CBSE Class 8 Mathematics A Story of Numbers Very Short Answer Questions with Answers

Very Short Answer Questions

Question 1. Give two reasons why people needed to count even as early as the Stone Age.
Answer:
Any two: to know how much food they had and how many animals were in their herd; to record trade of goods and ritual offerings; to keep track of the passing days so as to predict the new moon, the full moon or a change of season.

Teacher's Note:
1. Counting began with daily needs.
2. Food, animals, trade and time are the main reasons.

Question 2. Which group of people is mentioned in the chapter as having had all three forms of number representation, that is objects, names and written symbols?
Answer:
The Chinese, who represented numbers by physical objects, by names and by written symbols.

Teacher's Note:
1. The three forms are objects, names (sounds) and written symbols.
2. The Chinese had all three.

Question 3. How many numbers can be represented using the sounds of the 26 letters of the English alphabet in the way described in Method 2 of the chapter?
Answer:
Only 26 numbers, one for each letter. A collection of more than 26 objects cannot be counted this way.

Teacher's Note:
1. A good number system must never run out.
2. The letter method stops at 26.

Question 4. Apart from sticks and written symbols, name one other thing people have used as a standard counting sequence, and name a region where it is still used.
Answer:
Body parts (fingers and other parts of the hand and body). A group of people in Papua New Guinea still uses them.

Teacher's Note:
1. Body parts come in a fixed order, so they make a sequence.
2. This is why fingers shaped many number systems.

Question 5. What do the columns of notches on the Ishango bone possibly indicate?
Answer:
They possibly indicate a calendar (a calendrical system).

Teacher's Note:
1. Ancient tally marks often recorded days.
2. The Ishango bone is from the Democratic Republic of Congo.

Question 6. Name the two groups of people, other than the Gumulgal, who developed equivalent number systems by counting in 2s.
Answer:
The Bakairi of South America and the Bushmen of South Africa.

Teacher's Note:
1. Three far-apart groups counted in 2s the same way.
2. One theory is that they shared common ancestors.

Question 7. Up to what group size can most people see the number of objects in a collection at a single glance?
Answer:
Up to about 4 objects. A group of 5 or more is hard to judge at a glance.

Teacher's Note:
1. This is why tally marks are bunched in fives.
2. The fifth mark is often drawn across the first four.

Question 8. Which group sizes have been commonly used in the number systems of the world?
Answer:
Groups of 2, 5, 10 and 20.

Teacher's Note:
1. 5 and 10 come from fingers on one and two hands.
2. 20 comes from fingers and toes together.

Question 9. What is a landmark number? Give one example from the Roman system.
Answer:
A landmark number is a number given its own basic symbol and used as a reference point. Example: 50, written as L.

Teacher's Note:
1. Roman landmark numbers: 1, 5, 10, 50, 100, 500, 1000.
2. Any one of them is a correct example.

Question 10. What is another name for a base-10 number system?
Answer:
A decimal system.

Teacher's Note:
1. 'Deci' means ten.
2. Base 60 is called sexagesimal.

Question 11. What did the Mayans use as a placeholder symbol for zero?
Answer:
A symbol that looked like a seashell.

Teacher's Note:
1. The shell marks an empty position.
2. Mayans built place value independently of other cultures.

Question 12. How many digits does the Hindu number system use, and which are they?
Answer:
Ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9.

Teacher's Note:
1. Base 10 needs exactly ten digits.
2. 0 is one of them.

Question 13. What does the word ‘Hindu’ refer to in the name ‘Hindu numerals’?
Answer:
It refers to a geography and the people from whom these numbers came, not to a religion.

Teacher's Note:
1. The numerals came from the Indian subcontinent.
2. Correct names: Hindu, Indian or Hindu-Arabic numerals.

Question 14. By about which century had it become impossible for Europe to carry on without adopting the Indian numerals?
Answer:
By the 17th century, during the European Renaissance; not adopting them would have held back scientific progress.

Teacher's Note:
1. Science needed fast calculation.
2. Roman numerals could not keep up.

Question 15. From about which time has India had a symbol for 0?
Answer:
At least as early as 200 BCE.

Teacher's Note:
1. Zero as a symbol came long before zero as a number.
2. Brahmagupta codified 0 as a number in 628 CE.

Question 16. Out of which earlier number system, and in about which century, did the Roman system evolve?
Answer:
It seems to have evolved out of the ancient Greek number system, in about the 8th century BCE in Rome.

Teacher's Note:
1. Number systems grow from earlier ones.
2. The Roman system then kept developing over time.

Question 17. Besides XL, which other way of writing 40 was sometimes used in the Roman system, and why did this happen?
Answer:
40 was sometimes written as XXXX, because users did not always follow the practice of writing a number as one landmark number less than the next.

Teacher's Note:
1. XL means 10 less than 50.
2. The 'subtract before' rule was not always used.

Question 18. Which number does the Roman numeral MCMXLV stand for?
Answer:
MCMXLV = M + CM + XL + V = 1000 + 900 + 40 + 5 = 1945.

Teacher's Note:
1. CM is 100 less than 1000, i.e. 900.
2. Split the numeral into chunks before adding.

Question 19. Up to about which number could the Egyptian system represent numbers fairly efficiently, and what would have to be done to go beyond that?
Answer:
Up to about a crore, that is 10⁷. Beyond that, a new symbol would have to be invented for every higher power of 10.

Teacher's Note:
1. Each new power of 10 needs a new Egyptian symbol.
2. Place value removes this problem.

Question 20. Why can the number zero not be written in the base-5 system built in the chapter?
Answer:
Because that system has no symbol for zero; it has symbols only for the landmark numbers 1, 5, 25, 125, … and their repetitions.

Teacher's Note:
1. Like the Egyptian system, it is not a place value system.
2. Without a place value, there is no need for a zero symbol.

Question 21. Which calculating device did the users of Roman numerals take up in about the 11th century, and on which system was it built?
Answer:
The abacus, built on the decimal (base-10) system, each line standing for the next power of 10.

Teacher's Note:
1. Even Roman users calculated in base 10.
2. Lines stood for 1, 10, 100, 1000, …

Question 22. On that abacus, how was the number 7 shown on the line standing for 1?
Answer:
By one counter just above the line (worth 5) and two counters on the line: 5 + 2 = 7.

Teacher's Note:
1. A counter above a line is worth 5 of that line.
2. This reduces the number of counters needed.

Question 23. Before the placeholder symbol was invented, what did the Mesopotamians write in a position where a power of 60 did not occur?
Answer:
They simply left a blank space there.

Teacher's Note:
1. Blank spaces caused misreading.
2. This led to a placeholder symbol.

Question 24. Until about which century were the Chinese rod numerals in use, and which base did they use?
Answer:
Until about the 17th century; they used base 10 (decimal).

Teacher's Note:
1. Rod numerals were a place value system.
2. Zong and Heng forms alternated between places.

Question 25. Write the base-10 number 2026 in a base-7 place value system.
Answer:
Landmark numbers: 343, 49, 7, 1. 2026 = 5 × 343 + 6 × 49 + 2 × 7 + 3.
So 2026 = 5623 in base 7. Check: 1715 + 294 + 14 + 3 = 2026.

Teacher's Note:
1. Divide by the largest landmark number first.
2. Each digit must be less than 7.

Question 26. The numeral 314 is written in a base-5 place value system. Write the number it stands for in base 10.
Answer:
Places: 25, 5, 1. 314 = 3 × 25 + 1 × 5 + 4 × 1 = 75 + 5 + 4 = 84.

Teacher's Note:
1. Multiply each digit by its place value and add.
2. Place values are powers of the base.

Question 27. A number is written as 47 in a base-8 place value system. Write the same number in base 2.
Answer:
47 (base 8) = 4 × 8 + 7 = 39. 39 = 32 + 4 + 2 + 1.
So in base 2 it is 100111.

Teacher's Note:
1. Go through base 10 when changing bases.
2. Mark a 0 for every missing power of 2.

Question 28. How many different digits are needed in a base-n place value system, and which are they?
Answer:
Exactly n digits: 0, 1, 2, …, n − 1. No place can hold n or more, and 0 is needed for an empty place.

Teacher's Note:
1. Base 10 needs 0 to 9; base 2 needs only 0 and 1.
2. The largest digit is one less than the base.

Question 29. A number written in a base-n place value system ends in the digit 0. What does this tell you about the number? Illustrate your answer in base 5.
Answer:
The number is a multiple of n: the units place adds nothing and every other place is a multiple of n.
Example: 230 in base 5 = 2 × 25 + 3 × 5 + 0 = 65 = 13 × 5.

Teacher's Note:
1. Like base-10 numbers ending in 0 being multiples of 10.
2. Only the units place can make a number not divisible by n.

Question 30. Using powers, explain why the product of any two landmark numbers of a base-n system is again a landmark number.
Answer:
The landmark numbers are powers of n. If two of them are n^a and n^b, then n^a × n^b = n^a+b, again a power of n, so again a landmark number.

Teacher's Note:
1. Multiplying powers of the same base adds the exponents.
2. This is why multiplication is easy in a base system.