Mathematics · 13 min read
Numbers and Operations
Understanding the Number System
The number system is built in nested sets. Natural numbers (counting numbers) are 1, 2, 3, 4, …. Whole numbers add zero: 0, 1, 2, 3, …. Integers add the negatives: …, −3, −2, −1, 0, 1, 2, 3, ….
Rational numbers can be written as a fraction p/q, where p and q are integers and q ≠ 0. Every integer is rational (5 = 5/1), and so is every terminating decimal (0.25 = 1/4) and every repeating decimal (0.333… = 1/3). Irrational numbers cannot be written as a fraction of two integers; their decimals never terminate and never repeat (π, √2, √5). Together, the rational and irrational numbers make up the real numbers, which fill the entire number line.
Place Value and Base-Ten System
The base-ten system uses ten digits (0–9) and positional notation: each place is worth 10 times the place to its right and 1/10 of the place to its left. Moving left from the decimal point: ones (10⁰), tens (10¹), hundreds (10²), thousands (10³). Moving right: tenths (10⁻¹), hundredths (10⁻²), thousandths (10⁻³).
| Digit | Place | Value | Expanded form |
|---|---|---|---|
| 4 | Thousands | 4 × 1,000 | 4,000 |
| 5 | Hundreds | 5 × 100 | 500 |
| 2 | Tens | 2 × 10 | 20 |
| 7 | Ones | 7 × 1 | 7 |
| 3 | Tenths | 3 × 0.1 | 0.3 |
| 8 | Hundredths | 8 × 0.01 | 0.08 |
Expanded form writes a number as the sum of its place values: 4,527.38 = 4,000 + 500 + 20 + 7 + 0.3 + 0.08. In the same number the 5 is worth 500 and the 8 is worth 0.08, so the same digit's value depends entirely on its position. Place value is the basis for comparing, rounding, and every multi-digit algorithm.
Rounding and Estimation
To round, find the place you are rounding to and look at the digit immediately to its right. If that digit is 5 or greater, round up (add 1 to the rounding digit); if it is 4 or less, keep the rounding digit. Replace the digits to the right with zeros (whole numbers) or drop them (decimals).
- 47,862 to the nearest thousand: the hundreds digit is 8 (≥ 5), so round up → 48,000
- 3.276 to the nearest tenth: the hundredths digit is 7, so round up → 3.3
- 3.276 to the nearest hundredth: the thousandths digit is 6, so round up → 3.28
- 6,349 to the nearest hundred: the tens digit is 4, so keep 3 → 6,300
Estimation replaces exact numbers with friendly ones to get a quick, approximate answer. Common strategies: rounding (398 × 21 ≈ 400 × 20 = 8,000), front-end estimation (use the leading digits, then adjust), compatible numbers (numbers that divide easily: 358 ÷ 6 ≈ 360 ÷ 6 = 60), and benchmarks (0.48 is close to ½, so 0.48 × 61 ≈ 30).
Reasonableness: always compare an exact answer with an estimate. The exact product 398 × 21 is 8,358, close to the estimate of 8,000, so it is reasonable; an answer such as 835.8 or 83,580 would signal a place-value or decimal-point error. Rounding both factors up gives an overestimate, and rounding both down gives an underestimate — useful when deciding whether a budget is enough.
Properties of Operations
| Property | Rule | Example |
|---|---|---|
| Commutative (addition) | a + b = b + a | 3 + 5 = 5 + 3 = 8 |
| Commutative (multiplication) | a × b = b × a | 4 × 7 = 7 × 4 = 28 |
| Associative (addition) | (a + b) + c = a + (b + c) | (2 + 3) + 4 = 2 + (3 + 4) = 9 |
| Associative (multiplication) | (a × b) × c = a × (b × c) | (2 × 3) × 4 = 2 × (3 × 4) = 24 |
| Distributive | a × (b + c) = a × b + a × c | 3 × (4 + 5) = 12 + 15 = 27 |
| Additive identity | a + 0 = a | 9 + 0 = 9 |
| Multiplicative identity | a × 1 = a | 9 × 1 = 9 |
| Additive inverse | a + (−a) = 0 | 6 + (−6) = 0 |
| Multiplicative inverse | a × (1/a) = 1, a ≠ 0 | 4 × ¼ = 1 |
| Zero property of multiplication | a × 0 = 0 | 15 × 0 = 0 |
Operations with Whole Numbers
Addition combines quantities. Align place values and regroup (carry) when a column sum is 10 or more. Subtraction finds a difference; regroup (borrow) when the digit in the minuend is smaller than the digit being subtracted in that place.
Multiplication can be modeled as repeated addition, equal groups, or an array/area model. Multi-digit multiplication multiplies by each digit of the multiplier and adds the partial products. Division is the inverse of multiplication; long division repeats divide, multiply, subtract, bring down. A remainder can be reported as a remainder, a fraction, or a decimal, depending on the context.
- Compensation: 298 + 145 = (300 + 145) − 2 = 443
- Partial products: 23 × 14 = (20 × 10) + (20 × 4) + (3 × 10) + (3 × 4) = 200 + 80 + 30 + 12 = 322
- Partial quotients: 156 ÷ 12 → 10 groups of 12 = 120, leaving 36; 3 groups of 12 = 36; quotient = 10 + 3 = 13
- Interpreting a remainder: 50 students ride in vans that hold 8 each. 50 ÷ 8 = 6 R2, so 7 vans are needed (round up).
Operations with Fractions
Equivalent fractions name the same amount: multiply or divide the numerator and denominator by the same nonzero number (3/4 = 6/8 = 9/12). A fraction is in simplest form when the GCF of the numerator and denominator is 1. To compare fractions, rewrite them with a common denominator or cross-multiply: 5/8 vs. 3/5 → 5 × 5 = 25 and 8 × 3 = 24, so 5/8 > 3/5.
Adding and subtracting: find the least common denominator (LCD), rewrite each fraction, then add or subtract the numerators and keep the denominator. Multiplying: multiply numerators and multiply denominators, then simplify. Dividing: multiply by the reciprocal of the divisor (“keep, change, flip”). Mixed numbers: convert to improper fractions before multiplying or dividing.
- 2/3 + 1/4: LCD = 12, so 8/12 + 3/12 = 11/12
- 2/3 × 4/5 = (2 × 4)/(3 × 5) = 8/15
- 2/3 ÷ 4/5 = 2/3 × 5/4 = 10/12 = 5/6
- 2½ × 1⅓ = 5/2 × 4/3 = 20/6 = 10/3 = 3⅓
Operations with Decimals
Adding and subtracting: line up the decimal points (annex zeros if needed) and compute as with whole numbers. Multiplying: multiply as whole numbers, then give the product as many decimal places as the factors have in total. Dividing: move the decimal point in the divisor to make it a whole number, move the decimal point in the dividend the same number of places, then divide.
- 2.5 × 1.3 = 3.25 (1 + 1 = 2 decimal places)
- 7.56 ÷ 0.4 → 75.6 ÷ 4 = 18.9
- 12.4 − 3.75 → 12.40 − 3.75 = 8.65
Fractions, Decimals, and Percents
Fractions, decimals, and percents are three ways to write the same rational number. Fraction → decimal: divide the numerator by the denominator (3/8 = 3 ÷ 8 = 0.375). Decimal → percent: multiply by 100 (0.375 = 37.5%). Percent → decimal: divide by 100 (6% = 0.06). Decimal → fraction: read the place value and simplify (0.45 = 45/100 = 9/20).
| Fraction | Decimal | Percent |
|---|---|---|
| 1/2 | 0.5 | 50% |
| 1/3 | 0.333… | 33⅓% |
| 2/3 | 0.666… | 66⅔% |
| 1/4 | 0.25 | 25% |
| 3/4 | 0.75 | 75% |
| 1/5 | 0.2 | 20% |
| 1/8 | 0.125 | 12.5% |
| 1/10 | 0.1 | 10% |
| 1/100 | 0.01 | 1% |
Factors, Multiples, and Divisibility
Factors divide a number evenly (factors of 12: 1, 2, 3, 4, 6, 12). Multiples are the products of a number and the counting numbers (multiples of 4: 4, 8, 12, 16, …). A prime number has exactly two factors, 1 and itself (2, 3, 5, 7, 11, 13); 2 is the only even prime. A composite number has more than two factors (4, 6, 8, 9, 10). The number 1 is neither prime nor composite.
Prime factorization writes a number as a product of primes (36 = 2² × 3²); a factor tree is a common tool. The greatest common factor (GCF) is the largest factor two numbers share: use the primes they have in common, with the smaller exponent. The least common multiple (LCM) is the smallest multiple they share: use every prime, with the larger exponent.
- 24 = 2³ × 3 and 36 = 2² × 3²
- GCF = 2² × 3 = 12 (largest equal groups: 24 pencils and 36 erasers make 12 identical kits)
- LCM = 2³ × 3² = 72 (first time both counts line up again)
- Buses leave every 12 and every 18 minutes together at 8:00 AM; LCM(12, 18) = 36, so they next leave together at 8:36 AM.
Ratios, Rates, and Percents
A ratio compares two quantities: 3 to 4, 3:4, or 3/4. A rate compares quantities with different units (150 miles in 3 hours), and a unit rate has a denominator of 1 (50 miles per hour). A proportion states that two ratios are equal; solve it with cross-multiplication (if a/b = c/d, then a × d = b × c) or by scaling. In a proportional relationship, the ratio between the two quantities stays constant.
- Unit price: 12 oz for $4.20 → $4.20 ÷ 12 = $0.35 per oz; 16 oz for $5.44 → $5.44 ÷ 16 = $0.34 per oz. The 16-oz size is the better buy.
- Proportion: A recipe uses 3 cups of flour for 24 cookies. For 40 cookies: 3/24 = x/40 → 24x = 120 → x = 5 cups.
Percent means “per hundred.” To find a percent of a number, multiply by the percent written as a decimal (25% of 80 = 0.25 × 80 = 20). Percent change = (new − original) ÷ original × 100%. For a discount or a tax, multiply by (1 − rate) or (1 + rate).
- Percent increase: $40 → $50: change = 10; 10 ÷ 40 = 0.25 = 25% increase
- Percent decrease: $50 → $40: change = 10; 10 ÷ 50 = 0.20 = 20% decrease
- Discount then tax: an $80 jacket is 25% off with 6% sales tax: 80 × 0.75 = $60; 60 × 1.06 = $63.60
Integers and Absolute Value
Integers are the whole numbers and their opposites. Absolute value is a number's distance from zero on the number line, so it is never negative: |−5| = 5 and |5| = 5. On a number line, numbers increase to the right, so −8 < −3 even though 8 > 3.
Order of Operations
When an expression has more than one operation, evaluate in this order (often remembered as PEMDAS):
- Parentheses and other grouping symbols (brackets, fraction bars)
- Exponents and roots
- Multiplication and Division, left to right
- Addition and Subtraction, left to right
Exponents and Roots
An exponent shows repeated multiplication: 5³ = 5 × 5 × 5 = 125 (not 5 × 3). A square root √a is the nonnegative number that, multiplied by itself, gives a (√25 = 5). A cube root ∛a is the number used as a factor three times to give a (∛27 = 3). Perfect squares to know: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144.
| Law | Rule | Example |
|---|---|---|
| Product of powers | aᵐ × aⁿ = aᵐ⁺ⁿ | 2³ × 2² = 2⁵ = 32 |
| Quotient of powers | aᵐ ÷ aⁿ = aᵐ⁻ⁿ | 10⁶ ÷ 10² = 10⁴ |
| Power of a power | (aᵐ)ⁿ = aᵐⁿ | (3²)³ = 3⁶ = 729 |
| Zero exponent | a⁰ = 1 (a ≠ 0) | 7⁰ = 1 |
| Negative exponent | a⁻ⁿ = 1/aⁿ | 2⁻³ = 1/8 |
Scientific Notation
Scientific notation writes a number as a factor from 1 up to (but not including) 10, times a power of 10. Large numbers have positive exponents (45,000,000 = 4.5 × 10⁷); small numbers have negative exponents (0.00032 = 3.2 × 10⁻⁴). To multiply or divide, work with the factors and the powers of 10 separately, then adjust so the factor is between 1 and 10.
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