Overview of Cross-Division

Cross-division is an analogue of cross-multiplication that can be used to calculate an arbitrary number of digits of a quotient, even when the divisor is too large to work with directly.

We illustrate the method with:

12,345,678 ÷ 4,321

Why This Works

Cross-division is essentially cross-multiplication run in reverse.
It relies on the same distributive principles, but instead of building a product digit-by-digit, we remove the effect of the divisor digit-by-digit.

1. Set Up the Numbers

For cross-division, it does not matter whether the problem is written vertically or horizontally:

  12,345,678
÷      4,321

or

12,345,678 ÷ 4,321

Each digit of the dividend interacts with each digit of the divisor, but instead of column-by-column subtraction, we work along place-value diagonals, just as in cross-multiplication.

2. Split the Divisor into Two Components

• Take the leftmost digits of the divisor that you are comfortable multiplying by a one-digit number.

• Most people start with two digits, though starting with a single digit is also common.

In this example, the divisor 4321 can be split as:

• 4 | 321 (one-digit start), or

• 43 | 21 (two-digit start)

Unlike cross-multiplication, using smaller starting groups increases the likelihood of needing a correction later, but makes the early steps easier.

3. Calculating the First Digit (Using Two Digits)

• The dividend has 8 digits, the divisor has 4 digits, and the leftmost digits of the dividend are smaller than those of the divisor.

• Therefore, the integer component of the quotient will have 4 digits.

Take the leftmost digits of the dividend: 123

Now estimate the largest digit y such that:

43 × y < 123

The largest such value is y = 2.

So the first digit of the quotient is 2.

Subtract:

123 − (2 × 43) = 37

4. Calculating Subsequent Digits

From here on, we repeat a cycle:

1. Bring down the next digit of the dividend

2. Apply cross-multiplication corrections

3. Estimate the next digit of the quotient

Second Digit

• Bring down the next digit (4) → 374

• Apply the cross-multiplication correction:

  • Multiply the first quotient digit (2) by the first digit of the second component (2) → 4

• Subtract: 374 − 4 = 370

Now estimate:

43 × y < 370 → y = 8

Subtract:

370 − (8 × 43) = 26

Multiply by 10 and bring down the next digit → 265

Apply cross-multiplication corrections:

• 8 × 2 = 16

• 2 × 1 = 2

Subtract:

265 − 16 − 2 = 247

Third Digit

Estimate:

43 × y < 247 → y = 5

Subtract:

247 − (5 × 43) = 32

Multiply by 10 and bring down the next digit → 326

Apply corrections:

• 5 × 2 = 10

• 8 × 1 = 8

326 − 10 − 8 = 308

Fourth Digit

Estimate:

43 × y < 308 → y = 7

Subtract:

308 − (7 × 43) = 7

At this point, we have the integer part of the quotient:

2857

Decimal Digits

Multiply by 10 and continue:

• 7 → 77

• Apply corrections:

  • 7 × 2 = 14

  • 5 × 1 = 5

77 − 14 − 5 = 58

Since 43 × 1 < 58, the first digit after the decimal point is 1.

The quotient is now:

2857.1

This process can continue indefinitely if the decimal expansion repeats.

5. What If the Initial Estimate Is Incorrect?

Estimation errors (or overflows) will occur more frequently when starting with a single digit. Fortunately, they are easy to fix.

Suppose we had started with 4 | 321 instead of 43 | 21.

Correcting an Overflow

• Start: 4 × y < 12 → y = 2

• 12 − (2 × 4) = 4

• Bring down next digit → 43

• Apply correction: 3 × 2 = 6

• 43 − 6 = 37

Continue:

• 4 × y < 37 → y = 8

• 37 − (8 × 4) = 5

• Bring down next digit → 54

• Apply corrections:

  • 8 × 3 = 24

  • 2 × 2 = 4

54 − 24 − 4 = 26

At this point, one might incorrectly try y = 6, but the correct digit is 5.

Proceeding with 6 produces a negative result, immediately signaling an overflow.

To correct:

• Reduce the estimate from 6 to 5

• Adjust using the divisor components

• Resume from the last correct remainder

This correction is quick and local — no prior work is lost.