NCERT Solutions Class 8 Maths Ganita Prakash Chapter 3 A Story of Numbers on Padhaiwise

NCERT Solutions for Class 8 Maths Ganita Prakash Chapter 3 A Story of Numbers Figure it Out Page 54

Mon, 01 Jan 0001 00:00:00 +0000

NCERT Solutions for Class 8 Maths Ganita Prakash Chapter 3 — Figure it Out (Page 54)#

In Method 1 a number is a collection of sticks, one stick for each object. In Method 2 a number is a letter: a for 1, b for 2, …, z for 26. These questions ask how far each method can go.

1. Suppose you are using the number system that uses sticks to represent numbers, as in Method 1. Without using either the number names or the numerals of the Hindu number system, give a method for adding, subtracting, multiplying and dividing two numbers or two collections of sticks. #

Solution

Take the two collections of sticks, A and B.

NCERT Solutions for Class 8 Maths Ganita Prakash Chapter 3 A Story of Numbers Figure it Out Page 59

Mon, 01 Jan 0001 00:00:00 +0000

NCERT Solutions for Class 8 Maths Ganita Prakash Chapter 3 — Figure it Out (Page 59)#

The Roman landmark numbers are:

I	V	X	L	C	D	M
1	5	10	50	100	500	1000

A number is split into landmark numbers, taking as many of the largest as possible first. A smaller symbol written before a larger one is subtracted: IV = 4, IX = 9, XL = 40, XC = 90, CD = 400, CM = 900.

NCERT Solutions for Class 8 Maths Ganita Prakash Chapter 3 A Story of Numbers Figure it Out Page 60

Mon, 01 Jan 0001 00:00:00 +0000

NCERT Solutions for Class 8 Maths Ganita Prakash Chapter 3 — Figure it Out (Pages 60–61)#

The Gumulgal people of Australia counted in 2s. They had two words, urapon for 1 and ukasar for 2, and joined them to make bigger numbers: ukasar-urapon is 3, ukasar-ukasar is 4, and so on.

1. A group of indigenous people in a Pacific island use different sequences of number names to count different objects. Why do you think they do this? #

Solution

Different objects were probably counted in different ways and for different purposes. Some possible reasons:

NCERT Solutions for Class 8 Maths Ganita Prakash Chapter 3 A Story of Numbers Figure it Out Page 62

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NCERT Solutions for Class 8 Maths Ganita Prakash Chapter 3 — Figure it Out (Page 62)#

The Egyptian landmark numbers are the powers of 10, each with its own hieroglyph:

Value	1	10	100	1000	10,000	100,000	1,000,000
Symbol	stroke	heel bone (∩)	coil of rope	lotus flower	finger	tadpole	kneeling man

A number is written by repeating each symbol as many times as that power of 10 occurs in it. Below, each answer lists how many of each symbol to draw.

NCERT Solutions for Class 8 Maths Ganita Prakash Chapter 3 A Story of Numbers Figure it Out Page 63

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NCERT Solutions for Class 8 Maths Ganita Prakash Chapter 3 — Figure it Out (Page 63)#

In the base-5 system made in the chapter, each landmark number is 5 times the one before. The symbols (Table 2 of the book) are:

Value	$5^0 = 1$	$5^1 = 5$	$5^2 = 25$	$5^3 = 125$	$5^4 = 625$	$5^5 = 3125$
Symbol					(wave)

A number is grouped starting from the largest landmark number that fits, just as $143 = 125 + 5 + 5 + 5 + 1 + 1 + 1$ is written .

NCERT Solutions for Class 8 Maths Ganita Prakash Chapter 3 A Story of Numbers Figure it Out Page 65

Mon, 01 Jan 0001 00:00:00 +0000

NCERT Solutions for Class 8 Maths Ganita Prakash Chapter 3 — Figure it Out (Page 65)#

To add numerals in a system with a base, we put together the symbols of each kind, then regroup: in the Egyptian system 10 of a symbol make one of the next, and in our base-5 system 5 of a symbol make one of the next. This is the same as carrying in column addition.

NCERT Solutions for Class 8 Maths Ganita Prakash Chapter 3 A Story of Numbers Figure it Out Page 69

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NCERT Solutions for Class 8 Maths Ganita Prakash Chapter 3 — Figure it Out (Pages 69–70)#

1. Can there be a number whose representation in Egyptian numerals has one of the symbols occurring 10 or more times? Why not? #

Solution

No. Each Egyptian landmark number is 10 times the one before it. So 10 copies of any symbol are always replaced by one copy of the next symbol: 10 strokes make a heel bone, 10 heel bones make a coil, and so on. That is why each symbol appears at most 9 times, just like a digit from 0 to 9.

NCERT Solutions for Class 8 Maths Ganita Prakash Chapter 3 A Story of Numbers Figure it Out Page 73

Mon, 01 Jan 0001 00:00:00 +0000

NCERT Solutions for Class 8 Maths Ganita Prakash Chapter 3 — Figure it Out (Page 73)#

The Mesopotamian system is a base-60 system. Its landmark numbers are $1, 60, 60^2 = 3600, \ldots$ Inside each place, a number from 1 to 59 is written with two symbols: a wedge for 1 and a chevron (‹) for 10. For example, 23 is 2 chevrons and 3 wedges.

In the book’s notation, ◇1 marks the 60s and ◇2 marks the 3600s. In the compact (place value) form, these markers are dropped and the position alone tells the place.

NCERT Solutions for Class 8 Maths Ganita Prakash Chapter 3 A Story of Numbers Figure it Out Page 80

Mon, 01 Jan 0001 00:00:00 +0000

NCERT Solutions for Class 8 Maths Ganita Prakash Chapter 3 — Figure it Out (Page 80)#

In the Chinese rod numerals, the digits 1 to 9 have two forms: upright zongs (|, ||, |||, …) and flat hengs (—, =, ≡, …). Zongs are used for units, hundreds, tens of thousands, …, and hengs for tens, thousands, hundreds of thousands, …

1. Why do you think the Chinese alternated between the Zong and Heng symbols? If only the Zong symbols were to be used, how would 41 be represented? Could this numeral be interpreted in any other way if there is no significant space between two successive positions? #

Solution

Why they alternated: the rods for neighbouring places look different, so you can see where one place ends and the next begins. Without this, the digits of a number could run into each other.