Multiplication
Repeated addition into a product · 곱셈
1. What Is Multiplication?
Multiplication is repeated addition: 4 × 3 means “add 4 three times,” 4 + 4 + 4 = 12. The numbers being multiplied are factors, and the result is the product. As numbers grow, repeated addition is too slow, so we use place-value algorithms instead.
Multiplication also scales quantities (3 boxes of 12 eggs) and computes areas (a 4-by-3 grid has 12 cells), which is why the area model is such a powerful way to picture it.
곱셈은 반복 덧셈입니다(4 × 3 = 4 + 4 + 4 = 12). 곱하는 수를 인수(factor), 결과를 곱(product)이라 합니다. 양을 배수로 늘리거나 넓이(4×3 격자 = 12칸)를 구하는 데 쓰여, 넓이 모형으로 직관적으로 이해할 수 있습니다.
2. Long Multiplication & a Worked Example
Multiply by one digit of the bottom number at a time, shifting each partial product one place left for each higher place value, then add the partial products.
Worked example. Multiply 27 × 34:
- 27 × 4 = 108 (the ones partial product).
- 27 × 30 = 810 (the tens partial product — note the trailing zero).
- Add: 108 + 810 = 918.
So 27 × 34 = 918. Because multiplication is commutative, 34 × 27 gives the same 918 — a handy cross-check.
아래 수의 한 자리씩 곱해 부분곱을 만들고, 윗자리일수록 한 칸씩 왼쪽으로 밀어 더합니다. 예: 27 × 34 → 27×4 = 108, 27×30 = 810, 합 918. 교환법칙으로 34 × 27도 918입니다.
3. The Area Model & Mental Shortcuts
Every shortcut is the distributive law a(b + c) = ab + ac in disguise. Picture 27 × 34 as a rectangle split into pieces by place value: (20 + 7) × (30 + 4) = 600 + 80 + 210 + 28 = 918 — the four areas of the grid.
Two mental tricks fall out:
- Split by place value. 27 × 8 = 20 × 8 + 7 × 8 = 160 + 56 = 216.
- Doubling and halving. 16 × 25 = 8 × 50 = 4 × 100 = 400.
모든 단축은 분배법칙 a(b + c) = ab + ac입니다. 27 × 34를 (20+7)(30+4) = 600+80+210+28 = 918로 보는 넓이 모형, 자릿값 분리(27×8 = 160+56 = 216), 두 배·반(16×25 = 8×50 = 400)이 대표적입니다.
4. Key Properties
- Commutative. a × b = b × a.
- Associative. (a × b) × c = a × (b × c).
- Distributive over addition. a × (b + c) = a × b + a × c.
- Multiplicative identity. a × 1 = a.
- Zero property. a × 0 = 0 — any number times zero is zero.
곱셈은 교환(a × b = b × a)·결합((a × b) × c = a × (b × c))·분배(a × (b + c) = a × b + a × c) 법칙을 만족하고, 항등원 a × 1 = a, 영원소 a × 0 = 0 이 성립합니다.
5. Frequently Asked Questions
How do you do long multiplication? Multiply by each digit of the bottom number separately, shift each partial product left by its place value, then add the partial products.
Why does multiplying by 10 add a zero? Because base-10 place value shifts every digit one place left; appending a zero is exactly that shift.
Is multiplication commutative? Yes — a × b = b × a, which is why swapping the factors is a quick way to check a product.
아래 수의 각 자리로 따로 곱해 부분곱을 자릿값만큼 왼쪽으로 밀어 더합니다. 10을 곱하면 자릿수가 한 칸 이동하므로 0이 붙습니다. a × b = b × a 로 교환법칙이 성립합니다.
Ready to practice? Drill multiplication and the other operations on C:Arith, or review the full arithmetic reference and the related addition, subtraction, and division.