Division
Split into equal parts · 나눗셈
1. What Is Division?
Division splits a quantity into equal parts. In a ÷ b, the dividend a is shared by the divisor b to give the quotient — and whatever is left over is the remainder. Division answers two related questions: “how big is each share?” (sharing) and “how many groups fit?” (grouping).
Division is the inverse of multiplication: because 4 × 6 = 24, we have 24 ÷ 6 = 4. That link lets you check every division by multiplying back.
나눗셈은 양을 같은 크기로 나눕니다. a ÷ b에서 a를 피제수(dividend), b를 제수(divisor), 결과를 몫(quotient), 남은 것을 나머지(remainder)라 합니다. "한 묶음이 얼마인가"(등분)와 "몇 묶음인가"(묶음)에 답하며, 곱셈의 역(24 ÷ 6 = 4 ⇔ 4 × 6 = 24)입니다.
2. Long Division & a Worked Example
Work through the dividend one digit at a time, left to right: divide, multiply, subtract, then bring down the next digit.
Worked example. Divide 950 ÷ 4:
- 9 ÷ 4 = 2, remainder 1 (2 × 4 = 8; 9 − 8 = 1).
- Bring down 5 → 15 ÷ 4 = 3, remainder 3 (3 × 4 = 12; 15 − 12 = 3).
- Bring down 0 → 30 ÷ 4 = 7, remainder 2 (7 × 4 = 28; 30 − 28 = 2).
So 950 ÷ 4 = 237 remainder 2. Check: 237 × 4 + 2 = 948 + 2 = 950. ✓
피제수를 왼쪽부터 한 자리씩 처리합니다: 나누고·곱하고·빼고·내려쓰기. 예: 950 ÷ 4 → 9÷4=2(나머지1), 15÷4=3(나머지3), 30÷4=7(나머지2) → 몫 237, 나머지 2. 검산: 237 × 4 + 2 = 950.
3. Estimate and Refine
For mental division, bracket the answer, then refine. To compute 432 ÷ 18: twenty 18s make 360 and thirty make 540, so the quotient is between 20 and 30. Try 24: 24 × 18 = 24 × 20 − 24 × 2 = 480 − 48 = 432. Exact, so 432 ÷ 18 = 24.
This estimate–multiply–adjust loop is exactly what long division formalizes one digit at a time.
암산 나눗셈은 답의 범위를 잡고 보정합니다. 432 ÷ 18은 20~30 사이 → 24 시도 → 24 × 18 = 432, 따라서 24. 이 "추정·곱·조정" 과정이 곧 긴 나눗셈입니다.
4. Key Properties
- Not commutative. a ÷ b ≠ b ÷ a in general (12 ÷ 4 = 3 but 4 ÷ 12 = ⅓).
- Not associative. (a ÷ b) ÷ c ≠ a ÷ (b ÷ c) in general.
- Identity on the right. a ÷ 1 = a, and a ÷ a = 1 (for a ≠ 0).
- Division by zero is undefined. There is no number that, times 0, gives a nonzero a — so a ÷ 0 has no answer.
- Inverse of multiplication. a ÷ b = a × (1/b); dividing is multiplying by the reciprocal.
나눗셈은 교환·결합 법칙이 성립하지 않습니다. a ÷ 1 = a, a ÷ a = 1(a ≠ 0)이며, 0으로 나누는 것은 정의되지 않습니다. a ÷ b = a × (1/b)로 역수를 곱하는 것과 같습니다.
5. Frequently Asked Questions
How do you do long division? Repeat four steps left to right across the dividend: divide the current number by the divisor, multiply, subtract, then bring down the next digit.
What is a remainder? The amount left over when the divisor does not divide the dividend evenly — always smaller than the divisor.
Why can’t you divide by zero? Because a ÷ 0 would need a number that times 0 equals a; no such number exists for nonzero a, so the operation is undefined.
나누기·곱하기·빼기·내려쓰기 네 단계를 왼쪽부터 반복합니다. 나머지는 정확히 나누어떨어지지 않을 때 남는 양으로 항상 제수보다 작습니다. 0으로 나누면 곱해서 a가 되는 수가 없으므로 정의되지 않습니다.
Ready to practice? Drill division and the other operations on C:Arith, or review the full arithmetic reference and the related addition, subtraction, and multiplication.