Authoritative Content Ecosystem
Data Source and Methodology
Authoritative Source: OpenStax, Prealgebra 2e, Sections 5.5 (Divide Whole Numbers) and 6.2 (Divide Decimals), 2022. Direct link: https://openstax.org/details/books/prealgebra-2e
Tutti i calcoli si basano rigorosamente sulle formule e sui dati forniti da questa fonte.
The Formula Explained
Remainder form (integers):
LaTeX: Dividend = Divisor \times Quotient + Remainder,\quad 0 \le Remainder < |Divisor|
Decimal expansion:
LaTeX: Quotient = \frac{Dividend}{Divisor}
Decimal alignment (for inputs with decimals):
LaTeX: \frac{a \cdot 10^{-s_a}}{b \cdot 10^{-s_b}} = \left(\frac{a}{b}\right)\cdot 10^{\,s_b - s_a}, where s_a and s_b are the counts of decimal digits in the dividend and divisor respectively.
Rounding to p places:
LaTeX: \text{round}_p\!\left(\frac{a}{b}\right) = \frac{\left\lfloor \frac{a \cdot 10^{p}}{b} + \tfrac{1}{2} \right\rfloor}{10^{p}}
Glossary of Variables
- Dividend: The number being divided.
- Divisor: The number you divide by (must not be 0).
- Quotient: The result of the division.
- Remainder: What is left over in integer division, with 0 ≤ remainder < |divisor|.
- Decimal places: The number of digits shown to the right of the decimal point in the quotient (decimal mode).
Worked Example
How It Works: A Step-by-Step Example
Example: Compute 1234 ÷ 56 to 4 decimal places.
- Set up: 56 goes into 123 two times. Subtract 112, remainder 11. Bring down 4 → 114.
- 56 goes into 114 two times. Subtract 112, remainder 2. Integer quotient so far: 22.
- For decimals, append a decimal point and bring down 0: remainder 2 → 20. 56 goes into 20: 0 times. Append 0 to quotient → 22.0; bring down another 0 → 200.
- 56 goes into 200 three times (168). Remainder 32. Bring down 0 → 320 → 5 times (280). Remainder 40 → bring down 0 → 400 → 7 times (392). Remainder 8 → bring down 0 → 80 → 1 time (56). Remainder 24.
- Continue for 4 decimals, rounding the last digit if needed. The final result is approximately 22.0357.
This matches the formula Quotient = \frac{Dividend}{Divisor} with rounding to the desired number of places.
Frequently Asked Questions (FAQ)
Does the remainder have to be non-negative?
Yes. We follow the convention 0 ≤ remainder < |divisor| and apply the overall sign to the quotient.
Can I input negative numbers?
Yes. The sign of the result is determined by the signs of the inputs. The remainder (if shown) is non-negative by definition.
How does the tool handle decimals internally?
It temporarily removes decimal points by scaling both numbers by powers of 10, performs long division using integer arithmetic, and then places the decimal point correctly in the quotient.
How are repeating decimals indicated?
If a remainder repeats while generating digits, the repeating block is highlighted. Screen readers announce it as “repeating sequence.”
What precision should I choose?
For schoolwork, 3–6 places are common. For financial contexts, 2–4 places are typical. You can set from 0 to 50 places here.
Why can’t I use decimals in remainder mode?
Remainder is defined for integer division. If your inputs include decimals, switch to Decimal expansion mode.
What happens if I divide by zero?
The calculator prevents it and displays an accessible error message with suggestions to correct the input.