A Story of Numbers Class 8 Worksheet with Answers Maths Chapter 3

Complete the Ganita Prakash Class 8 Worksheet and NCERT Class 8 Maths Chapter 3 A Story of Numbers Worksheet with Answers before your unit tests for better preparation.

A Story of Numbers Worksheet Class 8

Class 8 Maths A Story of Numbers Worksheet

A Story of Numbers Class 8 Ganita Prakash Worksheet

Bunty’s Curiosity

At the community library, Bunty and Seema were browsing through old scrolls.
Bunty picked one and gasped,
“Seema , look! These marks look like secret codes.”
Seema tilted his head. “Maybe they’re drawings?”
The librarian laughed sofftly. “Not drawings, children. These are ancient numbers. Long ago, people used symbols like these to count animals, food, or even the days of the moon.”

Bunty’s eyes widened. “So people always needed numbers?” “Yes,” said the librarian, “but they didn’t always look like the numbers we write today.”

Question 1.
If you were living in the Stone Age, how would you count your things?
Answer:
Count with your fingers, scratches on stones, etc.

Question 2.
How are the numbers we use today different from those Bunty and Seema saw in the library?
Answer:
These numbers are symbolic, place-based and abstract

Question 3.
Could people manage life without numbers? Give one reason.
Answer:
No.

Question 4.
Answer:
Food or resources, tools or weapons

The Mechanism Of Counting

At the Sunday fair, Raunak and Ruhani volunteered at the community cycle stand. Each cycle had a hook-tag on a board. When a cycle went out, they moved its hook-tag to the OUT rail and dropped one token into ajar. When it returned, they shifted the tag back to IN and removed one token.

Raunak said, “Now the number of tokens in the jar always equals to the cycles currently out.” Ruhani added, “This is like pairing each cycle with one token-if every cycle has exactly one token, our counting can’t go wrong.”

Question 5.
What error could happen if a cycle returns but Raunak and Ruhani forget to remove a token.
Answer:
It will give an incorrect count

Question 6.
Design a method to count cycles without objects, using only two written symbols on a whiteboard Ce.g., I and ). Explain how you will add and subtract when cycles go out/return, so the method works for any large number.
Answer:
Do it yourself 7. Do it yourself

Question 7.
Create-your-own system by picking 5 simple doodle symbols  Decide which stands for “one”, and show how you will write 1-10 using only these doodle.
Answer:

Question 8.
List three places outside school where one-to-one mapping helps people to keep counting.
Answer:
Parking lots, supermarkets or stores, events or festivalsc

Some Early Number Systems

On a museum visit, Anika and Rehan entered a gallery titled “How people counted, in ancient times,” At one station, a guide traced a path along an outline of a human body. “Some communities count along body parts in a fixed order. The body becomes the standard sequence,” she explained.

Another section showed photos of ancient bones with carved lines. “These tally marks are among the -“ earliest recordings of quantity,” the label read.

They moved on to an audio booth that chanted numbers formed by twos: “2, 2 + 1,2 + 2…” The guide explained, the numbers were counted in 2s, using which the number names were formed: 3 = 2 + 1,4 = 2 + 2, 5 = 2 + 2 + 1,6 = 2 + 2 + 2. This is called Gumulgal number system and in this system they called any number greater than 6 ras.

Near the exit, a sign read Room XL. Rehan asked, “Is that forty?” The guide smiled: “Roman numerals use I, V, X, L, C, D, M with additive and subtractive patterns to represent any number, like IV = 4, IX = 9, XL = 40. They were fine for recording, but not so great for heavy calculations.

Question 9.
Count the number of objects in each of the following boxes and write their numbers using tally marks.

(a) Number of eggs = _____
(b) Number of pencils = _____
(c) Number of pets = _____
(d) Number of mangoes = _____
(e) Number of flowers in the vase = _____
(f) Number of people in the queue = _____
Answer:

Question 10.
How do tally marks record quantity without modern digits? Give one advantage.
Answer:
Do it yourself

Question 11.
Write the following Roman numbers in Hindu-Arabic system.
(a) CCXXVII = _____
(b) CDXLIV = _____
(c) MCCXLIX = _____
(d) MMCXXXVII = _____
Answer:
(a) 227
(b) 444
(c) 2137

Question 12.
Write the following Hindu Arabic numbers in Roman Numerals.
(a) 1999 = _____
(b) 3600 = _____
(c) 2075 = _____
(d) 2367 = _____
Answer:
(a) MCMXCIX
(b) MMMDC
(c) MMLXXV
(d) MMCCCIXVII

Question 13.
Evaluate the following without converting them into Hindu numerals.
(a) CCLIXVIII + LXXXVII = _____
(b) MCLXXXVI + DCXLV = _____
(c) X × II = _____
(d) IX × V = _____
Answer:
(a) CCCLV (355)
(b) MDCCXXXI (1831)
(c) XX (20)
(d) XLV (45)

Question 14.
Consider the extension of the Gumulgal number system beyond 6 in the same way of counting by 2s. Come up with ways of performing the different arithmetic operations (+, -, ×, ÷) for numbers occurring in this system, without using the Hindu numerals. Use this to evaluate the following:

(a) (ukasar-ukasar-ukasar-urapon) + (ukasar-ukasar-ukasar-ukasar-urapon)
(b) (ukasar-ukasar-ukasar-urapon) – (ukasar-ukasar-urapan)
(c) (ukasar-ukasar-ukasar-urapon) x (ukasar-ukasar-ukasar)
(d) (ukasar-ukasar-ukasar-ukasar-ukasar-ukasar-ukasar-urapan) + (ukasar-ukasar-urapan)
Answer:
(a) ukasar-ukasar-ukasar-ukasar-ukasar-ukasar-tikasar-ukasar.
(b) ukasar
(c) ukasar-ukasar-ukasar-ukasar-ukasar-ukasar-ukasar-ukasar-ukasar kasar-ukasar-ukasa r-u kasar-ukasar-ukasa r-u kasa r-u kasar-u kasar-ukasar-ukasar-ukasar
(d) ukasar-urapon

The Idea Of Base

The Egyptian number system
Raksha and MoKit were spending a quiet evening with their grandfather, whom they lovingly call Dadaji. He is a professor who loves weaving history into mathematics.

RaksKa: “Shall we learn about ancient Egypt? The Egyptians were the first one to use a base-10 system called Egyptian number system. In the system their landmark numbers were powers of 10, and they had a distinct symbol for each power.”

Moklt: “So every time the number becomes ten times bigger than previous in this system.

Dadaji: Yes. Each landmark is ten times the previous. For example, 1, 10, 102, 103, and so on. And the symbol they used in this system changes to the next one?”

He drew a neat chart and paused so the children can see it.

Question 15.
Represent the following numbers in the Egyptian number system.:
(a) 5967
(b) 1905
(c) 3000
(d) 2948
(e) 46085
(f) 8080808
Answer:

Question 16.
What numbers do these numerals stands for?

Answer:
(a) 10110211
(b) 100321
(c) 22
(d) 1101011
(e) 11000301

Dadaji: We saw how Egyptians grouped ten collections to jump to the next landmark. Now, what if we grouped Jive collections each time instead of ten? Would that give us a new number system?

Mohlt: So, we start with 1, then group five 1s to get the next landmark number?

Raksha: So, every new landmark number is five times the previous one.

Dadadji: Exactly, lets decide the symbols

Question 17.
Represent these numbers in base 5 number system.
(a) 367
(b) 120
(c) 9025
Answer:

Question 18.
Add the following Egyptian numerals.

Answer:

Question 19.

Answer:
(a) 10⁶
(b) 10⁴
(c) 10⁹
(d) 10⁹
(e) 10¹⁰
(f) 10⁴

Question 20.

Answer:

Abacus

Abacus uses the decimal system. Each horizontal line stands for a place: ones, tens, hundreds, thousands. Put as many counters on a line as the digit in that place. When a line reaches 10 counters, trade them for 1 counter on the next higher line. Zero means no counter on that line.

Question 21.
Show 5,072 on the abacus. How many counters are on the 1000, 100, 10, and 1 lines?
Answer:
Do it yourself (1000 line = 5 counters, 100 line = O counter, 10 line = 7 counters, 1 line = 2 counters

Question 22.
The abacus has: 1000 line = 3 counters, 100 line = 6 counters, 10 line = 2 counters, 1 line = 9 counters. Write the number.
Answer:
3629

Question 23.
Start with 428 on the abacus. Add 7 tens. After addition, how many counters are on each line and what is the new number?
Answer:
100 lines → 4 counters, 10 lines →9 counters, 1 line 7 → 8 counters New Number → 498

Question 24.
Start with 1,390. Add 240. Do the addition and give the final counters on each line and show the result.
Answer:
Sum → 1630; 1000 line → 1 counter, 100 line → 6 counters, 10 line → 3 counters, 1 lines → 0 counters

The mesopotamian number system

Mesopotamians were first to develop a written number system (base-60 system), used for trade and astronomy. This number system is also called Sexagesimal system or Babyloynian number system.

Here, the symbols for their landmark numbers are as follows:

This system used the symbol  for 1 and  for 10. In this system, using and , numbers 1 to 59 can be represented as

Question 25.
Represent the following numbers in the Mesopotamian system,
(a) 1000
(b) 3600
(c) 5790
Answer:

The Mayan Number System

The Mayan number system used a vigesimal (base-20) positional system written with just three symbols a dot for 1, a horizontal bar for 5, and a shell for zero.

They also had a true zero symbol (the shell), used both as a number and as a placeholder-an independent development in the Americas, likely by the 4th-5th Century.

In this system, in order to write a number, the horizontal bar (—) represents the quantity 5, the dot (•) represents the quantity 1, and the special symbol seashell represents zero.

Question 26.
Write these numbers in Mayan Number system.
(a) 108
(b) 156
(c) 2000
(d) 2006
Answer:

Question 27.
Convert these Mayan numerals to Hindu Arable system.

Answer:
(a) 1442
(b) 6986
(c) 1809

The Chinese Number System

The Chinese used two number systems — a written system for recording quantities, and a system making use of rods for performing computations. The numerals in the rod-based number system are called rod numerals. It was a decimal system (base-10). The symbols for 1 to 9 were as follows:

On a counting board, digits 1-9 were shown with little rods; to avoid confusion the shapes alternated orientation (vertical in ones, horizontal in tens, vertical in hundreds, …)

Question 28.
Convert the fallowing numbers into Chinese Number System.
(a) 4⁷
(b) 248
(c) 2438
Answer:

The Hindu Number System

Hindu-Arabic numerals Is a base-10 place-value system using the ten symbols 0-9; a numeral like 375 Is read as 3 × 10² + 7 × 10 + 5.

Question 29.
Expand 52603 as a sum of powers of 10. (Show each digit’s place).
Answer:
5 x 10⁴ + 2 × 10³ + 6 5 10² + 3 × 10¹;
3 → ones, 0, → Tens, 6 → Hundreds, 2 → Thousands,5 → ten Thousands

Question 30.
Convert XCIV and LXXV to Hindu-Arabic and add them in columns. Why is this column method is natural in a base-10 place-value system?
Answer:
XCIV → 94, LXXV → 75, Sum → 169

Worksheet On A Story of Numbers Class 8

A. Choose the correct option.

Question 1.
Which feature most clearly distinguishes a place value system from earlier systems?
(a) Use of body parts for counting.
(b) Use of landmark numbers only.
(c) Use of position to denote powers and a placeholder for empty places.
(d) Use of tally marks on bones.
Answer:
(c) Use of position to denote powers and a placeholder for empty places.

Question 2.
In a base-n system, the landmark numbers are:
(a) Multiples of n
(b) Powers of n
(c) Factors of n
(d) Prime numbers less than
Answer:
(b) Powers of n

Question 3.
The Mesopotamian system (in its developed form) is best described as:
(a) Base-10, non-positional
(b) Base-60, positional
(c) Base-5, positional
(d) Base-20, non-positional
Answer:
(b) Base-60, positional

Question 4.
A In the Roman system, XL usually represents:
(a) 14
(b) 40
(c) 400
Answer:
(b) 40

Question 5.
Which system is non-positional?
(a) Egyptian
(c) Hindu-Arabic
Answer:
(a) Egyptian

Question 6.
What is the Egyptian numeral for 23?

Answer:

Question 7.
What does the Egyptian symbol represent?
(a) 1
(b) 10
(c) 100
(d) 1000
Answer:
(d) 1000

Question 8.
Which of the following is NOT a symbol in the Egyptian number system?
(a) A heel bone for 1
(b) A coiled rope for 100
(c) A lotus flower for 1000
(d) A pointing finger for 10000
Answer:
(a) A heel bone for 1

Directions (For Q.9 to 11): In the following questions, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct option as:
(a) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
(b) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).
(c) Assertion (A) is true but Reason (R) is false.
(d) Assertion (A) is false but Reason (R) is true.

Question 9.
Assertion (A): The developed Mesopotamian system is a place value system.
Reason (R): The position of a group shows whether it counts 1s, 60s, 3600s, etc., and a placeholder symbol was later used for blanks.
Answer:
(a) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

Question 10.
Assertion (A): The Hindu number system enables unambiguous writing of all numbers using finitely many symbols.
Reason (R): It uses digits 0-9 with place value and treats 0 as a positional digit and as a number.
Answer:
(a) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

Question 11.
Assertion (A): In the Egyptian system, ten of a lower landmark regroup into the next higher landmark.
Reason (R): The Egyptian system is a base-10 system with landmark numbers 1,10, 100, …
Answer:
(d) Assertion (A) is false but Reason (R) is true.

B. Fill in the blanks.

1. The symbols used to write numbers in a written number system are called ______ .
Answer:
Digit or numerals

2 In a base-2 system, each landmark number is ______ times the previous one.
Answer:
Two

3. The Egyptian number system groups numbers by powers of ______.
Answer:
10

4. The Mayans used a seashell-like symbol as a placeholder for ______.
Answer:
zero

5. 144 can be expressed into base-5 system as ______.
Answer:

C. State whether the following statements are true (T) or false (F).

1. The Egyptian system is a base-10 system and also a full place value system.
Answer:
false

2. Early Mesopotamian numerals lacked a placeholder symbol; later a special symbol was introduced mainly for internal blanks.
Answer:
True

3. Chinese rod numerals formed a decimal system and used blank spaces to skip a place value.
Answer:
True

4. In a base-n system, the product of two landmark numbers is again a landmark number.
Answer:
True

5. In the Hindu number system, 0 is treated as a number with arithmetic properties as well as a positional digit.
Answer:
True

D. Match the following.

Question 1.

Answer:

E. Solve the following.

Question 1.
Write the following numbers in Roman numerals: (i) 1222 (ii) 2999 (iii) 302 (iv) 715
Answer:
(i) MCCXXII
(ii) MMCMXCIX
(iii) CCCII
(iv) DCCXV

Question 2.
Write the number 2310 in the Babylonian (base-60) system using place values.
Answer:

Question 3.
A Babylonian merchant sells jars of dates. If represents 60 and  represents 10, how many jars did he sell? (Assume a base-60 system where the Left-r symbol is the most significant.
Answer:
210

Question 4.
What numbers do these numerals stands for.

Answer:
(a) 210025
(b) 32000242
(c) 2000022

Question 5.
A Mayan astronomer observes a celestial event. He records the time as (using • for 1 and – for 5, and placing them vertically with the most significant level at the top). What is the time in our modern decimal system?
Answer:
12:00