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Unit 1 · Topic 05 · Number System & Notation

Roman Numerals I to C — Rules of Formation

Hook

Kabir's grandfather's clock had IIII where every other clock has IV.

Kabir told him the clock was wrong.

His grandfather said the clock was ninety years old and had never once been wrong.

Watch the Lesson

The Story

The clock in Kabir's grandfather's hall was old, wooden, and had Roman numerals around its face. Kabir had just learned Roman numerals at school and he spotted it immediately.

"Dada, your clock is broken. Look at the four. It says IIII. Four is IV."

His grandfather did not look up from his newspaper. "That clock has kept time since before your mother was born."

Kabir brought it to Ms. Rao, who was delighted rather than annoyed. She wrote both forms on the board: IIII and IV.

"Kabir is right about the rule," she said. "In Roman numerals a symbol may repeat at most three times. So four is written IV — one before five, meaning one less than five."

Kabir looked satisfied.

"And his grandfather is right about the clock," she added. "Clockmakers have used IIII on clock faces for centuries, because it balances the VIII on the opposite side. It is a design choice, not a maths one. The rule is still the rule."

Then she gave the class something harder. "Write forty-nine."

Kabir thought: forty-nine is one less than fifty. One is I, fifty is L. So IL.

Ms. Rao shook her head. "You may only subtract I from V and from X. Never from L, never from C. So you cannot write IL."

"Then how?"

"Break it the way you say it. Forty, then nine. Forty is XL. Nine is IX. Put them together."

XLIX. Kabir wrote it out and counted it back: fifty minus ten is forty, then ten minus one is nine. Forty-nine.

"The trick," said Ms. Rao, "is that you never subtract across a whole new symbol. You break the number into the pieces you would say out loud, and write each piece."

Kabir went home and checked the clock again. The IIII was still there. He decided not to mention it a second time.

Why forty-nine is XLIX and not ILILI may not follow LXLIXcorrect formSay it aloud: forty + nineXL = 40IX = 9Write each part in turn, largest first.
Break the number into the parts you say aloud, then write each part.

So What Just Happened?

Roman numerals have no zero and no place value. A symbol is worth the same wherever it appears, and the number is built by adding and occasionally subtracting.

The symbols to know are I for 1, V for 5, X for 10, L for 50 and C for 100.

Three rules control everything. First, only I, X and C may repeat, and never more than three times in a row — which is why IIII is not the standard form of 4. Second, V, L and D are never repeated at all, because two of them would simply make the next symbol up. Third, subtraction is limited: I may be taken from V and X only, and X from L and C only.

For any awkward number, break it into the parts you would say aloud and write each part in turn. Forty-nine is forty and nine, so XL then IX, giving XLIX.

Which symbols may repeat and which may be subtractedI = 1 V = 5 X = 10 L = 50 C = 100MAY repeat: I, X, Cat most three in a rowNEVER repeat: V, L, DVV would just be XSubtraction is limited:I only from V and X X only from L and CSo IV and IX are fine, but IL and IC are not.
I, X and C repeat up to three times. V, L and D never repeat.

Remember This

  • I = 1, V = 5, X = 10, L = 50, C = 100.
  • Only I, X and C may repeat, and never more than three times.
  • V, L and D are never repeated at all.
  • I may be subtracted only from V and X; X only from L and C.
  • There is no symbol for zero in the Roman system.
The numbers most often written wrongly4 = IVnot IIII9 = IXnot VIIII40 = XLnot XXXX90 = XCnot LXXXX49 = XLIXnot IL
Four, nine, forty, ninety and forty-nine.

Try It Yourself

Write your birth year and this year in Roman numerals, then swap with someone and read theirs back as ordinary numbers.

Find a Roman numeral in the real world — a clock face, a book chapter, a building date — and check it against the three rules.

Word Bank

Roman numeral
A number written with letters such as I, V, X, L and C.
Repetition rule
I, X and C may repeat at most three times in a row.
Subtraction rule
A smaller symbol before a larger one means subtract.
Addition rule
A smaller symbol after a larger one means add.
Place value
The idea that a digit's position changes its worth — which Roman numerals do not use.

Questions

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