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Old ways to write numbers

Numbers · level 2, lesson 7 of 7

A question to start

In ancient Egypt, about 5000 years ago, scribes wrote numbers with little pictures. A stroke meant one, a shape like an upside-down U meant ten, and a coil of rope meant a hundred.

To write 999, an Egyptian scribe had to draw 9 coils, 9 upside-down U's and 9 strokes: 27 pictures. We write the same number with just 3 digits.

Now try this: work out XLVIII + LXXIV. Those are Roman numerals. Can you add them without turning them into our numbers first? It is very hard.

People have written numbers in many clever ways. In this lesson you will learn four of them: tally marks, Egyptian hieroglyphs, Roman numerals and Maya numerals. Along the way you will see why almost the whole world now uses the ten digits 0 to 9.

The idea

Tally marks

The oldest way to keep count is to make one mark for each thing. Today we bundle the marks in fives: four straight marks, and the fifth goes across them.

= 17

Three bundles and two more: 5, 10, 15, 16, 17. Tally marks are great for keeping score in a game or counting cars as they go by, because you just add one mark at a time. But they are terrible for big numbers: 999 would need 999 marks!

Very old bones with rows of cut marks have been found, and many people think they were used for counting, perhaps the days of the moon. Nobody can be sure, since nobody wrote down what they meant.

Egyptian hieroglyph numerals

The ancient Egyptians had a picture for each power of ten: one, ten, a hundred, a thousand, and so on.

1101001000
Sign stroke heel bone (an arch shape) coil of rope lotus flower finger tadpole a kneeling god with raised arms
Worth 1 10 100 1000 10,000 100,000 1,000,000

To write a number, draw each sign as many times as you need and add them all up. Here is 243: 2 coils (200), 4 arches (40) and 3 strokes (3).

200 + 40 + 3 = 243

The Egyptians did not need a zero. For 307 they drew 3 coils and 7 strokes, with no arches. Nothing is drawn for "no tens", and nothing goes wrong, because each sign shows its own value. But they needed lots of signs: up to 9 of each kind. That is why 999 needs 27 pictures.

Roman numerals

The Romans wrote numbers with letters. You can still see them on clocks, in book chapters, and after the names of kings and queens.

Letter I V X L C D M
Worth 1 5 10 50 100 500 1000

The rules, in the form usually taught today:

  1. Write the letters from biggest to smallest and add them up. XVI is 10 + 5 + 1 = 16. CCLXII is 100 + 100 + 50 + 10 + 1 + 1 = 262.
  2. Do not use the same letter more than three times in a row. Instead, put a smaller letter in front of a bigger one to mean "take this away". IV is 5 − 1 = 4. IX is 10 − 1 = 9. XL is 50 − 10 = 40. XC is 90. CD is 400. CM is 900.
  3. Only I, X and C go in front like this, and only in front of the next two letters up: I before V and X, X before L and C, C before D and M. So 99 is not IC. It is 90 and 9: XC and IX, so XCIX.

To write a number in Roman numerals, split it into thousands, hundreds, tens and ones, and write each part:

  • 2026 = 2000 + 20 + 6 = MM + XX + VI = MMXXVI.
  • 49 = 40 + 9 = XL + IX = XLIX.

Fun fact: many clock faces show 4 as IIII instead of IV. The Romans themselves were not always strict about the "take away" rule either; it became the usual way much later.

Like the Egyptian system, Roman numerals have no zero, and the value of a letter does not depend on where it sits (except for the "take away" pairs). That makes sums very hard. Try XLVIII + LXXIV: there is no simple column method, because the numbers do not line up into ones, tens and hundreds. People in Roman times did their sums on a counting board, with pebbles, and only wrote down the answer.

Maya numerals

The Maya of Central America used just three signs: a dot for one, a bar for five, and a shell for zero. They were among the very few peoples of the ancient world to invent a sign for zero.

Up to 19, use as many bars as you can, then dots for the rest, with the dots on top:

071319

Seven is a bar and two dots (5 + 2). Nineteen is three bars and four dots (5 + 5 + 5 + 4), the biggest number in one level.

For 20 and beyond, the Maya stacked levels on top of each other. The bottom level counts ones. The next level up counts twenties. The next level up counts four hundreds (twenty twenties). The Maya counted in twenties, perhaps because they counted fingers and toes. Counting in twenties is called base 20; our way, counting in tens, is base 10.

5 × 400 = 20001 × 20 = 206 × 1 = 6 202026

Twenty is one dot in the twenties level and a shell in the ones level: 1 twenty and 0 ones. Here the shell is doing exactly what our 0 does in 20: holding the ones place open. Without it, the dot would drop into the ones level and mean 1.

2026 is a bar (5) in the four-hundreds level, one dot in the twenties level, and a bar and a dot (6) in the ones level: 5 × 400 + 1 × 20 + 6 = 2000 + 20 + 6 = 2026.

So the Maya had place value and a zero, just like us, only with twenties instead of tens. (For counting days in their famous calendar, they changed the third level to count 360s instead of 400s, close to the number of days in a year.)

Why we use 0 to 9

Compare the four ways:

Signs needed Has a zero? Place value? Easy to add in columns?
Tally marks 1 no no no
Egyptian a new one for each power of ten no no not really
Roman 7 letters, more for bigger numbers no no no
Maya 3 yes yes (base 20) yes
Ours (Hindu-Arabic) 10 digits yes yes (base 10) yes

Our system, which came from India and spread through the Arabic-speaking world to Europe, has both big ideas: place value and zero. With them, ten digits are enough for any number, however big, and adding, subtracting, multiplying and dividing can all be done in neat columns. That is why it won.

One more trace of an old system is in your watch. The Babylonians of ancient Iraq counted in sixties, and that is why an hour has 60 minutes and a minute has 60 seconds.

Worked examples

1. Roman to ours. What number is MCMXCIV?

Split it, watching for small-before-big pairs: M, CM, XC, IV. That is 1000 + 900 + 90 + 4 = 1994.

2. Ours to Roman. Write 368 in Roman numerals.

300 + 60 + 8 = CCC + LX + VIII = CCCLXVIII.

3. Egyptian. An Egyptian number has 1 lotus, 3 coils, 0 arches and 5 strokes. What is it?

1000 + 300 + 5 = 1305. In our system we need a 0 in the tens place; the Egyptians simply drew no arches.

4. Maya, one level. A Maya number has two bars and three dots in one level. What is it?

5 + 5 + 3 = 13.

5. Maya, two levels. The twenties level has one bar and one dot, and the ones level has a shell. What is it?

The twenties level shows 5 + 1 = 6, so that is 6 × 20 = 120. The ones level shows 0. The number is 120.

Try it

  1. Draw tally marks for 23. How many full bundles are there?
  2. What numbers are XXVII and CDXL?
  3. Write 99 and 444 in Roman numerals.
  4. How many Egyptian signs are needed for 432? For 1111?
  5. Draw (or describe) the Maya numeral for 17.
  6. A Maya number has one dot in the twenties level and two dots in the ones level. What is it?
  7. Why can't you write 20 in Maya numerals without the shell?
  8. Which needs more signs: 888 in Egyptian or 888 in Roman numerals?
Answers
  1. Four full bundles (20) and 3 more marks: 5, 10, 15, 20, then 21, 22, 23.
  2. XXVII = 10 + 10 + 5 + 1 + 1 = 27. CDXL = CD (400) + XL (40) = 440.
  3. 99 = 90 + 9 = XC + IX = XCIX. 444 = 400 + 40 + 4 = CD + XL + IV = CDXLIV.
  4. 432: 4 coils, 3 arches, 2 strokes = 9 signs. 1111: 1 lotus, 1 coil, 1 arch, 1 stroke = 4 signs.
  5. 17 = 5 + 5 + 5 + 2: three bars with two dots on top.
  6. 1 twenty and 2 ones: 20 + 2 = 22.
  7. Without the shell, the single dot would sit in the ones level and mean 1. The shell holds the ones level open (it says "0 ones"), so the dot stays in the twenties level and means 20.
  8. Egyptian: 8 coils, 8 arches, 8 strokes = 24 signs. Roman: 888 = 800 + 80 + 8 = DCCC + LXXX + VIII = DCCCLXXXVIII, which is 4 + 4 + 4 = 12 letters. Egyptian needs more.

Common mistake

Putting any small letter before any big one. It is tempting to write 99 as IC (100 − 1) or 45 as VL. The usual rule only allows I before V and X, X before L and C, and C before D and M. Split the number into tens and ones first: 99 = 90 + 9 = XCIX, and 45 = 40 + 5 = XLV.

Reading Maya levels from the top as ones. In Maya numerals the bottom level is the ones, and each level up is worth twenty times more. (In our numbers, the ones are on the right.)

Where it leads