Ganita Prakash I · Chapter 3 — A Story of Numbers

We write 7, 25 or 1000 without thinking. But the digits 0 to 9 and the way we place them are an invention, and a long story lies behind them. Humans needed numbers long before they could write. Stone Age people wanted to know how much food they had stored, how many animals were in the herd, what was given in a trade, and how many days were left until the full moon or the next season.

Counting without number words. Imagine you live ten thousand years ago and look after a herd of cows. You have no number names and no digits. How do you make sure every cow has come home after grazing? Here are three ways people could do it.

• Objects. Keep one stick for each cow. In the evening, take one stick out of the pile for each cow that comes back. If sticks are left over, cows are missing.
• Sounds or names. Point to the cows one by one while saying a fixed list of sounds, such as the letters a, b, c, ... The last letter you say tells how many there are.
• Written marks. Use a fixed list of symbols, such as I, II, III, IV, V, ...

In each method, every cow is matched with exactly one stick, sound or symbol, and no two cows share one. This is called a one-to-one mapping. The same idea answers other questions too. To find out whether your neighbour has more cows, match your sticks with your neighbour’s sticks one by one; whoever has sticks left over has more. The leftover sticks tell how many more cows the other person would need.

What is a number system? All three methods use a standard sequence with a fixed order. We count a collection by matching it, in order, with this sequence. Such a sequence is called a number system, and the written symbols in it are called numerals.

The big challenge. Numbers never end, so a good number system needs a sequence that never ends and is still easy to use. Sticks never run out, but to count 5000 sheep you need 5000 sticks. The 26 letters are easy to say but stop at 26. The Roman symbols go further but need new symbols for bigger numbers. The rest of this chapter follows how people met this challenge, step by step, until the system we use today.

Using the body. Many groups of people have counted on their fingers, and some have used the whole body. In parts of Papua New Guinea people count by touching fingers, wrist, elbow, shoulder, ear, eye and so on in a fixed order. The body becomes the standard sequence.

Tally marks. One of the oldest ways to record a number is to cut a notch for every object on a bone, a stick or a cave wall. Archaeologists have found bones with notches that seem to record numbers. The Lebombo bone from southern Africa, about 44,000 years old, has 29 notches; it may have been a tally stick or a record of the days of the Moon. The Ishango bone from Central Africa, 20,000 to 35,000 years old, has notches arranged in columns.

Counting in twos. The Gumulgal people of Australia had only two basic number words: urapon (1) and ukasar (2). Other numbers were built from them:

Anything above 6 was simply called ras. Surprisingly, the Bakairi of South America and the Bushmen of southern Africa used systems that work in the same way, even though these groups lived very far apart with no known contact.

Why group at all? Look at a set of dots. You can tell 3 or 4 at a glance, but most people cannot see 7 or 8 at once without counting. This limit may have led people to group their marks. For example, instead of eight marks in a row, write a symbol for a group of 5 followed by three marks. Common group sizes in history have been 2, 5, 10 and 20; our fingers and toes make 5, 10 and 20 natural.

The weakness. Counting in one group size helps, but not enough. In a system with only 1 and ‘a group of 5’, the number 1345 needs 269 group symbols. Larger numbers soon become impossible to read. People needed something better.

Try it. Write 13 in Gumulgal style: 13 = 2 + 2 + 2 + 2 + 2 + 2 + 1, so it would be six ‘ukasar’ followed by one ‘urapon’ (if the system were extended).

The Romans, and Europe for many centuries after them, used a system with special symbols for certain numbers. We will call numbers that get a new basic symbol landmark numbers.

Writing a number. Break the number into landmark numbers, taking as many of the largest as possible, then as many of the next, and so on. Then write the symbols from largest to smallest.

Example: 27 = 10 + 10 + 5 + 1 + 1, so 27 is XXVII.

Example: 2367 = 1000 + 1000 + 100 + 100 + 100 + 50 + 10 + 5 + 1 + 1, so 2367 is MMCCCLXVII.

The subtraction habit. To save symbols, a smaller symbol written before a larger one is subtracted: IV = 4, IX = 9, XL = 40, XC = 90, CD = 400, CM = 900. So 2999 is MMCMXCIX (2000 + 900 + 90 + 9). The Romans were not always consistent; 4 was sometimes written IIII and 40 as XXXX.

More examples: 1222 = MCCXXII, 302 = CCCII, 715 = DCCXV.

A big improvement. Grouping by a whole sequence of landmark numbers, rather than by one group size, makes numbers far shorter. 2367 needs only 10 symbols instead of 2367 tally marks.

Adding. To add CCXXXII (232) and CCCCXIII (413), collect like symbols: 6 C’s, 4 X’s and 5 I’s. Now regroup: 5 I’s make V, and 5 C’s make D. So the total is D C XXXX V, that is DCXLV = 645. Notice that the regrouping is not always by the same amount: 5 I’s make V, but 2 V’s make X.

Multiplying is hard. What is V × L? It is 250 = CCL. What is L × D? It is 25,000, which needs 25 M’s. Products of landmark numbers are often not landmark numbers, so there is no simple pattern. Long multiplication with Roman numerals is very difficult, and Roman calculators used a counting board called an abacus instead.

Another weakness. To write larger and larger numbers you would keep needing new symbols beyond M.

About 5000 years ago (around 3000 BCE) the Egyptians wrote numbers with landmark numbers too, but they chose them in a very special way. Start with 1. Ten 1s make the next landmark, 10. Ten 10s make the next, 100. Ten 100s make 1000, and so on. Every landmark number is 10 times the one before, so the landmarks are 1, 10, 10², 10³, ..., the powers of 10.

To write a number, group it into powers of 10 and repeat each picture as often as needed. So 324 = 100 + 100 + 100 + 10 + 10 + 1 + 1 + 1 + 1 is three coils, two arches and four strokes. The order of the pictures does not change the value.

Other bases. Instead of 10, we could group in fives. Start with 1; five 1s make 5; five 5s make 25; five 25s make 125; then 625, 3125 and so on. The landmark numbers are 5⁰ = 1, 5¹, 5², 5³, .... To write 143, take the largest landmark that fits: 143 = 125 + 5 + 5 + 5 + 1 + 1 + 1, which is one 125, three 5s and three 1s.

Definition. A number system is a base-n system if its first landmark number is 1 and every next landmark is n times the previous one. Its landmarks are then n⁰, n¹, n², n³, .... The Egyptian system is base 10, also called a decimal system. In base 7 the landmarks are 1, 7, 49, 343, 2401, ....

No symbol needs 10 copies. In the Egyptian system no picture is ever repeated 10 or more times, because ten of any landmark can be swapped for one of the next landmark. In base 5, no symbol appears 5 or more times.

Another example. 2660 = 1000 + 1000 + 100 × 6 + 10 × 6, so it is 2 lotuses, 6 coils and 6 arches. And 70707 = 7 × 10,000 + 7 × 100 + 7 × 1 needs 7 fingers, 7 coils and 7 strokes.

Why choose landmark numbers that are powers of one number? Because it makes calculation regular.

Adding. Add two Egyptian numbers: 47 (four arches, seven strokes) and 56 (five arches, six strokes). Put them together: 9 arches and 13 strokes. Ten strokes make one arch, so we have 10 arches and 3 strokes. Ten arches make one coil. The answer is 1 coil and 3 strokes, which is 103. This is exactly the ‘carry’ in our column addition, 47 + 56 = 103. Every exchange is ‘ten of these for one of the next’, the same rule at every level. In Roman numerals the exchange changes from level to level (five I’s make V, two V’s make X), so you must be careful each time.

Multiplying landmarks. In the Egyptian system, 10 × 100 = 1000 and 100 × 1000 = 1,00,000. Because the landmarks are powers of 10, the product of two landmarks is always another landmark: 10^a × 10^b = 10^{a + b}. This is true in every base: in base 5, 25 × 125 = 5² × 5³ = 5⁵ = 3125.

Multiplying any number by 10. Since multiplying by 10 turns each landmark into the next one, you simply replace every stroke by an arch, every arch by a coil, and so on. Using the distributive property, (100 + 10 + 1) × 10 = 1000 + 100 + 10. In base 5, multiplying by 5 moves every symbol up one landmark in the same way.

The abacus. From about the 11th century, even Europeans still using Roman numerals calculated on a counting board. It had horizontal lines for 1, 10, 100, 1000, ..., the powers of 10. Counters were placed on a line, one for each time that power was used, and a counter just above a line stood for 5 of that line. To show 3426: 3 counters on the thousands line, 4 on the hundreds, 2 on the tens, and for 6 ones, one counter above the ones line (5) and one on it.

To add 2907 + 43, bring the counters of each line together. The ones give 7 + 3 = 10, which become 1 counter on the tens line. Then the tens hold 0 + 4 + 1 = 5. The result is 2950.

The remaining problem. The Egyptian system still needs a new picture for every new power of 10. To write ever larger numbers, ever more symbols must be invented.

About 4000 years ago, in Mesopotamia (much of present-day Iraq), people pressed wedge shapes into clay tablets. In later times their system became base 60, also called sexagesimal. The landmark numbers were 1, 60, 60² = 3600, 60³ = 2,16,000, .... We still use their 60 today: 1 hour = 60 minutes and 1 minute = 60 seconds.

Writing 1 to 59. They had just two basic marks: a narrow upright wedge for 1 and a broad corner wedge for 10. So 23 was two corner wedges and three upright wedges. Any number from 1 to 59 can be made this way.

The great step: position. Take 640. Group it: 640 = 10 × 60 + 40. Instead of drawing a special symbol for 60 ten times, the Mesopotamians wrote just two groups side by side: ‘10’ then ‘40’. The group on the right counts 1s; the group to its left counts 60s; the next counts 3600s. The position of a group tells which landmark it belongs to.

If a group ever reached 60, sixty of that landmark would be exchanged for one of the next, so each group always lies between 1 and 59.

A system with a base that uses the position of a symbol to show its landmark number is called a positional or place value system. This was a wonderful idea: no new symbols are needed for bigger landmarks, so a few symbols can write every number.

The trouble with gaps. What if a place is empty? 3605 = 1 × 3600 + 0 × 60 + 5. The Mesopotamians left a blank space for the missing 60s. But spaces were hard to keep even on clay. Then ‘1 | blank | 5’ (3605) could be misread as ‘1 | 5’ (65). Even 60 (‘1’ in the 60s place) looked just like 1, and 3600 looked the same too.

A placeholder. Later Mesopotamian scribes used a special mark to show an empty place, an early kind of zero. However, they used it mainly in the middle of a number, not at the end, so numbers like 3600 could still be confused with 1 or 60. Reading a number often depended on its context.

Place value was invented more than once. Two other civilisations built their own versions.

The Maya of Central America. Between about the 3rd and 10th centuries CE the Maya used a dot for 1 and a bar for 5. With dots and bars they wrote 1 to 19; for example 17 is three bars and two dots. Larger numbers were written in levels stacked one above another, read from the bottom up. The bottom level counted 1s, the next counted 20s, and the next counted 360s (not 400). Further levels went up by 20 each time. For an empty level they drew a shell-like sign that acted as zero.

Because the third landmark is 360 and not 20 × 20 = 400, the Mayan system is not a true base-20 system, so it loses some of the ease of calculation that a base gives. Many scholars link the 360 to their calendar. Still, their place value idea and their zero were remarkable achievements, made independently of Asia.

Chinese rod numerals. In China, from at least the 3rd century CE until the 17th century, people calculated with small rods laid on a board. The rod numerals were base 10 with place value. The digits 1 to 9 had two forms: zong, with upright rods, and heng, with lying-down rods. The forms alternated from place to place: zong for units, heng for tens, zong for hundreds, heng for thousands, and so on. So 2634 was written as 2 in heng form, 6 in zong, 3 in heng and 4 in zong, meaning 2 × 10³ + 6 × 10² + 3 × 10 + 4.

Why alternate? If 41 were written with upright rods only, the four rods and one rod might run together and look like 5, or be split differently. Switching between upright and lying rods shows clearly where one place ends and the next begins.

For an empty place the Chinese left a blank space. Since their digit shapes were all of a similar size, blanks were easier to see than in Mesopotamian writing, but the gap could still be missed. With a symbol for zero, this would have been a fully developed place value system, very close to ours.

The system we use today combines every good idea in this story. It is base 10, it uses place value, and it has ten digits 0, 1, 2, ..., 9, with 0 treated as a digit like any other. In 375, the 3 stands for 3 × 10², the 7 for 7 × 10 and the 5 for 5 × 1. In 305 the 0 fills the tens place, so 305 can never be confused with 35. Only one digit sits in each place, so there are no spacing problems at all.

Where it came from. Ancient Indian texts such as the Yajurveda Samhita already named numbers by powers of 10: eka (1), dasha (10), shata (100), sahasra (1000), ayuta (10,000), up to 10¹² and beyond. Writing numbers with ten digits developed in India about 2000 years ago. The Bakhshali manuscript (about the 3rd century CE) is the earliest known text to write numbers with ten digits including zero, which it shows as a dot. Aryabhata (499 CE) used the system for large scientific calculations.

Zero as a number. Indian mathematicians did more than use zero as a placeholder. They treated it as a number in its own right. Aryabhata used rules such as ‘a number plus 0 is the same number’ and ‘a number times 0 is 0’. In 628 CE Brahmagupta, in his Brahmasphutasiddhanta, wrote down rules for calculating with zero and with negative numbers. Adding, subtracting or multiplying any two such numbers always gives another number of the same kind. These ideas laid foundations for algebra.

How it travelled.

• By about 800 CE the system reached the Arab world. Al-Khwarizmi (about 825 CE), after whom the word ‘algorithm’ is named, wrote a book on calculating with Hindu numerals, and Al-Kindi (about 830 CE) wrote another.
• By about 1100 CE it had reached Europe and parts of Africa. Around 1200 the Italian mathematician Fibonacci argued strongly for it.
• Roman numerals held on for centuries, but by the 17th century science and trade could not do without the new numerals.

Europeans, who learnt the digits from Arab scholars, called them ‘Arabic numerals’, while the Arab scholars themselves called them ‘Hindu numerals’. Today the names ‘Hindu numerals’, ‘Indian numerals’ and ‘Hindu-Arabic numerals’ are used. Here ‘Hindu’ refers to the land and people of India, not to a religion.

The five big ideas. (1) Count in groups of one size. (2) Group with several landmark numbers. (3) Choose landmarks that are powers of one number: a base. (4) Let position show the landmark: place value. (5) Use 0 as a digit and as a number. The Indian system brings all five together.

We use base 10 probably because we have ten fingers. But place value works with any base. In base n we need n digits, 0 to n − 1, and the places stand for 1, n, n², n³, ... from the right.

Changing 25 into other bases. Find how many of each landmark fit, starting with the largest.

• Base 8 (places 1, 8, 64): 25 = 3 × 8 + 1, so 25 is written 31 in base 8.
• Base 5 (places 1, 5, 25): 25 = 1 × 25 + 0 × 5 + 0, so it is 100 in base 5.
• Base 2 (places 1, 2, 4, 8, 16): 25 = 16 + 8 + 1 = 1 × 16 + 1 × 8 + 0 × 4 + 0 × 2 + 1, so it is 11001 in base 2.

Notice that in base 5 the digit 0 appears twice. Without zero we could not tell 1, 10 and 100 apart in any base.

Reading a number in another base. Multiply each digit by its place value and add. In base 8, 157 means 1 × 64 + 5 × 8 + 7 = 111. In base 2, 1011 means 8 + 0 + 2 + 1 = 11.

Base 2 is everywhere. Computers store everything using only two states, on and off, so they work in base 2, called the binary system, with digits 0 and 1. The Gumulgal also had just two words, but they added them (ukasar-ukasar = 4). A base-2 place value system with digits 0 and 1 is far more powerful: with 5 places it reaches 11111 = 16 + 8 + 4 + 2 + 1 = 31.

Making your own base-4 system. Use the digits 0, 1, 2, 3. The places are 1, 4, 16, 64. Then 16 is written 100, 15 is 33 (3 × 4 + 3) and 9 is 21 (2 × 4 + 1).

A rule for multiplying by the base. In base 10, multiplying by 10 puts a 0 at the end. In base 5, multiplying by 5 does the same, because each digit moves one place to the left. For example 13 is written 23 in base 5 (2 × 5 + 3); 13 × 5 = 65, and 65 in base 5 is 230 (2 × 25 + 3 × 5 + 0).

Check your answer. Always convert back: base 8 ‘31’ is 3 × 8 + 1 = 25.