Revision and important questions: A Story of Numbers

7.1 Summary box

A number system is a fixed ordered sequence of objects, names or symbols. Numerals are the written symbols. A landmark number is a reference number with its own symbol. A base-n system uses the powers of n as landmarks; a positional system uses the position of a symbol to show its landmark. The Hindu system is base-10, uses ten digits including zero, and is positional. Roman numerals use fixed values with landmarks 1, 5, 10, 50, 100, 500 and 1000. Egyptian numerals are base-10 but not positional. Mesopotamian numerals are base-60 and positional, with a placeholder for blanks. Mayan numerals use places 1, 20, 360 and a shell for zero. Chinese rod numerals are base-10 positional.

7.2 Common mistakes

• Calling a base-10 system "the natural system". Base-10 is a choice, likely linked to ten fingers; base-5, base-2 and base-60 have all been used.
• Forgetting that landmark numbers grow by multiplying by the base, not by adding a fixed amount.
• Treating the Mesopotamian system as a perfect base-60 system. Its blank spaces and missing final placeholder create ambiguity.
• Assuming the Mayan system is a true base-20 system. Its third landmark is 360, not 400.
• Thinking zero is only a placeholder. In the Hindu system zero is also a number with its own arithmetic.
• Writing an Egyptian numeral with ten or more copies of one symbol. Ten copies always regroup into the next symbol.
• Mixing Roman subtractive and additive forms carelessly. 4 is IV, 9 is IX, 40 is XL and 90 is XC.
• Forgetting that a place value system allows only finitely many symbols. This is its greatest advantage over systems with a new symbol for every landmark.

7.3 Important questions with answers

Question 1. Write 1984 in Roman numerals.

Answer. MCMLXXXIV: 1000 + 900 + 50 + 30 + 4.

Question 2. What are the landmark numbers of base 3 up to four places?

Answer. 1, 3, 9, 27, 81.

Question 3. Write 45 in base-5 notation.

Answer. 45 = 25 + 4 x 5, so 140 in base-5.

Question 4. Write 100 in the Mayan system.

Answer. 5 x 20 + 0, so five dots in the 20s place and a shell in the units place.

Question 5. Why is the Hindu number system unambiguous?

Answer. Each position holds exactly one of ten digits, and zero marks an empty place, so every numeral has exactly one reading.

Question 6. What is the largest coefficient allowed in a Mesopotamian place?

Answer. 59; sixty of a place would regroup into one of the next place.

Question 7. Convert 11001 in base 2 to base 10.

Answer. 16 + 8 + 1 = 25.

Question 8. Why could the Egyptian system not write very large numbers compactly?

Answer. It needed a new symbol for every new power of ten, so the symbol set would have to be infinite.

Question 9. What did Brahmagupta contribute to the number system?

Answer. He codified zero as a number, along with negative numbers, and described their arithmetic in 628 CE.

Question 10. Which is easier to compute with, Roman or Hindu numerals? Why?

Answer. Hindu numerals, because place value and zero allow carrying and borrowing with a small fixed set of digits; Roman numerals have fixed values and no zero, so multiplication and division are hard.

7.4 Key definitions to memorise

A number system is a fixed ordered sequence used for counting. Numerals are its written symbols. A landmark number has its own name or symbol. A base-n system uses the powers of n as landmarks. A place value system uses the position of a symbol to decide which landmark it represents. A placeholder is a symbol that marks an empty position; zero began in that role.

Keep the systems distinct. Roman numerals are fixed-value landmarks without place value. Egyptian numerals are base-10 but repeat symbols, so they have no place value. Mesopotamian numerals are base-60 and positional. Mayan numerals are positional with an irregular third landmark. Chinese rod numerals are base-10 positional. Hindu numerals are base-10 positional with ten digits including zero.

A useful revision habit is to write each system as a row: base, landmarks, place value or not, and zero or not. That row answers most comparison questions in one look.

7.5 Mixed practice

Practice 1. Write 444 in Roman numerals. Answer: CDXLIV.

Practice 2. List the first four landmarks of base 6. Answer: 1, 6, 36, 216.

Practice 3. Write 85 in base-5. Answer: 85 = 3 x 25 + 2 x 5, so 320 in base-5.

Practice 4. Write 3600 in the Mesopotamian system. Answer: 1 x 3600, written as a 1 in the 3600 place.

Practice 5. Explain why a numeral such as 400 needs a placeholder in base-60. Answer: 400 = 6 x 60 + 40, so the 60-place holds 6 and the units place holds 40; if a place were empty, a blank would be ambiguous without a placeholder.