Tally marks and Roman numerals

Why this exists

The oldest way to keep count is to make one mark for each thing. One sheep, one scratch. One vote, one line. This is One-to-one correspondence: each thing gets exactly one mark, so the number of marks is the number of things. These marks are called tally marks, and people still use them to count votes, laps or birds.

One mark per thing works, but a long row of marks is hard to read. Is it 17 marks or 18? So tally marks are drawn in bundles of five. Then you can count the bundles by fives and only the last few marks one at a time.

The Romans took the next step. Instead of drawing ten marks, they wrote one symbol that means ten. Instead of a hundred marks, one symbol that means a hundred. Their symbols, called Roman numerals, are still all around us. They show the year on monuments and cornerstones and at the end of movie credits. They number the chapters of some books and the hours on many clock faces and sundials. Reading them is a useful skill, and seeing how they work shows, by contrast, what is so clever about our own digits.

Definition

Tally marks. One upright mark stands for one thing. The fifth mark in each group is drawn slanting across the four before it, which closes a bundle of five. A tally is read by counting the bundles by fives and then counting the single marks by ones.

Roman numerals. Seven symbols, each with a fixed value:

A Roman numeral is a row of these symbols, normally written from largest to smallest. Its value is found by adding the values of the symbols. For example \mathrm{CLV} = 100 + 50 + 5 = 155.

There is one exception, the subtractive pairs. When a smaller symbol is written just before a larger one, the smaller one is subtracted from the larger. Six pairs are used:

So \mathrm{XC} means 100 - 10 = 90, but \mathrm{CX} means 100 + 10 = 110.

The standard modern form follows these rules:

• \mathrm{I}, \mathrm{X}, \mathrm{C} and \mathrm{M} may appear at most three times in a row.
• \mathrm{V}, \mathrm{L} and \mathrm{D} never repeat, since two fives make a ten (\mathrm{VV} would just be \mathrm{X}), two fifties make a hundred, and two five-hundreds make a thousand.
• Only the six pairs above subtract: \mathrm{I} before \mathrm{V} or \mathrm{X}, \mathrm{X} before \mathrm{L} or \mathrm{C}, \mathrm{C} before \mathrm{D} or \mathrm{M}.

Intuition

Here is a tally of the votes one movie got when Mandla’s class voted on what to watch. Each slanted mark closes a bundle of five.

Roman numerals grew out of the same idea of bundling. Think of \mathrm{I} as a single mark. Five of them are bundled into \mathrm{V}, two fives into \mathrm{X}, five tens into \mathrm{L}, two fifties into \mathrm{C}, five hundreds into \mathrm{D}, and two five-hundreds into \mathrm{M}. The bundles go five, then two, then five, then two, over and over.

In this way the Romans could write four as \mathrm{IIII} and nine as \mathrm{VIIII}, and old Roman inscriptions often do. The short forms \mathrm{IV} (one less than five) and \mathrm{IX} (one less than ten) save symbols. The strict rules used today were settled long after Roman times, and older habits survive: many clock faces still show four as \mathrm{IIII}.

A Roman numeral is like a pile of coins. A coin worth ten is worth ten wherever it sits in the pile. To find the total, you add up the coins. Writing the coins from largest to smallest just makes the pile easy to count, and it is what lets a reader notice the one break in the pattern: a small symbol sitting in front of a bigger one, which means “take this away”.

Results

Worked examples

The lesson is taught in two parts. Examples 1 to 3 belong to the first: tally marks, and Roman numerals that only add. Examples 4 to 7 belong to the second: the subtractive pairs, years, and why the system has no zero.

Boundaries

Tally marks are for counting things one at a time. A tally can only grow by whole marks, so it shows counting numbers, not parts of a thing. Large tallies are slow to draw and to read.

The standard form covers 1 to 3{,}999. Zero has no Roman numeral, and the standard symbols cannot write 4{,}000 or more. People who needed bigger numbers used other marks, such as a bar drawn over a symbol to mean a thousand times as much, but those are outside the standard form.

Only six subtractive pairs, one small symbol each. \mathrm{IL}, \mathrm{IC}, \mathrm{VX} and \mathrm{XM} are not standard, and neither is two small symbols in front, such as \mathrm{IIX} for eight. \mathrm{V}, \mathrm{L} and \mathrm{D} are never subtracted.

Real inscriptions do not always follow the modern rules. Clock faces with \mathrm{IIII} and old carvings with \mathrm{VIIII} are still read by adding. The rules here describe the standard modern form, not every Roman numeral ever written.

No place value. A Roman symbol’s worth does not depend on its position. Place value, where the same digit is worth ten times more one place to the left, is what our system has and the Roman one does not. Place value also works with bundles of sizes other than ten, as in Numbers in other bases.

Where this goes next

Tally marks become tally charts, which record counts for several categories side by side and lead to Data tables, tallies and frequency tables and graphs. Roman numerals are worth reading for their own sake, on clocks (Telling time), monuments and chapter headings. But their main lesson is the contrast. Our digits reach any size with just ten symbols, because a digit’s place sets its worth and a zero holds an empty place. That idea carries Numbers to one million and beyond, and it is what makes rounding to a place possible: rounding needs places to round to.

Exercises

Exercises 1 to 11 go with the first part of the lesson (Examples 1 to 3), Exercises 12 to 21 with the second (Examples 4 to 7), and the reasoning group draws on both.

Part 1: tally marks and adding symbols

1. A tally shows 3 bundles of five and 4 single marks. What number does it show?
2. Malie tallies 32 birds. How many bundles of five and how many single marks does her tally have?
3. A tally has 8 bundles of five and no single marks. Mandla adds 6 more marks. Describe the new tally and give its total.
4. Write \mathrm{XXVII} in our digits.
5. Write \mathrm{LXVI} in our digits.
6. Write \mathrm{CCLXXXIII} in our digits.
7. A monument shows the year \mathrm{MDCCLXXVI}. Which year is it?
8. Write 38 in Roman numerals.
9. Write 152 in Roman numerals.
10. Write 2{,}615 in Roman numerals.
11. Write 3{,}333 in Roman numerals.

Part 2: subtractive pairs and years

12. Write \mathrm{XIV} in our digits.
13. Write \mathrm{XCIV} in our digits.
14. Write \mathrm{CDLX} in our digits.
15. A cornerstone shows \mathrm{MCMLXIX}. Which year is it?
16. A movie’s credits end with \mathrm{MMXIX}. Which year is it?
17. Write 44 in Roman numerals.
18. Write 279 in Roman numerals.
19. Write the year 1904 in Roman numerals.
20. Write 2{,}444 in Roman numerals.
21. Write the twelve hours of a clock, 1 to 12, in standard Roman numerals. Which of them use a subtractive pair?

Reasoning

22. Behrouz writes 45 as \mathrm{VL}. Explain why that is not the standard form, and write the standard form.
23. A sundial shows four o’clock as \mathrm{IIII}. Explain why it can still be read, and why the standard form is \mathrm{IV}.
24. Write 505 in Roman numerals. Explain why our number 505 needs a 0 and the Roman numeral does not, and what each 5 in 505 is worth.
25. Write 3{,}888 in Roman numerals. Explain why no standard Roman numeral for a number from 1 to 3{,}999 has more symbols.