Chapter 6.4 — Exercise 6.3 — Division Method

The Division Method, Conceptually

Start by grouping the digits of the number in pairs, working outward from the decimal point (so 7744 splits into 77 | 44, and 2.56 splits into 2 . 56). Then, working one pair at a time from left to right:

• Find the largest digit whose square is less than or equal to the leftmost group — this becomes the first digit of the answer, and its square is subtracted from that group.
• Bring down the next pair of digits to sit beside the remainder.
• Double the answer found so far, and find the largest next digit d such that (twice the current answer, with d appended) × d fits into the new remainder.
• Repeat, bringing down each remaining pair and finding one new digit of the answer at a time, until every pair has been used.

This is the same idea as long division, just adapted so each new "trial divisor" grows to include the digit currently being found.

Worked Examples

• √1089: grouping as 10 | 89, working through the steps gives 33 (check: 33² = 1089).
• √2304: grouping as 23 | 04, gives 48 (48² = 2304).
• √7744: grouping as 77 | 44, gives 88 (88² = 7744).
• √6084: grouping as 60 | 84, gives 78 (78² = 6084).
• √9025: grouping as 90 | 25, gives 95 (95² = 9025).

Extending the Method to Decimals

For a decimal number, group digits in pairs on both sides of the decimal point separately, then run the same process, placing a decimal point in the answer at the same position it appears in the grouped number.

• √2.56 = 1.6 (1.6² = 2.56)
• √18.49 = 4.3 (4.3² = 18.49)
• √68.89 = 8.3 (8.3² = 68.89)
• √84.64 = 9.2 (9.2² = 84.64)

When the Number Isn't a Perfect Square

The division method doesn't stop working just because a number has no exact square root — it reveals exactly how far off the nearest perfect squares are, which is useful in its own right.

Least number to subtract from 4000: running the division method shows 4000 sits just past 63² = 3969, with a remainder of 4000 − 3969 = 31 — subtracting 31 leaves the perfect square 3969.

Least number to add to 6412: the method shows 80² = 6400 is just below 6412, while 81² = 6561 is the next perfect square up. The number needed is 6561 − 6412 = 149.

Least four-digit perfect square: 31² = 961 is the largest three-digit perfect square, so the next one up, 32² = 1024, is the smallest four-digit perfect square.

Word Problems Using the Division Method

• Square side from area: a square of area 4489 sq. cm has side √4489 = 67 cm.
• Plants left over: 8289 plants with 8 left over means 8281 were planted in a perfect square arrangement; √8281 = 91 plants per row.

Estimating a Square Root Without Full Division

When only an approximate answer is needed, it's often faster to sandwich the number between two known consecutive squares and pick whichever is closer:

• √97: 97 lies between 9² = 81 and 10² = 100; it's closer to 100, so √97 ≈ 10.
• √250: 250 lies between 15² = 225 and 16² = 256; closer to 256, so √250 ≈ 16.
• √780: 780 lies between 27² = 729 and 28² = 784; closer to 784, so √780 ≈ 28.

This estimation shortcut trades precision for speed — useful when a rough answer is all that's needed, but it's the division method that gives an exact result (or an exact decimal expansion) when precision matters. Knowing both is worth having: a quick sandwich-estimate first can also serve as a sanity check on a division-method answer, catching an arithmetic slip if the two disagree by more than a rounding difference should reasonably allow for, before committing to the more detailed division-method answer as the final, officially reported result.

Why the Division Method Beats Prime Factorisation for Large Numbers

Prime factorisation needs the number's prime factors identified first, which gets genuinely difficult once a number is large or has large prime factors buried inside it — there's no shortcut for finding that a big number's smallest prime factor is, say, 89, other than testing candidates one at a time. The division method sidesteps this entirely: it never needs to know a single prime factor, working instead directly with pairs of digits and a running trial divisor. This is exactly why it remains practical for numbers where factorising would take a very long time by hand, and why it's the method that extends cleanly to decimals and non-perfect squares, where prime factorisation doesn't apply at all.

Practising the Method on a Number You Choose Yourself

As with every shortcut in this chapter, the division method is best tested on a number picked at random rather than only the ones already worked through here. Choose any four- or five-digit number, group its digits in pairs from the decimal point outward, and work through the steps by hand before checking the result by squaring the answer back. Doing this once with a genuinely unfamiliar number is worth more than re-reading the worked examples above a second time, since it forces every step of the process to be applied rather than simply recognised.

Grouping and Doubling — Where It Falls Apart

• Grouping digits incorrectly. Pairs are always formed working outward from the decimal point, not from the leftmost digit of the whole number — a number with an odd number of digits will have a lone single digit in its leftmost group.
• Forgetting to double the running answer at each step. The trial divisor at each stage is built from twice the answer found so far, not the answer itself.
• Picking the nearer square incorrectly when estimating. Compare the actual difference to both neighbouring squares — the number partway between them isn't always obviously closer to the one that looks bigger.

What the Remainder at Each Step Actually Means

At every stage of the division method, the running remainder represents "how far the number built so far is from a perfect square using the digits seen up to this point." This is why the method can be stopped at any stage to answer a slightly different question — finding the least number to subtract to reach a perfect square, for instance, is just running the whole division method to completion and reading off whatever remainder is left at the very end, rather than needing a separate calculation. Seeing the remainder this way, as a running "distance to the nearest perfect square using the digits so far," ties together several of this exercise's seemingly different question types into one underlying process — subtracting to reach a square, adding to reach one, and finding the nearest four-digit square are all just different readings of the same final remainder, produced by running exactly the same process to the end and simply reading the leftover value once every digit pair has been used.

From Squares to Irrational Roots, and to Cubes

The division method's ability to handle decimals and non-perfect squares is exactly what makes irrational square roots like √2 approachable once Real Numbers introduces them formally in Class 9. Within this chapter, the same "repeated product" idea now shifts from squares to cubes in Exercise 6.4.