π’ Arithmetic Study Guide
1. Place Value
Place value tells us the value of a digit based on its position in a number. Each position is ten times the value of the position to its right.
| Millions | Hundred Thousands | Ten Thousands | Thousands | Hundreds | Tens | Ones |
|---|---|---|---|---|---|---|
| 1,000,000 | 100,000 | 10,000 | 1,000 | 100 | 10 | 1 |
\(3{,}456 = 3 \times 1{,}000 + 4 \times 100 + 5 \times 10 + 6 \times 1\)
\(= 3{,}000 + 400 + 50 + 6\)
2. Addition
Addition combines two or more numbers to find their sum. When adding multi-digit numbers, align digits by place value and work from right to left, carrying when a column sum exceeds 9.
If you have 3 apples and add 5 more, you get 8. Addition is just a fast way to count combined groups.
- Commutative: \(a + b = b + a\) (order does not matter: \(3 + 5 = 5 + 3\))
- Associative: \((a + b) + c = a + (b + c)\) (grouping does not matter)
- Identity: \(a + 0 = a\) (adding zero leaves a number unchanged)
3 8 7 + 4 5 6 ------- 8 4 3
7 + 6 = 13 β write 3, carry 1
8 + 5 + 1 = 14 β write 4, carry 1
3 + 4 + 1 = 8
3. Subtraction
Subtraction finds the difference between two numbers. When the digit on top is smaller than the digit below, you must borrow (regroup) from the next column.
6 0 3 - 2 4 8 ------- 3 5 5
3 < 8, so borrow: 13 β 8 = 5. The tens digit becomes 9 (after borrowing from hundreds).
9 < 4 is false β 9 β 4 = 5. The hundreds digit is now 5 (after lending to tens).
5 β 2 = 3.
4. Multiplication
Multiplication is repeated addition. The result is called the product.
| Γ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 2 | 2 | 4 | 6 | 8 | 10 | 12 | 14 | 16 | 18 | 20 | 22 | 24 |
| 3 | 3 | 6 | 9 | 12 | 15 | 18 | 21 | 24 | 27 | 30 | 33 | 36 |
| 4 | 4 | 8 | 12 | 16 | 20 | 24 | 28 | 32 | 36 | 40 | 44 | 48 |
| 5 | 5 | 10 | 15 | 20 | 25 | 30 | 35 | 40 | 45 | 50 | 55 | 60 |
| 6 | 6 | 12 | 18 | 24 | 30 | 36 | 42 | 48 | 54 | 60 | 66 | 72 |
| 7 | 7 | 14 | 21 | 28 | 35 | 42 | 49 | 56 | 63 | 70 | 77 | 84 |
| 8 | 8 | 16 | 24 | 32 | 40 | 48 | 56 | 64 | 72 | 80 | 88 | 96 |
| 9 | 9 | 18 | 27 | 36 | 45 | 54 | 63 | 72 | 81 | 90 | 99 | 108 |
| 12 | 12 | 24 | 36 | 48 | 60 | 72 | 84 | 96 | 108 | 120 | 132 | 144 |
- Commutative: \(a \times b = b \times a\)
- Associative: \((a \times b) \times c = a \times (b \times c)\)
- Distributive: \(a \times (b + c) = a \times b + a \times c\)
- Zero: \(a \times 0 = 0\)
- Identity: \(a \times 1 = a\)
\(47 \times 36\)
\(= 47 \times (30 + 6)\)
\(= 47 \times 30 + 47 \times 6\)
\(= 1{,}410 + 282 = 1{,}692\)
5. Division
Division splits a number into equal groups. The parts are: dividend Γ· divisor = quotient (with a possible remainder).
- Dividend: the number being divided (e.g., 17)
- Divisor: the number you divide by (e.g., 5)
- Quotient: the result (e.g., 3)
- Remainder: what is left over (e.g., 2), because \(5 \times 3 = 15\) and \(17 - 15 = 2\)
211 R 1
--------
4 ) 8 4 5
8
---
4
4
---
5
4
---
1
Check: \(4 \times 211 + 1 = 844 + 1 = 845\) β
6. Order of Operations (PEMDAS)
When an expression has multiple operations, you must follow a specific order. PEMDAS is the standard rule in the US.
- P β Parentheses (innermost first)
- E β Exponents (powers and roots)
- M/D β Multiplication and Division (left to right, equal priority)
- A/S β Addition and Subtraction (left to right, equal priority)
Memory trick: "Please Excuse My Dear Aunt Sally"
Multiplication first: \(3 + 8 = \mathbf{11}\) (NOT \(7 \times 2 = 14\))
Step 1 β Parentheses: \(10 - 4 = 6\) β \(2^3 + 6 \div 2\)
Step 2 β Exponents: \(2^3 = 8\) β \(8 + 6 \div 2\)
Step 3 β Division: \(6 \div 2 = 3\) β \(8 + 3\)
Step 4 β Addition: \(= \mathbf{11}\)
7. Factors and Multiples
Factors are numbers that divide evenly into a given number. Multiples are what you get when you multiply a number by counting numbers (1, 2, 3, β¦).
The largest factor shared by two or more numbers. Used to simplify fractions.
Example: GCF of 24 and 36Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
Common factors: 1, 2, 3, 4, 6, 12 β GCF = 12
The smallest multiple shared by two or more numbers. Used to find common denominators.
Example: LCM of 4 and 6Multiples of 4: 4, 8, 12, 16, 20, 24, β¦
Multiples of 6: 6, 12, 18, 24, β¦
LCM = 12
\(24 = 2^3 \times 3\) \(36 = 2^2 \times 3^2\)
GCF = product of lowest powers of common primes = \(2^2 \times 3 = 12\)
LCM = product of highest powers of all primes = \(2^3 \times 3^2 = 72\)
8. Practice Problems
- Write 2,047,385 in expanded form.
- Calculate: \(4{,}008 - 1{,}739\)
- Calculate: \(325 \times 47\)
- Calculate: \(1{,}296 \div 12\) (find the quotient and remainder)
- Evaluate: \(5 + 3^2 \times (8 - 5) \div 3\)
- Find the GCF and LCM of 18 and 30.
- A store sells 24 bottles of juice in one box. How many full boxes can be made from 580 bottles, and how many bottles are left over?
- Write true or false: \(15 \div 3 \times 5 = 15 \div (3 \times 5)\). Explain why.
- Order of operations: PEMDAS (Parentheses, Exponents, Multiplication/Division, Addition/Subtraction)
- Check: subtraction is NOT commutative (aβb β bβa)
- Division by zero is undefined
- Verify: divisor Γ quotient + remainder = dividend
Review this material at increasing intervals to commit it to long-term memory.