Exercise practice 1: A Story of Numbers

3.1 Exercise practice 1: number systems, Roman numerals and Gumulgal

This page covers the first group of Figure it Out sections.

3.1.1 E1.1 Arithmetic with sticks

Question. Using only sticks, give methods for adding, subtracting, multiplying and dividing two collections, without Hindu number names or numerals.

Answer. Count each collection; add by combining and recounting; subtract by removing sticks and recounting the rest; multiply by making equal groups and counting them; divide by forming groups of the required size and counting the groups and leftovers.

Key steps. Each operation is a physical action on the sticks. Addition puts both collections into one heap. Subtraction takes away the smaller heap. Multiplication repeats a heap of fixed size a given number of times. Division asks how many groups of a chosen size fit into the heap, with the remainder left ungrouped.

3.1.2 E1.2 Extending the letter system

Question. Method 2 uses single letters a to z for 1 to 26. Give a way to extend it to all numbers.

Answer. Use strings of letters in order: a to z, then aa, ab, ac, ..., az, ba, bb, and so on, with a fixed rule that orders the strings.

Key steps. A single letter gives 26 numbers. Strings of two letters give 26 x 26 = 676 more if all pairs are used. Longer strings give more numbers. Any coding is a valid extension as long as every collection gets exactly one string and the order is fixed. The difficulty is that the names become long, which is why letter-based systems are not practical for large numbers.

3.1.3 E1.3 Make your own number system

Question. Design your own number system.

Answer. One design uses symbols for 1, 5, 10, 50 and 100, groups each number from the largest symbol downwards, and repeats a symbol at most four times.

Key steps. Choose the landmark numbers and a symbol for each. Write a number by taking as many of the largest landmark as possible, then the next, and so on. Test it on 27, 49 and 134. This system is base-5-like for the small landmarks and base-10-like at 10 and 100, which shows that a number system is a design choice, not a fixed rule.

3.1.4 E1.4 Roman numerals

Question. Write 1222, 2999, 302 and 715 in Roman numerals.

Answer. 1222 = MCCXXII; 2999 = MMCMXCIX; 302 = CCCII; 715 = DCCXV.

Key steps. Break each number into landmark values and use the subtractive forms for 900, 400, 90, 40, 9 and 4. For 2999: take MM (2000), then CM (900), then XC (90), then IX (9). For 302: CCC + II. For 715: D (500) + CC (200) + X (10) + V (5).

3.1.5 E1.5 Roman multiplication of landmarks

Question. Find V x L, L x D, V x D and VII x IX without converting to Hindu numerals.

Answer. V x L = CCL; L x D is M repeated 25 times; V x D = MMD; VII x IX = LXIII.

Key steps. Multiply the values and write the Roman form: 5 x 50 = 250 = CCL; 50 x 500 = 25000, which is 25 M symbols; 5 x 500 = 2500 = MMD; 7 x 9 = 63 = LXIII. Roman multiplication is laborious because the landmarks do not follow a single repeat rule.

3.1.6 E1.6 Different number-name sequences

Question. A Pacific island group uses different sequences of number names for different objects. Why might they do this?

Answer. Different sequences can mark different classes of objects, preserve ritual or customary ways of counting, or help speakers remember which kind of object is being counted.

Key steps. The chapter says the reason is not fully known and invites more ideas. A possible explanation is that counting systems often carry cultural information beyond quantity: the choice of sequence can signal what is being counted, in the same way that some languages use different counting words for long thin objects and for flat objects. The key point is that a number system can be more than a list of words; it can also carry context.

3.1.7 E1.7 Gumulgal arithmetic

Question. Extend the Gumulgal system beyond 6 using groups of 2, and evaluate: (i) (four ukasar + urapon) + (three ukasar + urapon); (ii) (four ukasar + urapon) - (three ukasar); (iii) (four ukasar + urapon) x (two ukasar); (iv) eight ukasar divided by two ukasar.

Answer. (i) eight ukasar (16); (ii) ukasar-urapon (3); (iii) eighteen ukasar (36); (iv) ukasar-ukasar (4).

Key steps. Replace each name with its value: urapon = 1 and ukasar = 2. (i) 9 + 7 = 16, which is 8 pairs, written as eight ukasar. (ii) 9 - 6 = 3 = ukasar-urapon. (iii) 9 x 4 = 36 = 18 pairs. (iv) 16 / 4 = 4 = ukasar-ukasar. The grouping by 2 is kept in the answer.

3.1.8 E1.8 Comparing Hindu and Roman systems

Question. Identify the features that make the Hindu number system more efficient than Roman numerals.

Answer. Hindu numerals use place value and a zero digit, so every number needs at most ten symbols and the four operations are easy. Roman numerals have fixed values, no zero, and become long and awkward for large numbers, so multiplication and division are difficult.

Key steps. Organise the comparison as a table: place value versus fixed value; zero versus no zero; easy computation versus difficult computation. The efficiency comes from a small set of symbols whose meaning depends on position, so a numeral can be read and computed with quickly.

3.1.9 E1.9 Refining your own number system

Question. Take the number system you designed earlier and improve it using the ideas from this section.

Answer. A refinement can replace repeated tally symbols with landmark symbols and then with place value. Start from a simple system such as a symbol for 1 and a symbol for 5; add a symbol for 10 and for 50; then let position encode the landmarks.

Key steps. Suppose the earlier system used only a unit mark. Group five marks into one new symbol, and five of those into another symbol. Write 27 as five marks plus two more with the new symbols: one 5-symbol and two units. To go further, give each position a landmark value so that one symbol in the tens place means ten. This mirrors the historical steps from tallies to landmarks to place value, and it removes the need for a new symbol at every step.