🔢 Arithmetic — Reference

Essential arithmetic concepts and operations that form the foundation of MATHCOUNTS success.

Basic Operations

Addition

Definition: Combining two or more numbers to find their sum Properties:

  • Commutative: $a + b = b + a$
  • Associative: $(a + b) + c = a + (b + c)$
  • Identity: $a + 0 = a$

Mental Math Tricks:

  • Adding 9: Add 10, subtract 1
  • Adding 11: Add 10, add 1
  • Adding near multiples of 10: Use compatible numbers

Subtraction

Definition: Finding the difference between two numbers Properties:

  • Not commutative: $a - b \neq b - a$ (unless $a = b$)
  • Not associative: $(a - b) - c \neq a - (b - c)$
  • Identity: $a - 0 = a$

Mental Math Tricks:

  • Subtracting 9: Subtract 10, add 1
  • Subtracting 11: Subtract 10, subtract 1
  • Subtracting near multiples of 10: Use compatible numbers

Multiplication

Definition: Repeated addition or finding the product of two numbers Properties:

  • Commutative: $a \times b = b \times a$
  • Associative: $(a \times b) \times c = a \times (b \times c)$
  • Distributive: $a \times (b + c) = a \times b + a \times c$
  • Identity: $a \times 1 = a$
  • Zero: $a \times 0 = 0$

Mental Math Tricks:

  • Multiplying by 5: Multiply by 10, divide by 2
  • Multiplying by 25: Multiply by 100, divide by 4
  • Multiplying by 11: Add digits and place in middle

Division

Definition: Finding how many times one number fits into another Properties:

  • Not commutative: $a \div b \neq b \div a$ (unless $a = b$)
  • Not associative: $(a \div b) \div c \neq a \div (b \div c)$
  • Identity: $a \div 1 = a$
  • Zero: $a \div 0$ is undefined

Mental Math Tricks:

  • Dividing by 5: Multiply by 2, divide by 10
  • Dividing by 25: Multiply by 4, divide by 100
  • Dividing by 4: Divide by 2 twice

Number Properties

Even and Odd Numbers

Even numbers: Divisible by 2 (end in 0, 2, 4, 6, 8) Odd numbers: Not divisible by 2 (end in 1, 3, 5, 7, 9)

Properties:

  • Even + Even = Even
  • Odd + Odd = Even
  • Even + Odd = Odd
  • Even × Even = Even
  • Odd × Odd = Odd
  • Even × Odd = Even

Prime and Composite Numbers

Prime numbers: Natural numbers greater than 1 with exactly two factors (1 and itself) Composite numbers: Natural numbers greater than 1 with more than two factors

First 25 primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97

Divisibility Rules

  • 2: Last digit is even (0, 2, 4, 6, 8)
  • 3: Sum of digits is divisible by 3
  • 4: Last two digits form a number divisible by 4
  • 5: Last digit is 0 or 5
  • 6: Divisible by both 2 and 3
  • 8: Last three digits form a number divisible by 8
  • 9: Sum of digits is divisible by 9
  • 10: Last digit is 0
  • 11: Alternating sum of digits is divisible by 11

Order of Operations

PEMDAS/BODMAS

Parentheses/Brackets: Do operations inside parentheses first Exponents/Orders: Calculate powers and roots Multiplication/Division: From left to right Addition/Subtraction: From left to right

Example: $2 + 3 \times 4^2 - (6 + 2) \div 2$

  1. Parentheses: $6 + 2 = 8$
  2. Exponents: $4^2 = 16$
  3. Multiplication: $3 \times 16 = 48$
  4. Division: $8 \div 2 = 4$
  5. Addition: $2 + 48 = 50$
  6. Subtraction: $50 - 4 = 46$

Number Sense

Place Value

Whole numbers: Ones, tens, hundreds, thousands, etc. Decimals: Tenths, hundredths, thousandths, etc.

Example: 3,247.56

  • 3 thousands
  • 2 hundreds
  • 4 tens
  • 7 ones
  • 5 tenths
  • 6 hundredths

Rounding

To the nearest ten: Look at the ones digit

  • 0-4: Round down
  • 5-9: Round up

To the nearest hundred: Look at the tens digit

  • 0-4: Round down
  • 5-9: Round up

Example: Round 3,247 to the nearest hundred

  • Tens digit is 4, so round down
  • Answer: 3,200

Estimation

Front-end estimation: Use the first digit of each number Compatible numbers: Use numbers that are easy to work with Benchmark estimation: Use known reference points

Example: Estimate $347 + 256$

  • Front-end: $300 + 200 = 500$
  • Compatible: $350 + 250 = 600$
  • Actual: $603$

Mental Math Strategies

Breaking Numbers Apart

Addition: $47 + 23 = (40 + 20) + (7 + 3) = 60 + 10 = 70$ Subtraction: $73 - 28 = (70 - 20) + (3 - 8) = 50 - 5 = 45$ Multiplication: $24 \times 5 = (20 \times 5) + (4 \times 5) = 100 + 20 = 120$

Using Properties

Commutative: $47 + 23 = 23 + 47 = 70$ Associative: $(47 + 23) + 15 = 47 + (23 + 15) = 47 + 38 = 85$ Distributive: $6 \times 47 = 6 \times (40 + 7) = 240 + 42 = 282$

Shortcuts

Doubling: $47 \times 2 = 94$ Halving: $94 \div 2 = 47$ Multiplying by 10: Add a zero Dividing by 10: Remove a zero

Common Mistakes

Careless Errors

  • Arithmetic mistakes: $2 + 3 = 6$ (should be 5)
  • Sign errors: Forgetting negative signs
  • Copying errors: Writing 47 instead of 74
  • Order of operations: $2 + 3 \times 4 = 20$ (should be 14)

Concept Errors

  • Using wrong operation: Adding instead of multiplying
  • Misapplying properties: $(a + b)^2 = a^2 + b^2$ (incorrect)
  • Forgetting rules: $a \div 0$ is undefined, not 0
  • Confusing operations: Subtraction is not commutative

Prevention Strategies

  • Slow down and double-check arithmetic
  • Show work clearly to avoid copying errors
  • Use estimation to check reasonableness
  • Practice regularly to build fluency

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