i
Powers of i follow a cyclic pattern:
- i¹ = i
- i² = -1
- i³ = i² · i = -i
- i⁴ = i² · i² = 1
- i⁵ = i⁴ · i = i (pattern repeats)
To find i^n, divide n by 4 and use the remainder: i²⁷ = i³ (since 27 = 4 × 6 + 3) = -i.
Complex solutions to quadratics: When the discriminant b² - 4ac < 0, solutions are complex conjugates. For x² + 4x + 13 = 0, using the quadratic formula: x = [-4 ± √(16 - 52)]/2 = [-4 ± √(-36)]/2 = [-4 ± 6i]/2 = -2 ± 3i.
Quantities and Units
Dimensional analysis converts measurements between units systematically. Common Core assessments require fluency with metric and customary units.
Conversion factor method:
- Identify the starting quantity and unit
- Determine the target unit
- Multiply by conversion factors as fractions equal to 1
- Arrange factors so units cancel
- Perform the calculation
Convert 65 miles per hour to feet per second:
- 65 mi/hr × (5280 ft/1 mi) × (1 hr/60 min) × (1 min/60 sec)
- = 65 × 5280 / 3600 ft/sec
- = 95.33 ft/sec
Unit consistency requires all terms in an equation to have compatible dimensions. Adding 5 meters and 3 seconds is meaningless; adding 5 meters and 300 centimeters (after conversion) equals 8 meters.
Level of precision in calculations follows significant figure rules:
- When multiplying/dividing: result has the same number of significant figures as the measurement with fewest
- When adding/subtracting: result has the same decimal place precision as the measurement with lowest precision
- Exact conversion factors (like 12 inches = 1 foot) do not limit precision
Vectors and Matrices
Vector representation expresses quantities with magnitude and direction. In two dimensions, vectors appear as ordered pairs ⟨x, y⟩ or column matrices [x; y].
Vector operations:
- Addition: ⟨a, b⟩ + ⟨c, d⟩ = ⟨a + c, b + d⟩
- Scalar multiplication: k⟨a, b⟩ = ⟨ka, kb⟩
- Magnitude: |⟨a, b⟩| = √(a² + b²)
Matrix dimensions are described as rows × columns. A 3 × 2 matrix has 3 rows and 2 columns.
Matrix addition and subtraction: Matrices must have identical dimensions. Add or subtract corresponding elements:
[1 2; 3 4] + [5 6; 7 8] = [6 8; 10 12]
Scalar multiplication of matrices: Multiply every element by the scalar:
3[2 -1; 0 5] = [6 -3; 0 15]
Matrix multiplication: The number of columns in the first matrix must equal the number of rows in the second. For matrices A (m × n) and B (n × p), the product AB is m × p.
For a 2 × 2 example: [a b; c d][e f; g h] = [ae+bg af+bh; ce+dg cf+dh]
Matrices as transformations represent geometric operations. A reflection matrix, rotation matrix, or dilation matrix applied to coordinate vectors produces transformed coordinates.
Worked examples
Example 1: Simplifying radical expressions with rational exponents
Question: Simplify (27x⁹)^(2/3) and express without negative exponents.
Solution:
Step 1: Apply the exponent to both the coefficient and variable. (27x⁹)^(2/3) = 27^(2/3) · (x⁹)^(2/3)
Step 2: Evaluate 27^(2/3) = (³√27)² = 3² = 9
Step 3: Apply the power rule to the variable. (x⁹)^(2/3) = x^(9 · 2/3) = x^(18/3) = x⁶
Step 4: Combine results. (27x⁹)^(2/3) = 9x⁶
Example 2: Complex number operations
Question: Simplify (4 - 3i)/(2 + i) and express in standard form a + bi.
Solution:
Step 1: Identify the conjugate of the denominator: 2 - i
Step 2: Multiply numerator and denominator by the conjugate. (4 - 3i)/(2 + i) · (2 - i)/(2 - i)
Step 3: Expand the numerator. (4 - 3i)(2 - i) = 8 - 4i - 6i + 3i² = 8 - 10i + 3(-1) = 5 - 10i
Step 4: Expand the denominator. (2 + i)(2 - i) = 4 - 2i + 2i - i² = 4 - (-1) = 5
Step 5: Simplify the fraction. (5 - 10i)/5 = 5/5 - 10i/5 = 1 - 2i
Example 3: Dimensional analysis with compound units
Question: A car travels at 88 feet per second. Express this speed in miles per hour.
Solution:
Step 1: Set up conversion factors to cancel feet and seconds, introduce miles and hours. 88 ft/sec × (1 mi/5280 ft) × (60 sec/1 min) × (60 min/1 hr)
Step 2: Arrange to show unit cancellation. 88 ft/sec × (60 × 60 sec/hr)/(5280 ft/mi) = 88 × 3600/(5280) mi/hr
Step 3: Calculate. 316,800/5280 = 60 mi/hr
The speed is 60 miles per hour.
Common mistakes and how to avoid them
Mistake: Treating i as a variable that can be simplified or canceled in expressions. Students write i/i = 1 but forget that i² ≠ i.
Correction: Remember i is defined as √(-1) with the fundamental property i² = -1. Use this definition when simplifying, not algebraic cancellation.
Mistake: Incorrectly applying exponent rules to addition, writing (a + b)² = a² + b². This extends to radicals: √(a + b) ≠ √a + √b.
Correction: Exponent rules apply to multiplication and division, not addition. Expand (a + b)² = a² + 2ab + b² using the distributive property.
Mistake: Forgetting to rationalize denominators completely, leaving answers like 2/(3 + √5) instead of simplifying.
Correction: Check that no radicals appear in any denominator. For binomial denominators, multiply by the conjugate (change the middle sign).
Mistake: Converting units incorrectly by multiplying when division is needed, or inverting conversion factors. Converting 5 feet to inches by calculating 5/12 instead of 5 × 12.
Correction: Write conversion factors as fractions equal to 1 (like 12 in/1 ft). Arrange factors so unwanted units are in opposite positions (numerator/denominator) and cancel.
Mistake: Assuming all number sets are closed under all operations, such as claiming integers are closed under division.
Correction: Test closure with specific examples. Integer ÷ integer doesn't always yield an integer (5 ÷ 2), so integers are not closed under division.
Mistake: Adding or multiplying matrices without checking dimension compatibility, or applying scalar operations element-by-element when matrix multiplication is required.
Correction: For addition/subtraction, matrices must have identical dimensions. For multiplication, inner dimensions must match: (m × n)(n × p) = (m × p).
Exam technique for Number and Quantity
Show all conversion steps in dimensional analysis problems. Examiners award method marks for correct setup even if arithmetic errors occur. Write conversion factors explicitly as fractions and show unit cancellation.
Express complex numbers in standard form a + bi as the final answer unless otherwise specified. Leaving an answer as (3 + 4i)/(1 - i) without simplifying loses marks even if algebraically equivalent.
Verify closure properties with counterexamples when proving a set is NOT closed. One counterexample suffices to disprove closure; proving closure requires showing the property holds for all elements.
Check answer reasonableness for unit conversions. Converting 100 meters to centimeters should yield a larger number (10,000 cm); a smaller result indicates an inverted conversion factor.
Quick revision summary
Number and Quantity covers the real number system (rationals, irrationals, closure properties), rational exponents and radical simplification, the complex number system (operations with i, complex conjugates), dimensional analysis with unit conversions, and vector/matrix representations. Master exponent rules, especially fractional exponents as roots. For complex numbers, remember i² = -1 and multiply by conjugates when dividing. Set up dimensional analysis with conversion factors as fractions ensuring proper unit cancellation. Verify matrix dimension compatibility before operations. Check whether number sets are closed by testing specific operations with examples.