Class 8 Mathematics Chapter 3: A Story of Numbers

A herd of cows, and no numbers

Imagine keeping cows ten thousand years ago, with no digits and no number names. Three questions still matter. Did every cow come back from grazing? Does the neighbour hold more than you? If so, how many more would you need to match them?

The stick answer

Pick up one stick for each cow as the herd leaves. In the evening put a stick aside for each cow that returns. A stick left in your hand is a cow left on the hill. You never counted, yet you know.

What the sticks are really doing

Each cow is tied to exactly one stick, and no stick does duty for two cows. That pairing is called a one-to-one mapping, and it is the whole mechanism of counting. The pile of sticks then stands for the number itself.

Comparing two herds

The sticks answer the other questions too. Set your pile beside the neighbour’s and pair them off one by one. Whoever still has sticks has the bigger herd, and the leftovers say by how many. No number name is ever spoken.

Sounds instead of sticks

Objects are heavy, so try sounds. Walk through the letters a, b, c and so on, one letter per object, and the letter you stop at names the count. This is the same mapping with the order carried in your head rather than in your hands.

Where the letters run out

The alphabet has an end. Twenty-six letters count twenty-six objects, and a twenty-seventh object has nothing left to pair with. A counting sequence that stops is a counting sequence that fails, sooner or later.

Written symbols

The third choice is to write symbols in a settled order: one stroke, two strokes, three, and then a fresh symbol when the strokes grow unreadable. That is roughly how the Roman marks I, II, III, IV and V came about, and Europe used them for centuries.

What all three have in common

Sticks, sounds and symbols differ in material and share a structure. Each is a standard sequence held in a fixed order. We can give that structure a name: a number system. Counting is pairing a collection against it, in order.

Two demands that pull apart

A good number system must never run out, and must stay convenient. Sticks never run out but a herd of three hundred needs three hundred sticks. Letters are light but stop at twenty-six. Every system in this chapter is an attempt to hold both demands at once.

Numerals

The written symbols of a number system have their own name: numerals. The marks 0, 1, 5, 36 and 193 are numerals of the system we use now. Spoken names and written numerals grew side by side, because small numbers were said aloud long before they were written.

Notches on bone

The oldest counting we can still hold in our hands is a row of notches. Bones more than twenty thousand years old carry them. The Ishango bone from the Democratic Republic of the Congo has notches set out in columns, which may have tracked a calendar.

The Lebombo bone

Older still is the Lebombo bone, found in South Africa, with twenty-nine notches cut into it. Estimates put it near forty-four thousand years old. Twenty-nine is close to the length of a lunar month, so a moon calendar is one reasonable guess.

Counting in twos

The Gumulgal of Australia named numbers using just two words. One was urapon and two was ukasar. Three became ukasar-urapon, four became ukasar-ukasar, and six became three ukasars in a row. Anything past six was simply called many.

The same idea, three times over

The Bakairi of South America and the Bushmen of southern Africa built their number names the same way, from a word for one and a word for two. The three peoples lived oceans apart with no known contact. Shared distant ancestors are one theory; nobody is certain.

Why grouping caught on

Counting in twos halves the length of a name or a row of marks. Fives do better still. Instead of cutting five notches, cut a different mark that means five, and a long row becomes short. The Roman V is exactly that.

The glance limit

There is a reason grouping feels natural. Put three objects in a box and you see three without counting. Put seven and you have to count. Most people cannot take in more than about four at a glance, so a mark standing for a whole group is easier to read than the group itself.

Scenario:

A museum is preparing labels for three objects. The first is a bundle of notched sticks from a cattle market. The second is a cast of the Lebombo bone, with its twenty-nine marks. The third is a card printed with the Gumulgal number words. The curator wants each label to say which idea its object carries, rather than only how old it is and where it was found. A school group is due on Friday, and their teacher has asked that no label call any of the three objects primitive, because the children will take that word to mean failed.

Analysis:

Each object carries a different stage of one argument: the pairing itself, a record of that pairing, and the first step away from it.

Title:

Build a two-word number system

Safety:

No equipment is involved, so no safety step is needed.

What You Notice:

Names up to six come easily. Eleven takes six words and becomes unreadable, which is exactly why larger group sizes were adopted.

The Lebombo bone, found in South Africa, carries 29 notches and is reckoned to be around 44,000 years old, which makes it one of the oldest surviving pieces of mathematics.

Three peoples separated by oceans, the Gumulgal of Australia, the Bakairi of South America and the Bushmen of southern Africa, all built number names by counting in twos.

One thing, one mark, one place in the row — Every counting method that ever worked pairs each object with exactly one item of a sequence held in a fixed order.

Giving a number its own symbol

A row of marks is unreadable past about four, so useful systems hand certain numbers a symbol of their own. Call such a number a landmark number. The Roman marks are I for one, V for five, X for ten, L for fifty, C for a hundred, D for five hundred and M for a thousand.

Building a Roman numeral

To write a number, take as many of the largest landmark as will fit, then as many of the next, and carry on down. Twenty-seven is ten and ten and five and one and one, which gives XXVII. The method is just grouping, done from the top.

A bigger one

Try 2,367. It is two thousands, three hundreds, one fifty, one ten, one five and two ones, so the numeral is MMCCCLXII. Compare that with 2,367 notches, and the gain from landmark numbers is obvious.

The subtractive trick

Rather than IIII for four, the Romans wrote IV, meaning one less than five. The same thought turns forty into XL, ten less than fifty. They were not consistent about it, and XXXX for forty appears in their records too.

Adding in Roman numerals

Add CCXXXII and CCCCXIII. Collect the symbols: six Cs, five Xs, five Is. Five Is are a V, and five Cs are a D, so the pile regroups into DCXXXV. It works, but every step needs its own rule about how many make the next mark.

Why that is a nuisance

In the Roman list, five ones make a five, two fives make a ten, five tens make a fifty, two fifties make a hundred. The step changes as you climb. Multiplication is worse still, which is why specially trained people with an abacus did the calculating.

The Egyptian choice

The Egyptians, around 3000 BCE, chose their landmarks differently. Start at 1. Gather ten of them and the next landmark is 10. Gather ten of those and the next is 100. Carry on and the landmarks are 1, 10, 100, 1,000 and upwards.

What is special about that list

Every landmark is ten times the one before, so all of them are powers of ten. A number is then written by grouping from the largest landmark down, exactly as in Rome. The number 324 is three hundreds, two tens and four ones.

Why not gather five instead of ten?

Nothing says the group size must be ten. Start at 1, gather five and the next landmark is 5. Gather five of those and the next is 25. Then 125, then 625, then 3,125. Every landmark is a power of five.

Writing a number in base five

Take 143. The largest landmark below it is 125, so start there. What is left is 18, which holds three 5s, and 3 is left over. So 143 is one 125, three 5s and three 1s.

The definition

A system is called a base-n system when its first landmark is 1 and every later landmark is n times the one before. Its landmarks are then the powers of n: 1, n, n squared, n cubed and so on. The Egyptian system is base ten and ours is too.

Decimal

A base-ten system has a second name: decimal. A base-seven system would have landmarks 1, 7, 49, 343 and upwards. Any whole number above one can serve as a base, and the choice of ten is a human accident, not a mathematical law.

Addition becomes uniform

Here is the first payment. In a base-ten system, ten of any landmark make the next one, every time. Fifteen tens regroup into one hundred and five tens without a thought. Contrast the Roman pile, where the number that makes the next mark keeps changing.

And multiplication becomes easy

Here is the larger payment. Multiply two landmarks in a base system and the answer is another landmark, because multiplying powers of one number adds their exponents. A hundred times a thousand is a lakh, which is on the list. In Rome, L times D is nowhere on the list.

Spreading it over a whole number

That is why a base makes long multiplication work. Any number is a sum of landmarks, and multiplying a sum means multiplying each part and adding, which is the distributive property. Each part lands on another landmark, so the answer assembles itself.

The abacus

By about the eleventh century, people writing Roman numerals were calculating on a board of lines. The lowest line stood for 1 and each line above it for the next power of ten, with counters laid on them. A counter placed above a line counted as five of it.

Where the Egyptian system still failed

Egyptian numerals handled anything up to a crore, which is ten to the seventh. Past that, a new symbol was needed for every higher power of ten, with no end in sight. The old problem had come back wearing different clothes.

Scenario:

A grain office keeps its records in Roman numerals. The clerk adds CCXXXII sacks to CCCCXIII sacks. He takes some time over it. He must remember that five Is make a V, while five Cs make a D. The supervisor then asks him to multiply fifty sacks by five hundred. He cannot do it on paper at all. A visiting trader suggests keeping the books with landmark numbers that are all powers of ten. He offers to show the clerk an abacus laid out the same way.

Analysis:

The two tasks test different things. Addition exposes the shifting regrouping step, while multiplication exposes landmarks that are not powers of one number.

Title:

Build your own base-four system

Safety:

No equipment is involved, so no safety step is needed.

What You Notice:

The landmarks come out as 1, 4 and 16. Sixteen needs just one symbol, while fifteen needs three 4s and three 1s, which shows how a landmark shortens a numeral.

The Romans were not consistent about writing forty as XL. The form XXXX turns up in their records too, which is a reminder that a living system is untidier than a textbook rule.

By the eleventh century even people still writing Roman numerals were calculating on an abacus laid out in powers of ten, so the decimal idea arrived before the decimal numerals did.

Powers in, trouble out — Choose landmarks that are powers of one number and both regrouping and multiplication fall into line. Choose them freely and every step needs its own rule.

Where we had got to

A base fixed arithmetic but left one hole. Every higher power needed a brand new symbol, so the symbols never stopped coming. Mesopotamia found the way out, and oddly enough it did so with base sixty rather than ten.

Why sixty?

Nobody is certain. Their lunar month ran to thirty days, sixty divides neatly for fractions, and their earlier landmarks had been 1, 10, 60, 600 and 3,600 before settling on powers of sixty alone. Any of these may have pushed them that way.

The system itself

They had one symbol for 1 and another for 10, and with those two they wrote every number from 1 to 59. The landmarks were 1, then 60, then 3,600, then 216,000, each sixty times the one before.

A number in base sixty

Take 640. Grouping gives ten 60s with 40 left over, so the numeral is ten 60s and one 40. Take 7,530. That is two 3,600s, five 60s and a 30. Each landmark is written with a count beside it.

A rule that always holds

No power of sixty can appear sixty or more times. If it did, sixty of them would simply group into the next power up. So one 3,600 with seventy 60s and a 2 is really two 3,600s with ten 60s and a 2.

The leap

Now the clever step. If the counts always sit in the same order, largest on the left, the symbols naming the powers carry no information at all. So drop them. The rightmost group counts the 1s, the next the 60s, the next the 3,600s.

What a position does

This is what the Mesopotamians actually did. A system with a base where the POSITION tells you which landmark is meant is called a place value system, or a positional system. The endless hunt for new symbols is over at a stroke.

But there was a gap

What if a power of sixty does not occur at all? They left a blank space. Spacing was not written as carefully as we print it, and different scribes spaced differently, so a reader could not always tell how many blanks sat between two groups. That made numerals genuinely ambiguous. Without knowing which group belongs to which power of sixty, one written number could be read as two different numbers. The system that solved one problem had quietly created another.

A symbol for nothing

Later Mesopotamians fixed most of it with a placeholder symbol standing for an empty position. It is the ancestor of our 0 in that role. But they used it mainly in the middle of a numeral, not at the end, so a trailing gap was still unclear.

Sixty on your wrist

Their choice outlived their civilisation. An hour has sixty minutes and a minute has sixty seconds. Every clock face is a surviving piece of the Mesopotamian system.

The Mayan system

In Central America, between the third and tenth centuries CE, the Mayans arrived at place value independently. A dot stood for 1 and a bar for 5, and with those they wrote 1 to 19. Their digits were stacked one above another.

Their curious third landmark

The lowest row counted 1s and the row above counted 20s, which suggests base twenty. But the next row up counted 360s, not 400s. Some scholars connect this to their calendars. It means the system is not a true base-twenty system. Because the landmarks are not all powers of one number, the Mayan system loses the arithmetic advantages a base gives. Their place value notation and their shell-shaped placeholder are still a real advance.

Chinese rod numerals

The Chinese kept two systems, one for recording and one for calculating with rods. The rod numerals were decimal, in use from at least the third century CE until the seventeenth, and they look strikingly like our own digits in how they work. They alternated two shapes of digit between positions, called zong and heng. A skipped place was still a blank, but because the neighbouring shapes differed, a missing one was far easier to spot than in Mesopotamia. With a zero symbol it would have been complete.

The Hindu system

The system we use is base ten with ten symbols, 0 to 9, and one digit in each position. The numeral 375 is three 100s, seven 10s and five 1s. Nothing is ambiguous, because a position is never blank: if a power is missing, a 0 is written there.

Zero twice over

That is the heart of it. In India zero was used as a digit at least as early as 200 BCE. The first known writing of ten digits including a zero is in the Bakhshali manuscript, from about the third century CE, where the zero is marked as a dot. Aryabhata computed with the ten symbols in 499 CE.

And as a number in its own right

Zero was not left as a mere gap-filler. Brahmagupta, in 628 CE, set out how it behaves as a number: adding it changes nothing, multiplying by it gives nothing. With zero and the negative numbers he had built a system closed under addition, subtraction and multiplication.

How it travelled

The system reached the Arab world by about 800 CE, carried by Al-Khwarizmi, whose name gave us the word algorithm, and by Al-Kindi. It reached Europe by about 1100 CE, and Fibonacci argued for it around 1200. Roman numerals were so entrenched that full adoption took until the seventeenth century.

A name that went astray

European scholars learned the numerals from Arab scholars and so called them Arabic, while Arab scholars had always called them Hindu numerals. Today Hindu, Indian or Hindu-Arabic numerals are the usual terms, and here Hindu names a region and its people rather than a religion.

Scenario:

A museum receives a Mesopotamian tablet with two groups of symbols on it. The first group plainly counts three and the second counts two. The curator reads it as three 60s and two 1s, which is 182. A visiting scholar reads the same tablet as three 3,600s and two 1s, which is 10,802, on the grounds that a blank position sits between the groups. Neither can settle it from the tablet alone. The museum then asks why a tablet written a few centuries later would not have the same trouble, and why one written in our own numerals could never have it.

Analysis:

The disagreement is not about the symbols, which both read the same way. It is about how many empty positions lie between them.

Title:

Build a clock in base sixty

Safety:

No equipment is involved, so no safety step is needed.

What You Notice:

The three counts come back as the hours, minutes and seconds you started with. A clock is a base-sixty place value system with colons instead of positions.

Base sixty is still on your wrist. An hour holds sixty minutes and a minute holds sixty seconds, which comes straight from Mesopotamia.

The first known writing of numbers with ten digits including a zero appears in the Bakhshali manuscript, from about the third century CE, where the zero is written as a dot.

The place says which, the digit says how many — In a place value system the position carries the landmark and the symbol carries only the count, which is why ten digits are enough for every number there is.