A Story of Numbers — Hard

Q1. A teacher writes 232 as CCXXXII and 413 as CCCCXIII on the board (no subtractive form used). A student adds them carefully by combining like landmarks and then re-grouping (5 Cs = D, 4 Xs = XL). The Roman numeral for the sum is

• A. DCXLV
• B. DCCXV
• C. MCXLV
• D. CDXLV

Q2. Using the constructed base-5 system with landmarks 1, 5, 25, 125,..., the number 143 is written as

• A. Two 125s and one 5
• B. One 125, three 5s, three 1s
• C. Five 25s and one 1
• D. One 125, one 25, three 1s

Q3. Statement A: In any base-n system, the k-th landmark number equals n raised to the power (k − 1). Statement B: This is why the base-10 landmarks 1, 10, 100, 1000 are powers of 10. Which is correct?

• A. Only A is true
• B. Only B is true
• C. Both A and B are true, and B follows from A
• D. Both A and B are false

Q4. A key advantage of base-n systems over the Egyptian-style listing: the product of any two landmark numbers is again a landmark number. The deep reason is

• A. Because landmarks are written using only one digit each
• B. Because n^a × n^b = n^(a+b), and a + b is still a non-negative integer
• C. Because the system uses 0 as a placeholder
• D. Because n is always taken as 10

Q5. Mayan landmark numbers as 1, 20, 360, 7200, 144000. A student claims the Mayan system is therefore a pure base-20 system. Which response is correct?

• A. Yes, because every landmark is a multiple of 20
• B. No, because in pure base-20 the landmark after 20 would be 400, not 360
• C. No, because the Mayans had no symbol for 0
• D. Yes, because 1, 20, 400, 8000 are the Mayan landmarks

Q6. Assertion (A): The later Mesopotamians introduced a special wedge symbol to mark an empty middle place in a sexagesimal number. Reason (R): Without this symbol, '2' (two units) and '120' (two 60s, zero units) could not be distinguished from the writing alone.

• A. Both A and R are true, and R is the correct explanation of A.
• B. Both A and R are true, but R is NOT the correct explanation of A.
• C. A is true, R is false.
• D. A is false, R is true.

Q7. The Chinese rod numerals (3rd century CE) is described as a base-10 system that uses two rod orientations — Zong (vertical) and Heng (horizontal) — alternating across places. The MAIN purpose of alternating Zong and Heng is to

• A. Show whether the number is odd or even
• B. Visually separate adjacent place values when no zero-symbol is used
• C. Indicate addition vs multiplication of digits
• D. Mark units of time as in the Babylonian system

Q8. The Hindu number system is an unambiguous positional system. Which pair of features is JOINTLY responsible for this?

• A. Use of letters of the alphabet and unending new symbols
• B. Use of the digit 0 as a place-holder and a fixed base of 10 for place values
• C. Use of subtractive shortcuts like IV = 4 and IX = 9
• D. Use of dot, bar and shell as in the Mayan system

Q9. Consider the following statements about Indian contributions to the number system: I. The Bakhshali manuscript (c. 3rd century CE) carries the earliest written 0 (as a dot). II. Aryabhata's Aryabhatiya (499 CE) carries out elaborate computations using a decimal place-value system. III. Brahmagupta's Brahma-sphuta-siddhanta (628 CE) codifies 0 as a number on which arithmetic operations can be performed. Which are correct?

• A. I and II only
• B. II and III only
• C. I and III only
• D. I, II and III

Q10. Match the scholar with the role assigns in the spread of Indian numerals: (1) Al-Khwarizmi, c. 825 CE; (2) Al-Kindi, c. 830 CE; (3) Fibonacci, c. 1200 CE. Roles: (a) wrote 'On Calculation with the Hindu Numerals' in the Arab world; (b) introduced Hindu numerals to Europe; (c) wrote 'On the Use of the Hindu Numerals' in the Arab world.

• A. 1-a, 2-c, 3-b
• B. 1-c, 2-a, 3-b
• C. 1-b, 2-a, 3-c
• D. 1-a, 2-b, 3-c

Q11. A Class 8 student writes: '0 is nothing, so 205 and 25 mean the same thing — there are 2 hundreds and 5 in both.' Using the chapter, the BEST teacher response is to

• A. Tell the student that 0 is nothing and the answer 205 = 25 is acceptable shorthand
• B. Show that 0 in the tens place is not nothing but a place-holder; in 205 it forces 2 to mean two hundreds, while in 25 the 2 means two tens
• C. Ask the student to memorise the Roman form CCV for 205
• D. Tell the student to drop the 0, because Mesopotamians wrote it as a blank

Q12. In a class test, four students write the Roman numeral for 40 as: Rahul — XL, Sita — LX, Imran — XXXX, Meera — XC. Using only, who is correct?

• A. Only Rahul
• B. Only Sita
• C. Rahul and Imran both
• D. Rahul and Meera both

Q13. Priya wants her Class 5 children to truly understand why we need the digit 0. Drawing on the chapter, the MOST effective first activity is to ask them to

• A. Memorise the values of the Roman landmark numerals from I to M
• B. Try writing 2, 120 and 7200 using only wedge-shaped marks with no symbol for zero, and see when the writings look the same
• C. Convert 1 to 100 from Hindu numerals into Egyptian symbols
• D. Recite the Yajurveda Samhita names eka, dasha, shata, sahasra in order

Q14. A teacher asks Class 8 students to build their own base-4 number system using fresh symbols @ for 1, # for 4, $ for 16. Using the base-n rule from, the number 27 should be written in this system as

• A. $ followed by ## followed by @@@
• B. ## followed by $ followed by @
• C. $$ followed by @@@
• D. ### followed by @@@

Q15. Using the additive rule for Roman numerals 1–39 (group 10s, then 5s, then 1s, largest landmark first), the correct Roman numeral for 38 is

• A. XXXVIII
• B. XXXIIX
• C. VIIIXXX
• D. IIXL