Short division
Short division is a formal written method in arithmetic for dividing a multi-digit dividend by a single-digit divisor, offering a more compact alternative to long division by minimizing written steps and focusing on digit-by-digit division.[1] It is particularly efficient for calculations where the divisor is between 2 and 9, allowing the quotient to be built progressively above a line while handling remainders by carrying them to the next digit.[2]
Often referred to as the "bus stop method" in primary education, especially within the UK curriculum, short division is introduced to children in Key Stage 2 (typically years 4–6) to develop fluency in division after mastering multiplication tables and basic partitioning.[3] The method begins by writing the divisor outside a bracket resembling a bus stop, with the dividend inside; each digit of the dividend is then divided sequentially from left to right, recording the largest possible multiple below and the remainder (if any) to combine with the next digit.[1] This approach supports exact divisions yielding whole numbers, as well as those producing remainders, which can be expressed as fractions, decimals, or simply noted (e.g., "r3" for a remainder of 3).[2]
Short division emphasizes conceptual understanding of division as sharing or repeated subtraction, linking to real-world applications like dividing resources or costs, and serves as a foundational skill before progressing to long division for two-digit or larger divisors.[3] Unlike chunking or grid methods used in earlier primary years, it provides a structured, algorithmic process that builds accuracy and speed for larger numbers, typically up to four digits.[1] For non-exact results, a decimal point is added to the quotient, with zeros appended to the dividend as needed to continue the process, and repeating decimals marked accordingly.[2]
Fundamentals
Definition and purpose
Short division is a streamlined arithmetic technique designed for dividing multi-digit numbers by a single-digit divisor (typically 1 through 9), employing a compact written format that emphasizes mental calculations over extensive recording. Unlike more elaborate methods, it avoids the full expansion of partial quotients and remainders on paper, making it suitable for straightforward computations where the divisor is small. This approach relies on partitioning the dividend into manageable parts, often visualized in a simplified layout that facilitates quick progression through the division process.[1][4]
The primary purpose of short division is to foster division fluency in elementary education, serving as an accessible entry point for students learning to handle larger numbers without overwhelming procedural complexity. It is particularly efficient for rapid mental or semi-written calculations in everyday scenarios, such as sharing quantities or computing ratios, and contrasts with long division by reducing verbosity when the divisor does not exceed one digit. In curricula, it builds conceptual understanding of division as repeated subtraction or grouping, preparing learners for advanced topics while reinforcing number sense.[1][5]
Short division was first described in 1491 by Filippo Calandri. It appeared in 19th-century arithmetic textbooks within British and American educational systems, where it was introduced as a practical method to teach number partitioning and place value application. Known in some British contexts as the "bus stop" method due to its visual resemblance, it has been a staple in primary mathematics since the late 1800s, evolving to align with modern standards like the UK National Curriculum for Key Stages 2.[5][6][7]
Effective use of short division presupposes foundational knowledge of multiplication tables up to 10×10 and a solid grasp of place value, enabling students to estimate quotients and handle remainders mentally.[1]
Notation
In short division, the tableau is structured with the divisor positioned to the left of a vertical line or outside a right-facing bracket, while the dividend is placed beneath a horizontal line or within the bracket, forming a compact "bus stop" shape that resembles a right angle.[2] The quotient is inscribed above the horizontal line or bracket, aligned directly over the corresponding digits of the dividend, and any remainder is typically noted at the end of the quotient or superscripted/subscripted adjacent to the relevant digit.[8] This layout eliminates the need for explicit subtraction lines or intermediate calculations visible in long division, emphasizing mental arithmetic for each step.[9]
Brackets or short vertical lines may also indicate the boundaries of the dividend, with no additional underlining or crossing out required, distinguishing it from more verbose methods.[2] For decimal extensions, a decimal point is aligned above and below the line, and repeating portions can be marked with overlines or dots.[8]
Variations in notation exist regionally; the "bus stop" method, prevalent in British education, uses a distinct bracket-like enclosure for the dividend with the divisor externally adjacent, promoting visual simplicity for young learners.[2] In contrast, American and some international notations (e.g., in Mexico) abbreviate long division by placing the quotient above a standard division bar (resembling ) without full bracketing, and remainders may be superscripted before the next digit or subscripted at the conclusion.[9] Remainder placement differs as well, with British styles often appending it directly after the quotient (e.g., 24 r2), while U.S. variants might integrate it inline or denote it separately as "remainder =".[8]
A key visual representation of the empty tableau in the bus stop method appears as follows:
_____
| |
d | dividend
-----
_____
| |
d | dividend
-----
Here, d denotes the divisor, the vertical line forms the "stop" side, and the horizontal line underlies the dividend, ready for quotient entry above.[2]
Method
Steps
Short division, also known as the bus stop method, follows a structured algorithm that relies heavily on mental arithmetic to compute the quotient digit by digit. The process begins by identifying the divisor and the dividend, with the dividend partitioned into chunks corresponding to the divisor's digit length—typically one digit for single-digit divisors. For each chunk, the quotient digit is determined through trial multiplication, where the largest multiple of the divisor that fits within the current partial dividend is selected. This quotient digit is then recorded above the corresponding dividend digit, and the product is subtracted mentally from the partial dividend to yield a remainder. The next dividend digit is brought down and appended to this remainder to form the subsequent partial dividend, repeating the process until all digits are processed.[2][10]
Mental strategies are central to short division, leveraging memorized multiplication facts such as times tables to quickly identify the appropriate quotient digit for each partial dividend. For instance, knowing that a divisor times a certain digit yields a product less than or equal to the partial dividend allows for rapid estimation, with adjustments made if the initial trial exceeds the partial dividend. Remainders, which are always less than the divisor, are carried over to the next step without explicit subtraction notation, maintaining the method's brevity and emphasis on internal calculation.[2][1]
If a partial dividend (or initial chunk) is smaller than the divisor, a quotient digit of zero is placed above it, and the process proceeds by incorporating the next dividend digit to form a larger partial dividend, effectively borrowing from subsequent digits to enable division. This handles cases where exact division is not possible at that stage, ensuring the algorithm progresses without interruption. For exactness, the method yields whole-number quotients when the division is precise, but remainders indicate inexactness, which may be expressed as fractions or decimals depending on the required precision. Mathematically, each quotient digit q is computed as q = \left\lfloor \frac{d}{v} \right\rfloor, where d is the current partial dividend and v is the divisor, with the remainder r = d - q \cdot v where $0 \leq r < v.[2][10]
Example
To illustrate the short division method, consider dividing 624 by 3. Set up the tableau with the divisor 3 to the left of a right-facing bracket and the dividend 624 inside the bracket, aligned by place value: hundreds, tens, and units digits.[2]
Begin with the hundreds digit: 3 goes into 6 exactly 2 times (since $3 \times 2 = 6), with no remainder; subtract 6 from 6 to get 0, and write the quotient digit 2 above the hundreds place. The tableau now appears as:
2
3 ) 6 2 4
-
0
2
3 ) 6 2 4
-
0
Next, bring down the tens digit 2 to form 02 (or simply 2); 3 goes into 2 zero times (since $3 \times [0](/page/0) = 0), leaving a remainder of 2 after subtracting 0 from 2, and write the quotient digit 0 above the tens place. The updated tableau is:
2 0
3 ) 6 2 4
- -
0 2
2 0
3 ) 6 2 4
- -
0 2
Finally, bring down the units digit 4 to form 24; 3 goes into 24 exactly 8 times (since $3 \times 8 = 24), with no remainder; subtract 24 from 24 to get 0, and write the quotient digit 8 above the units place. The completed tableau shows:
2 0 8
3 ) 6 2 4
- - -
0 0 0
2 0 8
3 ) 6 2 4
- - -
0 0 0
The quotient is 208, with a remainder of 0.[11]
To verify the result, multiply the quotient by the divisor and add the remainder: $3 \times 208 + 0 = 624.[2]
In this example, common pitfalls include misaligning the digits when bringing down, which can lead to incorrect place values in the quotient, or forgetting to bring down the next digit entirely, resulting in an incomplete division.[12]
Applications
Prime factorization
Short division can be adapted for prime factorization by iteratively applying the method to divide a given integer by the smallest prime numbers in sequence, recording each prime divisor until the quotient reaches 1. This process systematically decomposes the number into its prime factors, leveraging the efficiency of short division to handle successive quotients without full long-division notation.[13][14]
The detailed procedure begins with checking divisibility by 2, the smallest prime, by examining if the number is even; if so, perform short division to obtain the quotient and record 2 as a factor, then repeat with the new quotient. Once no further division by 2 is possible, proceed to the next smallest prime, 3, and continue with subsequent single-digit odd primes (5, 7), applying short division at each step only if the current quotient is divisible by that prime—determined via divisibility rules such as summing digits for 3 or ending in 0 or 5 for 5. For larger primes, divisibility rules are used to check, and if divisible, long division or other methods are applied to find the quotient. Each successful division yields a prime factor and updates the quotient as the new dividend; if a prime does not divide evenly (remainder nonzero), skip to the next prime. This iteration continues until the quotient is 1, at which point all prime factors have been identified and can be listed in ascending order, with exponents for repeated factors.[13][14]
For example, to factorize 360 using this method: first divide by 2 to get quotient 180 (factor: 2); divide 180 by 2 to get 90 (factor: 2); divide 90 by 2 to get 45 (factor: 2); 45 is odd, so try 3, dividing to get 15 (factor: 3); divide 15 by 3 to get 5 (factor: 3); finally, divide 5 by 5 to get 1 (factor: 5). The prime factorization is thus $2^3 \times 3^2 \times 5.[13][14]
This approach offers benefits over basic trial division, as short division enables quicker computations for small primes and integrates learning of divisibility rules, making it particularly effective for educational purposes with numbers up to several digits.[13][14]
Modulo division
Short division, as a method of performing Euclidean division, inherently yields both a quotient and a remainder, where the remainder r satisfies $0 \leq r < d (with d as the divisor), directly corresponding to the modulo operation a \mod d = r for dividend a. This alignment stems from the division algorithm, which guarantees unique integers q (quotient) and r (remainder) such that a = d \cdot q + r.
In practice, the procedure mirrors the standard short division steps—dividing digits sequentially from left to right and recording partial quotients above—but emphasizes capturing the final remainder after processing all digits of the dividend. This remainder-focused output is particularly useful for divisibility tests, where r = 0 confirms exact division, and for identifying cyclic patterns in modular arithmetic, such as repeating sequences in clock-like systems.[15][1]
For instance, dividing 25 by 4 proceeds as follows: the tens digit 2 divided by 4 gives quotient 0 with remainder 2 (brought down as 20); 20 divided by 4 gives quotient 5; the units digit 5 divided by 4 gives quotient 1 with remainder 1, yielding overall quotient 6 and remainder 1, satisfying $25 = 4 \times 6 + 1.[1]
In mathematics, this remainder computation supports applications like verifying divisibility (e.g., r = 0 implies d divides a) and algorithmic checksums, where modulo operations detect errors in data such as credit card numbers via the Luhn algorithm's mod 10 check. The foundational equation a = d \cdot q + r underpins these uses, ensuring the remainder's consistency across computations.[16]
Automaton representation
Short division for computing remainders modulo a divisor d can be modeled as a deterministic finite automaton (DFA), where the states represent the possible remainders at each step of the process. The DFA processes the decimal digits of the dividend from left to right (most significant to least significant), updating the current remainder incrementally without requiring the full number to be stored. This finite state model captures the essence of short division's digit-by-digit approach to modulo operations.[17]
In this automaton, the state set consists of the integers \{0, 1, \dots, d-1\}, corresponding to the possible remainders when dividing by d. The initial state is 0, reflecting the empty prefix of the number. For each input digit k (where $0 \leq k \leq 9), the transition function \delta from current state r to the next state is given by
\delta(r, k) = (r \times 10 + k) \mod d.
This update rule simulates the accumulation of the number's value in base 10 while tracking only the remainder, ensuring the automaton remains in one of the d states at all times. The final state after processing all digits yields the overall remainder of the dividend modulo d, which aligns with the remainder obtained via short division. No accepting states are predefined for general remainder computation; instead, the terminal state directly indicates the result. This structure ensures the model is memory-efficient, using constant space relative to d.[17]
To illustrate, consider the automaton for d = 3, with states \{0, 1, 2\}. Since $10 \equiv 1 \pmod{3}, the transitions simplify to \delta(r, k) = (r + k) \mod 3, but the general formula applies. The transition table for digits 0 through 9 is as follows:
| Current State | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
|---|
| 0 | 0 | 1 | 2 | 0 | 1 | 2 | 0 | 1 | 2 | 0 |
| 1 | 1 | 2 | 0 | 1 | 2 | 0 | 1 | 2 | 0 | 1 |
| 2 | 2 | 0 | 1 | 2 | 0 | 1 | 2 | 0 | 1 | 2 |