A Story of Numbers

Count up every landmark symbol on both sides together, then regroup starting from the smallest landmark upward — exactly the way the book's own CCXXXII + CCCCXIII example does it:

So LXXXVII + LXXVIII = CLXV. Check: 87 + 78 = 165, and CLXV = 100+50+10+5 = 165 ✓. Notice how much MORE careful regrouping this takes compared to just adding 87+78 with Hindu numerals — that's exactly the weakness the book is pointing out!

First convert each to Hindu numerals so we can check our work (the book uses this example just to show HOW hard it is to multiply in Roman numerals directly — there's no neat landmark-number shortcut like there is in a base system):

In Roman numerals, 427812 would be written out using CDXXVIICMDCCCXII-style repeated groupings — extremely long and easy to get wrong! This is exactly why the Romans needed a physical abacus (Section 3b) to actually calculate, rather than multiplying the numerals directly on paper.

Addition: put both groups of sticks together in one pile, then that combined pile IS the answer.

Subtraction: physically remove that many sticks from the pile; whatever's left over is the answer.

Multiplication: make several equal-sized groups of sticks (as many groups as the second number), then combine all the groups into one big pile.

Division: from the big pile, keep pulling out the "size of group" you're dividing by, over and over. The number of full groups you can make is the answer, and any leftover sticks are the remainder.

There are 26 letters (a–z), giving numbers 1–26. One neat way to keep going: after z, repeat each letter TWICE for the next 26 numbers, giving 'aa'=27, 'bb'=28, ... 'zz'=52 — then repeat each letter THREE times for 'aaa'=53, 'bbb'=54, and so on forever.

There are many valid ways to do this — any scheme that keeps the sequence going forever without repeating an earlier code works!

This is an open, creative "Try This" activity — there's no single fixed answer! A good number system needs a standard, fixed-order sequence that never runs out. Try picking your own symbols and a group size (like groups of 3, or of 6) and see if you can write, say, the number 50 in it.

Notice L×D=25,000 needs 25 copies of the BIGGEST available symbol M — there's no bigger landmark symbol to jump to, which is exactly the weakness a base system doesn't have!

This is an open discussion question. A likely reason: early counting often grew out of counting SPECIFIC, culturally important things (like canoes, or coconuts, or people) rather than "numbers" as an abstract idea — so different objects that mattered for different purposes ended up with their own separate counting words, before the idea of one single, general-purpose number sequence (that could count absolutely anything) was recognised.

Recall urapon = 1, ukasar = 2. So (ukasar-ukasar-ukasar-ukasar-urapon) = 2+2+2+2+1 = 9, and (ukasar-ukasar-ukasar-urapon) = 2+2+2+1 = 7, (ukasar-ukasar-ukasar) = 2+2+2 = 6, (ukasar-ukasar) = 2+2 = 4.

To add/subtract in this system without Hindu numerals: just count out (or remove) sticks/pairs directly in urapon/ukasar chunks! To multiply, repeat the group that many times; to divide, see how many times the smaller group fits into the bigger one.

Open activity — using the ideas from this chapter (landmark numbers, a fixed base/group size, and eventually place value), try to fix any weaknesses in the number system you invented earlier. Does it need new landmark symbols? Would grouping by a base help it handle bigger numbers more compactly?

Each part is written using that many copies of the matching landmark symbol — e.g. 70707 needs seven "10,000" symbols, seven "100" symbols, and seven "1" symbols.

(i) Two coil symbols (10² each) + six heel-bone symbols (10 each) + six stroke symbols (1 each):

(These are the official NCERT answers — 276 and 4322 — reverse-decoded from the count of each hieroglyph symbol shown in the textbook figure.)

Base-5 landmark numbers: 1, 5, 25, 125, 625...

Check 651: 5×125=625, +1×25=650, +1=651 ✓

Yes — zero! Our base-5 system (built only from powers of 5: 1, 5, 25, 125...) has no symbol representing "nothing." This is exactly the gap that the invention of zero as a placeholder later solved.

In general, a base-n system's landmark numbers are n⁰, n¹, n², n³...

The method (shown just above, in the book's own worked example) is: total up every symbol of each landmark size, then regroup any time you hit 10 copies of a symbol — 10 copies of one landmark ALWAYS becomes exactly 1 copy of the next landmark up.

Apply the same regroup-on-reaching-10 method to any pair of Egyptian numerals you're asked to add — count all symbols of each size, then carry upward through the landmark numbers exactly like carrying in column addition.

Same idea as Egyptian addition, but regroup every time a landmark symbol reaches 5 copies instead of 10 (since 5× a landmark number gives the next one up in base-5). E.g. adding a numeral with six "5"-symbols: 6 fives → regroup 5 of them into one "25"-symbol, leaving 1 "5"-symbol + 1 "25"-symbol.

Multiplying any landmark number 10ᵀ by 10 gives 10ᵀ⁺¹ — always the NEXT landmark number. Multiplying by 10² jumps two landmarks ahead, and so on.

Yes — this holds true in the base-5 system we built, and in fact holds for ANY number system with a base, because it's really just the exponent rule nᵃ×nᵇ=nᵃ⁺ᵇ in disguise, and that's true for every value of n.

This is the distributive-law walkthrough from Section 3 above — each landmark "chunk" is multiplied on its own, then the results are recombined.

No. If any symbol needed to repeat 10 times, those 10 copies would automatically combine into ONE copy of the next landmark number instead (since each landmark number is exactly 10× the one before it) — so no symbol is ever repeated 10 or more times.

Landmark numbers: 4⁰=1, 4¹=4, 4²=16. Using symbols □ (=1) and △ (=4):

16 = 4² needs a brand NEW landmark symbol, since it's the next power of 4!

Multiplying by 5 just shifts every symbol UP one landmark level — each "1" becomes a "5", each "5" becomes a "25", each "25" becomes a "125", and so on. (It's exactly like multiplying by 10 in our own system just appends a zero and shifts every digit one place left!)

Watch 60 carefully: it's written as ONE symbol in the 60s-place with NOTHING in the ones-place — exactly the kind of blank space that later caused so much reading confusion (see Section 4b)!

100 stacks as: "5" (one bar + nothing? — actually 5 dots, i.e. one bar) in the 20s-row, and the shell (0) in the 1s-row: 5×20+0=100 ✓

Why alternate: using zong for one place and heng for the very next place keeps neighbouring digits visually distinct, so you can tell where one digit ends and the next begins, even without a clear gap.

41 using only Zong symbols: 41 = 4 tens + 1 one → using zong for BOTH places would give "‖‖" (4) then "|" (1), i.e. ‖‖ |.

Yes, it could be misread! Without a clear gap, "‖‖|" (four strokes then a gap then one stroke) could just as easily be read as 2 then 3 (23), or 3 then 2 (32), or even all five strokes as one run read as 122 or similar — exactly the same kind of ambiguity that blank spaces caused for the Mesopotamians!

Let urapon = digit 0 and ukasar = digit 1 in base-2 place value:

Both systems have only TWO landmark building blocks (urapon/ukasar, or "0"/"1")! But base-2 is a true PLACE VALUE system — each symbol's value depends on its position — while Gumulgal just repeats "ukasar" the right number of times with no positional meaning. That's why base-2 can compactly represent ANY number, while Gumulgal only had names up to 6.

Hindu numerals and zero show up constantly: money and banking, telling time, measuring in science and engineering, computer programming (binary itself is built on the idea of 0!), medicine dosages, sports scores, exam marks, and more.

Without an efficient place-value system with zero, large-scale trade, accurate science, modern computing, and fast arithmetic in general would all have been dramatically slower and harder to develop — much like how multiplying in Roman numerals is so much harder than in our own system.

Try building 25 in the place-value builder in Section 4 above! (This question also asks: if humans had 8 fingers instead of 10, we'd likely have grown up using a base-8 system instead of base-10 — our whole number system is thought to trace back to counting on 10 fingers!)

(b) Brahmagupta

(c) 0

(b) 360 — a quirky exception, likely linked to their calendar, not a true power of 20 (which would be 400).

Any two of: the Mesopotamians/Babylonians (base-60, placeholder symbol), the Mayans (base-20-ish, shell symbol for zero), the Chinese (rod numerals, blank space), and the Indians (fully developed zero as both placeholder AND number).

A counter above a line is worth 5 times that line's power of 10. It's useful because it lets you represent a digit like 6, 7, 8, or 9 using just 2–4 counters instead of needing up to 9 separate counters on the line — very similar to how Roman numerals jump from IIII to V.

In a base-n system, every landmark number is a power of n, so multiplying any two landmark numbers always gives another exact landmark number (e.g., 10×100=1000, all powers of 10). This predictable pattern means multiplication can be done systematically using the distributive rule. In the Roman system, landmark numbers (I, V, X, L, C, D, M) don't follow one consistent multiplying rule — V×X doesn't cleanly become another "landmark" symbol — so there's no shortcut, making multiplication far harder.

All three numbers — 12, 602 (=10×60+2), and 36002 (=10×3600+2) — used the SAME "10-symbol, then 2-symbol" pattern, differing only in how many BLANK positions separated them. Since spacing wasn't written consistently, readers could easily misjudge how many positions were skipped, and so misread the number entirely. Later Mesopotamians fixed part of this by inventing a placeholder symbol (an ancestor of our 0) to explicitly mark an empty position instead of leaving a blank — though even that placeholder wasn't used at the very end of a number, so some ambiguity remained.

Landmark numbers: 4⁰=1, 4¹=4, 4²=16. Using symbols □ (=1) and △ (=4):

16 = 4² needs a brand NEW landmark symbol, since it's the next power of 4!

All three used place value (position determines the landmark number), but each had a gap:

Mesopotamian (base-60): even after inventing a placeholder symbol, it still wasn't used at the END of a number, so ambiguity remained.

Mayan: HAD a genuine zero (the shell symbol) used consistently, but wasn't a true base system since the third landmark jumped to 360 instead of 400, so it lost some of the clean multiplication benefits of a real base-n system.

Chinese (rod numerals): alternated zong/heng symbols to reduce (but not eliminate) blank-space confusion, but never developed its own symbol for zero at all. The Hindu system is the one that combined a true base-10 place value system WITH a fully-fledged zero used everywhere — which is why it's the system used worldwide today.

(a) Incomplete. Both are base-10, but that's not the whole picture.

(b) The Hindu system is a place value system using zero — the same 10 digits are reused, and their VALUE depends on their position. The Egyptian system is not positional; it just uses more copies of fixed symbols.

(c) Without place value, the Egyptians needed a brand-new unique symbol for every higher power of 10 (100, 1000, 10000...) forever — there's no limit to how many new symbols you'd eventually need for bigger and bigger numbers. Place value with zero solves this by reusing just 10 digits for ANY number, no matter how large.