In Roman Numerals the Same Symbol Always Means the Same Thing
Learn to read and write tally, Egyptian, Roman, Mesopotamian and Mayan numerals, identify the landmark numbers each system uses, and see what separates a positional system from a non-positional one.
What makes our number system better than Roman numerals?
Place value. In Roman numerals an means ten wherever it stands, so a big number needs many symbols. In our system the digit means three hundred in and just three in — the position does the work.
That one difference changes everything about how numbers are written. This page covers everything in the CBSE Class 8 Mathematics chapter's first part: early number systems, the landmark numbers each uses, and positional against non-positional systems.
That one difference changes everything about how numbers are written. This page covers everything in the CBSE Class 8 Mathematics chapter's first part: early number systems, the landmark numbers each uses, and positional against non-positional systems.
How do the early number systems write a quantity?
Each uses its own set of symbols, and most of them simply add the values up.
Tally marks. One stroke per item, grouped in fives with the fifth struck across the other four. So seven is a struck group of five followed by two strokes. Simple to record, but hopeless for large numbers.
Egyptian numerals. Separate picture symbols for each power of ten — a stroke for 1, a heel bone for 10, a coil of rope for 100 and a lotus for 1000. A number is written by repeating each symbol as many times as needed and adding.
So needs two coils, four heel bones and three strokes — nine symbols in all.
Roman numerals. Letters stand for values:
Symbols are added when written in decreasing order, so and . But when a smaller symbol precedes a larger one it is subtracted: , , , , , .
So — two hundreds, forty, five and three ones.
Mesopotamian numerals. Only two symbols, a wedge for 1 and a chevron for 10, combined to build the numbers up to 59 — and then grouped in a base 60 positional arrangement.
Mayan numerals. A dot for 1, a bar for 5 and a shell shape for zero, arranged vertically in a base 20 positional system.
The systems divide into two kinds, and the count of symbols shows why it matters. Writing took nine Egyptian symbols and takes only three digits in our system — because Egyptian numerals repeat a symbol for each unit while ours lets the position carry the size.
Tally marks. One stroke per item, grouped in fives with the fifth struck across the other four. So seven is a struck group of five followed by two strokes. Simple to record, but hopeless for large numbers.
Egyptian numerals. Separate picture symbols for each power of ten — a stroke for 1, a heel bone for 10, a coil of rope for 100 and a lotus for 1000. A number is written by repeating each symbol as many times as needed and adding.
So needs two coils, four heel bones and three strokes — nine symbols in all.
Roman numerals. Letters stand for values:
Symbols are added when written in decreasing order, so and . But when a smaller symbol precedes a larger one it is subtracted: , , , , , .
So — two hundreds, forty, five and three ones.
Mesopotamian numerals. Only two symbols, a wedge for 1 and a chevron for 10, combined to build the numbers up to 59 — and then grouped in a base 60 positional arrangement.
Mayan numerals. A dot for 1, a bar for 5 and a shell shape for zero, arranged vertically in a base 20 positional system.
The systems divide into two kinds, and the count of symbols shows why it matters. Writing took nine Egyptian symbols and takes only three digits in our system — because Egyptian numerals repeat a symbol for each unit while ours lets the position carry the size.
What are the landmark numbers of each system?
The landmark numbers are the quantities a system gives its own symbols to, and everything else is built from them.
- Tally — landmark 5, marked by the struck-through stroke
- Egyptian — landmarks 1, 10, 100, 1000, each a separate picture
- Roman — landmarks 1, 5, 10, 50, 100, 500, 1000, so both halves and whole powers of ten
- Mesopotamian — landmarks 1, 10 and then 60, giving a base 60 system
- Mayan — landmarks 1, 5 and then 20, giving a base 20 system
- Hindu system, the one we use — landmark 10, with ten digits from to
How they are combined:
- Additively, in Egyptian and Roman numerals — repeat and add, with Roman also subtracting for certain pairs
- Positionally, in Mesopotamian, Mayan and Hindu systems — the place a symbol sits multiplies its value
Traces of these survive in daily use. Roman numerals appear on clock faces and chapter numbers, and tally marks are still used by shopkeepers keeping a running count.
The reason a system picks 5 and 10 as landmarks is not mathematical but practical — they match the fingers of one hand and of two. And 20, the Mayan base, matches fingers and toes together, which is why base 20 appears independently in more than one place.
- Tally — landmark 5, marked by the struck-through stroke
- Egyptian — landmarks 1, 10, 100, 1000, each a separate picture
- Roman — landmarks 1, 5, 10, 50, 100, 500, 1000, so both halves and whole powers of ten
- Mesopotamian — landmarks 1, 10 and then 60, giving a base 60 system
- Mayan — landmarks 1, 5 and then 20, giving a base 20 system
- Hindu system, the one we use — landmark 10, with ten digits from to
How they are combined:
- Additively, in Egyptian and Roman numerals — repeat and add, with Roman also subtracting for certain pairs
- Positionally, in Mesopotamian, Mayan and Hindu systems — the place a symbol sits multiplies its value
Traces of these survive in daily use. Roman numerals appear on clock faces and chapter numbers, and tally marks are still used by shopkeepers keeping a running count.
The reason a system picks 5 and 10 as landmarks is not mathematical but practical — they match the fingers of one hand and of two. And 20, the Mayan base, matches fingers and toes together, which is why base 20 appears independently in more than one place.
What separates a positional system from a non-positional one?
In a positional (place value) system the value of a symbol depends on where it stands. In a non-positional system a symbol has the same value wherever it appears.
Non-positional — Egyptian and Roman:
In , each means exactly ten, so they simply add. Move them about and nothing changes, because position carries no value.
That is why needs eight symbols. Every ten and every one must be written out separately.
Positional — the Hindu system:
In :
- the is in the hundreds place, so it means
- the is in the tens place, so
- the is in the ones place, so
Now rearrange the same three digits as :
The digit meant three hundred in the first and three in the second. Same symbol, different value — which is impossible in a Roman numeral.
The consequences are large:
- A positional system needs only as many symbols as its base, ten in our case, however large the number
- A non-positional system needs more and more symbols as numbers grow
- Arithmetic is far easier positionally, since columns line up and can be added place by place. Try multiplying by and the difficulty is obvious
The piece that makes a positional system complete is a symbol for nothing, and that is why the Mayan shell and our zero matter. Without a zero there is no way to show an empty place, so could not be distinguished from — the subject of the next part of this chapter.
Non-positional — Egyptian and Roman:
In , each means exactly ten, so they simply add. Move them about and nothing changes, because position carries no value.
That is why needs eight symbols. Every ten and every one must be written out separately.
Positional — the Hindu system:
In :
- the is in the hundreds place, so it means
- the is in the tens place, so
- the is in the ones place, so
Now rearrange the same three digits as :
The digit meant three hundred in the first and three in the second. Same symbol, different value — which is impossible in a Roman numeral.
The consequences are large:
- A positional system needs only as many symbols as its base, ten in our case, however large the number
- A non-positional system needs more and more symbols as numbers grow
- Arithmetic is far easier positionally, since columns line up and can be added place by place. Try multiplying by and the difficulty is obvious
The piece that makes a positional system complete is a symbol for nothing, and that is why the Mayan shell and our zero matter. Without a zero there is no way to show an empty place, so could not be distinguished from — the subject of the next part of this chapter.
Exam tip
Exam tip: counting symbols to show efficiency
This chapter is marked on comparisons, so make them concrete.
When asked why our system is better, count the symbols for the same quantity: takes three digits but eight Roman symbols. A number beats an adjective here.
For Roman numerals, apply the subtractive rule correctly — a smaller symbol before a larger one subtracts, so and , while and .
Name the landmark numbers when asked about a system, and say whether it combines them additively or positionally.
For place value, show the expansion: **. That line is the definition of positional in action.
And give the same digit in two positions as your example of place value, since demonstrating that can mean 300 or 3 is exactly what the question is testing.
When asked why our system is better, count the symbols for the same quantity: takes three digits but eight Roman symbols. A number beats an adjective here.
For Roman numerals, apply the subtractive rule correctly — a smaller symbol before a larger one subtracts, so and , while and .
Name the landmark numbers when asked about a system, and say whether it combines them additively or positionally.
For place value, show the expansion: **. That line is the definition of positional in action.
And give the same digit in two positions as your example of place value, since demonstrating that can mean 300 or 3 is exactly what the question is testing.
Did you know
Why do so many number systems choose five, ten or twenty?
Because those are the numbers people can count on their own bodies.
Five is the fingers of one hand, which is why tally marks are grouped in fives and why Roman and Mayan numerals both give five its own symbol. Ten is the fingers of both hands, and it is the landmark of the Egyptian and Hindu systems. Twenty is fingers and toes together, and it is the base of the Mayan system.
The choice is therefore practical rather than mathematical — any base would work arithmetically. That is also why base 60 in the Mesopotamian system stands out as unusual, and why traces of it survive in the 60 minutes of an hour.
Five is the fingers of one hand, which is why tally marks are grouped in fives and why Roman and Mayan numerals both give five its own symbol. Ten is the fingers of both hands, and it is the landmark of the Egyptian and Hindu systems. Twenty is fingers and toes together, and it is the base of the Mayan system.
The choice is therefore practical rather than mathematical — any base would work arithmetically. That is also why base 60 in the Mesopotamian system stands out as unusual, and why traces of it survive in the 60 minutes of an hour.
Key takeaways
Early number systems: quick revision
- Tally groups strokes in fives; Egyptian numerals give separate pictures to 1, 10, 100 and 1000 and add them.
- Roman numerals use , adding in decreasing order but subtracting when a smaller symbol precedes a larger one — so and .
- Mesopotamian numerals use a wedge and a chevron in base 60; Mayan numerals use dot, bar and shell in base 20.
- Landmark numbers are the values a system gives its own symbols, and they are combined either additively or positionally.
- In a non-positional system a symbol's value never changes; in a positional system it depends on place, so the in means 300 while in it means 3.
- A positional system needs only as many symbols as its base and makes arithmetic far easier — and it needs a zero to mark an empty place.
You will remember all of this far better after answering five questions on it than after reading it twice.
- Roman numerals use , adding in decreasing order but subtracting when a smaller symbol precedes a larger one — so and .
- Mesopotamian numerals use a wedge and a chevron in base 60; Mayan numerals use dot, bar and shell in base 20.
- Landmark numbers are the values a system gives its own symbols, and they are combined either additively or positionally.
- In a non-positional system a symbol's value never changes; in a positional system it depends on place, so the in means 300 while in it means 3.
- A positional system needs only as many symbols as its base and makes arithmetic far easier — and it needs a zero to mark an empty place.
You will remember all of this far better after answering five questions on it than after reading it twice.