Chapter 11 — The World of Algorithms

Chapter 11, 'The World of Algorithms', introduces the concept of an algorithm — a precise, step-by-step procedure to solve a problem — using column addition and GCD computation as the main examples. It also covers algorithm efficiency, data structures, Euclid's subtraction algorithm, and Āryabhaṭa's division-based GCD algorithm.

The chapter defines an algorithm as a systematic, step-by-step procedure to arrive at an answer, consisting of basic steps that are assumed to be directly executable. Every algorithm must be justified (shown to be correct) and can be analysed for efficiency.

Numbers are written one below the other with digits right-aligned. Starting from the rightmost column, two digits are added; if the sum is less than 10 it is written below, and carry is set to 0; if the sum is 10 or more the units digit is written below and carry is set to 1. This repeats leftward; a final carry of 1 is written to the left of the result.

Counting dots grows proportionally to the value of the numbers — 4-digit numbers require 10 times as many dots as 3-digit numbers. The column method adds only one more column for each extra digit, so the work grows with the number of digits, not the size of the numbers.

Euclid's subtraction algorithm is based on the fact that gcd(m, n) = gcd(n, m – n) when m ≥ n. The steps are: (1) if m < n, reverse the numbers; (2) if n = 0, the answer is m; (3) otherwise reduce to gcd(n, m – n). This is repeated until n becomes 0.

Instead of subtracting n from m repeatedly, Āryabhaṭa's algorithm replaces step 3 with gcd(n, m mod n) — the remainder when m is divided by n. This is the 'long division' method recorded in the Āryabhaṭīya (499 CE). The number of reduction steps is now proportional to the number of digits in m and n, making it far more efficient.

Euclid's algorithm reduces gcd(m, n) to gcd(n, m – n) one subtraction at a time, which can be very slow (e.g. gcd(99, 2) requires about 49 steps). Āryabhaṭa's version replaces subtraction with the remainder (m mod n), so gcd(99, 2) converges in just 2 steps. The division version is proportional to the number of digits; the subtraction version is proportional to the value.

A data structure is a way to organise information during the execution of an algorithm. In this chapter, an ordered list is used to store divisors of a number. Using named data structures (like 'list-of-divisors') makes algorithms easier to write concisely and allows intermediate results to be referred to later.

The word comes from Algoritmi, the Latin form of the name of Al-Khwārizmī (780–850 CE), a Persian mathematician. His 12th-century Latin translation of a treatise on Indian numerals and arithmetic — Liber Algorismi de numero Indorum — was so influential in medieval Europe that 'algorismus' came to mean any Indian method of computation, eventually shortening to 'algorism' and then 'algorithm'.

gcd(375, 825) = 75. Using the division algorithm: since 375 < 825, start with gcd(825, 375); 825 mod 375 = 75, so reduce to gcd(375, 75); 375 mod 75 = 0, so gcd = 75. This matches the result obtained by the divisor-list method in the chapter.

Exercise Set 11.1 asks students to apply the column-addition algorithm to 4-digit numbers, consider how the algorithm handles numbers with different digit counts, explain why right-to-left alignment matters, justify why carry can never exceed 1, and explore what happens if the final carry step is omitted.

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