A Story of Numbers

 HEGP103 — A Story of Numbers

Complete Questions and Answers

This guide answers the exercises, “Figure it Out” questions, and the main activity questions in the attached Grade 8 chapter. For symbol-based systems, the numerical value and a clear textual representation are given where the PDF’s symbols are difficult to reproduce in plain text.

 

1. The mechanism of counting

Question 1. How can we ensure that all cows have returned safely after grazing without using Hindu numerals?

Make a one-to-one correspondence between the cows and an available collection of objects, such as sticks or pebbles. Keep one stick for every cow that leaves. When the cows return, match each cow with one stick. If every stick has exactly one cow matched to it, all cows have returned; an unmatched stick indicates a missing cow.

 

Question 2. How can we determine whether we have fewer cows than our neighbour?

Make one row of sticks for our cows and another row of sticks for the neighbour’s cows. Pair the sticks one-to-one. If our row has unpaired sticks after all the neighbour’s sticks are matched, we have more cows. If the neighbour’s row has unpaired sticks, we have fewer cows. If no sticks remain unpaired, both herds have the same number of cows.

 

Question 3. If our herd is smaller, how can we find how many cows are needed to equal the neighbour’s herd?

Pair the cows, or their corresponding sticks, one-to-one. Count the unpaired sticks in the neighbour’s group. That number is the additional number of cows required.

 

Question 4. How many numbers can be represented using the 26 sounds or letters of the English alphabet if each letter is used only once?

Only 26 numbers, namely 1 through 26, can be represented. The system can be extended by allowing strings of letters.

 

Question 5. How can a letter-based system be extended to represent all numbers?

One possible rule is to use one-letter names for 1–26, two-letter names for the next group, and then continue systematically. For example, after z, use aa, ab, ac, and so on, or use repeated-letter blocks such as aa, bb, cc, etc. The essential requirement is a fixed, unambiguous order in which every new string represents the next number.

 

Question 6. Figure it Out: How can sticks be used to add, subtract, multiply, and divide without Hindu numerals?

For addition, place two collections of sticks together and count the total collection. For subtraction, remove from one collection as many sticks as are present in the second collection; the remaining sticks give the difference. For multiplication, make equal groups: the number of groups and the number of sticks per group show the product. For division, repeatedly form equal groups of the divisor and count the groups; any leftover sticks are the remainder.

 

Question 7. Make an example of a number system.

One possible system uses coloured stones. Let one red stone represent 1, two red stones represent 2, and so on up to five. Then let one blue stone represent 5, one blue plus one red represent 6, and continue by grouping five stones of one level into one stone of the next level. This is a base-5 system with landmark values (1,5,25,125,\ldots).

 

2. Early number systems

Question 8. How are the Gumulgal number names formed?

The Gumulgal count in groups of two. urapon means 1 and ukasar means 2. Therefore:

 

Number	Gumulgal form	Meaning
1	urapon	1
2	ukasar	2
3	ukasar-urapon	2 + 1
4	ukasar-ukasar	2 + 2
5	ukasar-ukasar-urapon	2 + 2 + 1
6	ukasar-ukasar-ukasar	2 + 2 + 2

Numbers greater than 6 were called ras in the system described in the chapter.

 

Question 9. What difficulties arise in a system that counts only in groups of one fixed size?

Such a system may be convenient for small numbers but becomes cumbersome for large numbers. It may require many repeated symbols or words, and arithmetic becomes difficult. For example, representing 1345 by repeatedly grouping only in fives would require 269 groups of five, which is not compact.

 

Question 10. How would 1345 be represented in a system that counts only by fives?

[ 1345=269\times5. ]

 

Thus, it would require 269 groups of five, with no remainder. If the system has a symbol for a group of five, the representation is 269 repetitions of that group symbol.

 

3. Roman numerals

Question 11. Represent the following numbers in Roman numerals.

Number	Roman numeral	Decomposition
1222	MCCXXII	1000 + 200 + 20 + 2
2999	MMCMXCIX	2000 + 900 + 90 + 9
302	CCCII	300 + 2
715	DCCXV	500 + 200 + 15

Question 12. Add (LXXXVII+LXXVIII) without first converting to Hindu numerals.

Combine the symbols:

 

[ LXXXVII+LXXVIII=CLXV. ]

 

Numerically, this is 87 + 78 = 165, and the Roman form is CLXV.

 

Question 13. What are the products of the Roman landmark numbers (V\times L), (L\times D), (V\times D), and (VII\times IX)?

[ V\times L=5\times50=250=\textbf{CCL}, ] [ L\times D=50\times500=25,000=\textbf{\overline{XXV}}. ]

 

Without overline notation, 25,000 can be written as 25 symbols for 1000. Also,

 

[ V\times D=5\times500=2500=\textbf{MMD}, ] [ VII\times IX=7\times9=63=\textbf{LXIII}. ]

 

Question 14. Why is the Hindu number system more efficient than the Roman system?

The Hindu system is a place-value system using only ten symbols, including zero. The value of a digit depends on its position, so arbitrarily large numbers can be written with a finite set of symbols. Addition, subtraction, multiplication, and division are also systematic.

 

Roman numerals use symbols with fixed values, have no ordinary zero digit, and do not have a fully developed place-value structure. Larger numbers require additional notation, and arithmetic operations are more difficult.

 

Question 15. Why might a Pacific island community use different sequences of names for different objects?

Different objects may be counted in different customary groupings. For example, people may count coconuts singly, fish in pairs, and bundles in groups of five. Such systems may reflect the objects’ physical forms, traditional trade practices, or the way the community commonly handles them.

 

4. Egyptian numerals

Egyptian landmark symbols represent powers of 10: 1, 10, 100, 1000, 10,000, 100,000, 1,000,000, and so on. Egyptian notation is additive: repeated symbols are added together.

 

Question 16. Represent the following numbers in Egyptian additive form.

Number	Egyptian additive decomposition
10,458	(10,000+400+50+8)
1,023	(1,000+20+3)
2,660	(2,000+600+60)
784	(700+80+4)
1,111	(1,000+100+10+1)
70,707	(70,000+700+7)

To draw the Egyptian numeral, use one 10,000 symbol, four 100 symbols, five 10 symbols, and eight 1 symbols for 10,458, and similarly for the other rows.

 

Question 17. What numbers do the two Egyptian numerals in the exercise represent?

The two illustrated numerals represent:

 

1      276, which is (200+70+6).

2      4322, which is (4000+300+20+2).

 

Question 18. Can an Egyptian numeral have one symbol occurring ten or more times?

No. Ten copies of any landmark symbol can be regrouped as one symbol of the next higher landmark value. For example, ten 10-symbols become one 100-symbol. Therefore, after regrouping, no symbol needs to occur ten or more times.

 

Question 19. Add the two Egyptian numerals in Figure it Out Question 1(i).

The first numeral represents

 

[ 9(1000)+6(100)+8=9608. ]

 

The second represents

 

[ 5(100)+7=507. ]

 

Therefore,

 

[ 9608+507=\boxed{10115}. ]

 

The Egyptian result contains one 10,000-symbol, one 100-symbol, one 10-symbol, and five 1-symbols.

 

Question 20. Add the two Egyptian numerals in Figure it Out Question 1(ii).

The first numeral represents

 

[ 1000+8(10)=1080. ]

 

The second represents

 

[ 4(10)+6=46. ]

 

Therefore,

 

[ 1080+46=\boxed{1126}. ]

 

The Egyptian result contains one 1000-symbol, one 100-symbol, two 10-symbols, and six 1-symbols after regrouping.

 

5. Base-5 number system

Question 21. What are the landmark numbers in the base-5 system?

The landmark numbers are the powers of 5:

 

[ 5^0=1,\quad5^1=5,\quad5^2=25,\quad5^3=125,\quad5^4=625,\quad\ldots ]

 

Question 22. Express 143 in the base-5 system.

[ 143=125+5+5+5+1+1+1. ]

 

In ordinary base-5 notation,

 

[ 143=1033_5. ]

 

In the chapter’s additive symbol notation, this is one 125-symbol, three 5-symbols, and three 1-symbols.

 

Question 23. Write the following numbers in the base-5 system.

Decimal number	Base-5 form	Additive form
15	(30_5)	3 groups of 5
50	(200_5)	2 groups of 25
137	(1022_5)	1×125 + 2×5 + 2×1
293	(2133_5)	2×125 + 1×25 + 3×5 + 3×1
651	(10101_5)	1×625 + 1×25 + 1×1

Using the chapter’s symbols, replace the digits by the corresponding number of 1-, 5-, 25-, 125-, and 625-symbols.

 

Question 24. Is there a number that cannot be represented in the base-5 system described in the chapter?

In the version described in the chapter, which has no symbol for zero, zero cannot be represented directly. Every positive number can be represented additively using the landmark symbols. A fully developed positional base-5 system would introduce a zero symbol and could represent zero as well.

 

Question 25. Find the landmark numbers of a base-7 system.

[ 7^0=1,\quad7^1=7,\quad7^2=49,\quad7^3=343,\quad7^4=2401,\ldots ]

 

Thus, the landmark numbers are 1, 7, 49, 343, 2401, ….

 

Question 26. What are the landmark numbers of a base-(n) system?

They are

 

[  n^0=1,\ n^1=n,\ n^2,\ n^3,\ n^4,\ldots ]

 

In general, the landmark numbers are all non-negative integer powers of the base.

 

Question 27. Add the two base-5 numerals shown in the exercise.

The first numeral contains one 1-symbol, two 5-symbols, one 25-symbol, and two 125-symbols:

 

[ 1+2(5)+25+2(125)=291. ]

 

The second contains three 1-symbols, one 5-symbol, two 25-symbols, and two 125-symbols:

 

[ 3+5+2(25)+2(125)=308. ]

 

Therefore,

 

[ 291+308=599. ]

 

In base 5,

 

[ 599=4(125)+3(25)+4(5)+4=4344_5. ]

 

Thus, the answer is 599 in decimal, or (4344_5). The result contains four 125-symbols, three 25-symbols, four 5-symbols, and four one-symbols.

 

6. Addition and multiplication in base systems

Question 28. What is the advantage of using a base system for addition?

When a landmark symbol occurs as many times as the base, it can be regrouped into one symbol of the next landmark value. In base 10, ten 1s become one 10, ten 10s become one 100, and so on. In base 5, five symbols of one level become one symbol of the next level. This gives a systematic carrying procedure.

 

Question 29. What is any Egyptian landmark number multiplied by 10?

It becomes the next landmark number:

 

[ 10^k\times10=10^{k+1}. ]

 

For example:

 

[10\times10=100,
\quad100\times10=1000,
\quad1000\times10=10,000,
\quad10,000\times10=100,000.]

 

Question 30. Find the products of the Egyptian landmark numbers with 10.

The four illustrated products are:

 

[ 10\times10=100, ] [ 100\times10=1000, ] [ 1000\times10=10,000, ] [ 10,000\times10=100,000. ]

 

Question 31. What is any Egyptian landmark number multiplied by (10^2)?

Multiplication by (10^2=100) increases the exponent by 2:

 

[ 10^k\times10^2=10^{k+2}. ]

 

Thus:

 

[10\times100=1000,
\quad100\times100=10,000,
\quad1000\times100=100,000,
\quad10,000\times100=1,000,000.]

 

Question 32. Find the products of the illustrated Egyptian landmark pairs.

The products are:

 

[ 10\times100=1000, ] [ 100\times100=10,000, ] [ 1000\times100=100,000, ] [ 10,000\times100=1,000,000. ]

 

Question 33. Does the product of two landmark numbers remain a landmark number in the base-5 system and in any base system?

Yes. In a base-(n) system, landmark numbers are powers of (n). Therefore,

 

[ n^a\times n^b=n^{a+b}, ]

 

which is another landmark number. This holds for every positive integer base.

 

Question 34. What is the product of a number and 10 in the Egyptian system?

Multiplying any Egyptian number by 10 moves every landmark component one level higher. For example,

 

[(2\times100+3\times10+4)\times10
=2\times1000+3\times100+4\times10.]

 

The result is obtained by replacing every symbol with the next higher landmark symbol.

 

Question 35. Find the four illustrated Egyptian landmark products on page 67.

Using the Egyptian symbols for 10, 100, 1000, 10,000, 100,000, and 1,000,000:

 

[ 10\times100=1000, ] [ 100\times1000=100,000, ] [ 1000\times1000=1,000,000, ] [ 10,000\times1,000,000=10^{10}=10,000,000,000. ]

 

Question 36. Does the distributive law hold in Egyptian numerals?

Yes. Egyptian symbols represent ordinary numbers, so the distributive law applies:

 

[ (a+b)\times n=a\times n+b\times n. ]

 

For example, if a numeral represents (100+100+1), then multiplying by 10 gives

 

[ (100+100+1)\times10=1000+1000+10. ]

 

7. Base-4 system

Question 37. Construct a base-4 system and represent 1 through 16.

Let the symbols be (u) for 1, (v) for 4, and (w) for 16. Then:

 

Decimal	Base-4 form	Symbolic form
1	(1_4)	u
2	(2_4)	uu
3	(3_4)	uuu
4	(10_4)	v
5	(11_4)	v + u
6	(12_4)	v + 2u
7	(13_4)	v + 3u
8	(20_4)	2v
9	(21_4)	2v + u
10	(22_4)	2v + 2u
11	(23_4)	2v + 3u
12	(30_4)	3v
13	(31_4)	3v + u
14	(32_4)	3v + 2u
15	(33_4)	3v + 3u
16	(100_4)	w

Question 38. Give a simple rule for multiplying a base-5 number by 5.

Multiplying by 5 shifts every base-5 place one position to the left and adds a zero in the units position. For example,

 

[ 23_5\times5=230_5. ]

 

In additive notation, each symbol is replaced by the next higher landmark symbol.

 

8. Mesopotamian or Babylonian base-60 system

Question 39. Represent the following numbers in the Mesopotamian system.

Write the base-60 digits from largest place to smallest place, separated by commas.

 

Decimal number	Calculation	Base-60 representation
63	(1\times60+3)	((1,3)_{60})
132	(2\times60+12)	((2,12)_{60})
200	(3\times60+20)	((3,20)_{60})
60	(1\times60+0)	((1,0)_{60})
3605	(1\times3600+0\times60+5)	((1,0,5)_{60})

The zero in the middle is a placeholder showing that no 60s occur in 3605.

 

Question 40. How would 3600 be represented, and what difficulty did the original Babylonian system have?

In a fully specified place-value notation,

 

[ 3600=1\times60^2+0\times60+0, ]

 

so it would be written as ((1,0,0)_{60}). The original system often used blank spaces rather than a consistently written zero, making it difficult to distinguish 60, 3600, and other numbers. A later placeholder symbol reduced some of this ambiguity, but the system still had limitations.

 

9. Mayan number system

The Mayan system used dots for 1, bars for 5, and a shell-like placeholder for zero. The place values in the exercise are 1, 20, 360, and then higher values.

 

Question 41. Represent 77 in the Mayan system.

[ 77=3\times20+17. ]

 

Use three 20s in the upper level and 17 in the units level. The 17 is represented by three bars (15) and two dots (2).

 

Question 42. Represent 100 in the Mayan system.

[ 100=5\times20+0. ]

 

Use five 20s in the upper level and the shell-like zero symbol in the units level.

 

Question 43. Represent 361 in the Mayan system.

[ 361=1\times360+0\times20+1. ]

 

Use one unit of the 360-level, a zero placeholder in the 20-level, and one dot in the units level.

 

Question 44. Represent 721 in the Mayan system.

[ 721=2\times360+0\times20+1. ]

 

Use two units of the 360-level, a zero placeholder in the 20-level, and one dot in the units level.

 

10. Chinese rod numerals and the Hindu number system

Question 45. Why did Chinese rod numerals alternate between Zong and Heng symbols?

The symbols alternated orientations so that adjacent place values could be distinguished clearly. Zong symbols represented units, hundreds, ten-thousands, and so on, while Heng symbols represented tens, thousands, hundred-thousands, and so on. Alternating the orientations prevented neighbouring positions from being confused.

 

Question 46. If only Zong symbols were used, how might 41 be represented and misread?

The four tens would be followed by one unit symbol. Without clear spacing or alternating orientations, the same sequence could be regrouped in several ways and might be interpreted as 23, 32, 122, or another value depending on where the place boundaries were assumed. This illustrates why positional systems need clear place markers, spacing, or a zero placeholder.

 

Question 47. Form a base-2 place-value system using ukasar and urapon as digits. Compare it with the Gumulgal system.

Assign urapon to 0 and ukasar to 1. The base-2 representations are:

 

Decimal	Base-2 form	Using urapon = 0 and ukasar = 1
1	1	ukasar
2	10	ukasar urapon
3	11	ukasar ukasar
4	100	ukasar urapon urapon
5	101	ukasar urapon ukasar
6	110	ukasar ukasar urapon
7	111	ukasar ukasar ukasar
8	1000	ukasar urapon urapon urapon

Both systems use two basic ideas, but their structures differ. The Gumulgal system uses repeated additive groups of 2 and 1, whereas the base-2 system uses place value with powers of 2. A base-2 system therefore represents arbitrarily large numbers compactly and supports systematic arithmetic.

 

Question 48. Where are Hindu numerals and zero used in daily life and professions?

They are used in money, prices, banking, accounting, calendars, addresses, telephone numbers, measurements, science, engineering, medicine, computing, education, transport, and commerce. Without zero and a place-value system, large-number notation and modern arithmetic would be far more complicated, and the development of science, technology, and digital computing would have been severely hindered.

 

Question 49. What would happen if humans had eight fingers instead of ten?

A base-8 system would be natural. Its digits would be 0 through 7, and the place values would be powers of 8:

 

[ 1,8,64,512,4096,\ldots ]

 

Digits 8 and 9 would not be needed as single symbols.

 

Question 50. Write decimal 25 in base 8, base 5, and base 2.

For base 8:

 

[ 25=3\times8+1=31_8. ]

 

For base 5:

 

[ 25=1\times5^2+0\times5+0=100_5. ]

 

For base 2:

 

[ 25=16+8+1=1\times2^4+1\times2^3+0\times2^2+0\times2+1=11001_2. ]

 

Therefore,

 

[ \boxed{25_{10}=31_8=100_5=11001_2}. ]

 

Key concepts and formulas

Concept	Rule
Base-(n) landmark numbers	(n^0,n^1,n^2,n^3,\ldots)
Place-value expansion	(a_k n^k+a_{k-1}n^{k-1}+\cdots+a_1n+a_0)
Base-5 carrying	Five units of one place become one unit of the next place
Base-10 carrying	Ten units of one place become one unit of the next place
Base-60 expansion	(a_2\times60^2+a_1\times60+a_0)
Base-2 digits	Only 0 and 1
Base-8 digits	0 through 7