Methods and worked examples: A Story of Numbers
A Story of Numbers22 min readLesson 2 of 8
2.1 Bases and powers
A base-n number system has landmark numbers that are powers of n: n^0 = 1, n, n^2, n^3, and so on. The Egyptian system is base-10 because its landmarks are 1, 10, 100, 1000, each ten times the previous one. A system built on 5s is base-5, with landmarks 1, 5, 25, 125, 625.
A number is represented in a base by grouping it from the largest landmark downwards. In base-5, 143 is one 125, three 5s and three 1s, because 125 is the largest landmark below 143. In base-10, 324 means three hundreds, two tens and four ones.
2.2 The Egyptian system
The Egyptians used base-10 landmarks with a separate symbol for each power of ten. They grouped a given number into as many of each landmark as possible, starting from the largest, and wrote the matching symbols. This made addition easy: collect all symbols of the same kind and regroup ten of them into one symbol of the next landmark.
The system still needed a new symbol for every new power of ten. For very large numbers, the list of symbols would never end, so the representation was not a complete solution.
2.3 Mesopotamian place value
Mesopotamia used base 60. Its landmark numbers are 1, 60, 3600, 216000 and so on. A number such as 640 is 10 x 60 + 40, written as ten 60-symbols and forty 1-symbols. The later Mesopotamians made the writing compact by dropping the landmark symbols and using position: the rightmost group gives units, the next group 60s, the next 3600s. This is the beginning of place value.
Base 60 also survives in time measurement: one hour contains sixty minutes, and each minute contains sixty seconds. The Mesopotamian symbols were wedge-shaped, and their notation could be ambiguous because blank spaces for missing powers were hard to keep consistent. A placeholder symbol was introduced for blanks, an early form of zero.
2.4 Mayan and Chinese systems
The Mayan civilisation of Central America used a dot to mean one and a bar to mean five, with positions written vertically. Its landmarks are 1, 20 and 360, then multiples of 360. The third landmark is 360, not 400, which scholars link to their calendar. The Mayans used a shell-like placeholder for zero, an important advance, but the system is not a true base for computation.
The Chinese used rod numerals for computation, in base 10. Rods for 1 to 9 alternated between two forms, zong and heng, to distinguish adjacent places. For example, a numeral with positions 10^3, 10^2, 10 and 1 is read as (2 x 1000) + (6 x 100) + (3 x 10) + 4 = 2634. The Chinese used a blank space for a missing place; with a zero symbol it would have been a complete place value system.
2.5 The Hindu number system
The Hindu system uses just ten symbols 0 to 9 and base-10 place value. The numeral 375 means (3 x 100) + (7 x 10) + 5. Indian texts such as the Yajurveda Samhita listed number names for powers of ten; the Bakhshali manuscript (around the 3rd century CE) shows ten digits including a dot for zero; Aryabhata (499 CE) used the system for scientific computation. Al-Khwarizmi popularised it in the Arab world around 825, and Fibonacci argued for it in Europe around 1200.
The decisive step was treating zero as a number in its own right, rather than as a blank. Aryabhata used the arithmetic of zero, and Brahmagupta codified zero and negative numbers in 628 CE. With zero as a digit, no ambiguity arises: each position has exactly one digit, and the whole infinite sequence of numbers uses only ten symbols. That is why the Hindu system spread worldwide.
2.6 The abacus
Roman users performed calculations on an abacus: a board with lines, where each line stood for a power of ten. Counters on a line recorded how many of that landmark a number contained, and a counter above a line counted five. Adding two numbers meant combining counters line by line, then regrouping ten counters on one line into one counter on the line to the left. The abacus shows that even a fixed-value system could be used efficiently with a tool, but only trained people could operate it.
2.7 Worked examples
Example 1. Write 7530 in the Mesopotamian place value system. 7530 = 2 x 3600 + 5 x 60 + 30, so the numeral shows 2 in the 3600 place, 5 in the 60 place and 30 in the units place.
Example 2. Represent 77 in the Mayan system. 77 = 3 x 20 + 17, so the upper group is three dots (3 in the 20s place) and the lower group is three bars and two dots (17).
Example 3. Write 10458 in the Egyptian system. 10458 = 10000 + 4 x 100 + 5 x 10 + 8, so it uses one 10000-symbol, four 100-symbols, five 10-symbols and eight 1-symbols.