Math Detours · Grade 5 · Around the corner · 4

Numbers before zero

Roman numerals, Egyptian strokes, Babylonian sixties. Try adding in them, and see why our ten digits won.

Name ____________________Date ____________

Tonight's idea

People wrote numbers for thousands of years with no zero. How?

In ancient Egypt each size had its own sign: a stroke for 1, a hoop for 10, a coil of rope for 100. You drew as many of each as you needed.

= 1 = 10 = 100 = 243 = 203
Egyptian signs. For 203 you simply draw no hoops. Nothing is needed to say "no tens".

Roman numerals, as we write them today, use letters:

I = 1   V = 5   X = 10   L = 50   C = 100   D = 500   M = 1000
A smaller letter in front means take it away: IV = 4, IX = 9, XL = 40.

In both systems a sign stands for the same number wherever it sits. X always means ten, even when the Romans subtract it.

Our system is different. We have only ten digits, and the place of a digit says what it is worth. The 2 in 203 means two hundreds. That trick needs a sign for "nothing here". Without the 0, 203 and 23 would look the same. Zero holds the empty place open.

Watch one: XXVII + XVI

Tip all the letters into one pile, biggest first: XXX VV III.

Now swap. Two Vs make an X: XXXX III. Four Xs are written XL. The answer is XLIII.

In our digits that was 27 + 16 = 43. Adding in Roman numerals is collecting and swapping. It works, but try it for multiplying. With places and a zero, the same short method works for numbers of any size.

Places do not have to be worth ten. See a system with only two digits: mathdetours.com/demos/binary.html.

Your turn

  1. Write in our digits: (a) XIV   (b) XXIX   (c) LXVI   (d) CXL
  2. Write in Roman numerals: (a) 8   (b) 19   (c) 44   (d) 2026
  3. Count Egyptian signs. (a) How many signs does 99 need? (b) How many does 100 need? (c) Which number below 1000 needs the most signs, and how many?
  4. (a) Write 407 in Roman numerals. (b) Draw 407 in Egyptian signs. (c) Neither needed a zero. Why not? (d) Rub the zero out of 407, and out of 470. What goes wrong?

Stretch: sixties

In ancient Babylon, scribes used places worth sixty. "2, 5" meant 2 sixties and 5, which is 125. What did "3, 20" mean? For a long time they had no zero. Why would "1" alone be hard to read?

Answers

  1. (a) 14. (b) 29. (c) 66. (d) 140.
  2. (a) VIII. (b) XIX. (c) XLIV. (d) MMXXVI.
  3. (a) 18: nine hoops and nine strokes. (b) 1: a single coil. A bigger number can need far fewer signs. (c) 999, with nine of each sign: 27.
  4. (a) CDVII. CD is 500 take away 100. (b) Four coils and seven strokes. (c) Each sign carries its own size, so you just leave the tens out. (d) Both turn into 47. With places, only the zero tells you which place each digit is in.

Stretch. "3, 20" is 3 sixties and 20: \(3 \times 60 + 20 = 200\). A lone "1" could mean 1, or 1 sixty, or 1 lot of sixty sixties (3600). With no zero to fill the empty places, the reader had to guess from the story. We still count in sixties for time: 60 seconds in a minute, 60 minutes in an hour.

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