Accessable NCERT Class VI Mathematics text book for Blind and Visually Impaired Students.

Now, we shall try to arrange numbers in elementary shapes made up of dots. The shapes we take are
(1) a line
(2) a rectangle
(3) a square and
(4) a triangle.
Every number should be arranged in one of these shapes. No other shape is allowed.

Every number can be arranged as a line;

The number 1 is shown as ●
The number 2 is shown as ● ●
The number 3 is shown as ● ● ●
and so on.

Some numbers can be shown also as rectangles.
For example,
The number 6 can be shown as
●●●
●●●
a rectangle.

Note there are 2rows of ● ● ●
and 3 columns.

Some numbers like 4 or 9 can also be arranged as squares;
For example:
Number 4 as,
●●
●●

Number 9 as,
●●●
●●●
●●●

Which looks like squares.

Some numbers can also be arranged as triangles.
For example,
Number 3 may be represented as,
●
●●

Number 6 may be represented as,
●
●●
●●●
A triangle.

Note that the triangle should have its two sides equal. The number of dots in the rows starting from the bottom row should be like 4, 3, 2, 1. The top row should always have 1 dot.

Now, complete the table :

1 is a special number

 

Number
Line
Rectangle
Square
Triangle

2
Yes
No
No
No

3
Yes
No
No
Yes

4
Yes
Yes
Yes
No

5
Yes
No
No
No

6
 
 
 
 

7
 
 
 
 

8
 
 
 
 

9
 
 
 
 

10
 
 
 
 

11
 
 
 
 

12
 
 
 
 

13
 
 
 
 

 

Help yourself

1. Which numbers can be shown only as a line?
2. Which can be shown as squares?
3. Which can be shown as rectangles?
4. Write down the first seven numbers that can be arranged as triangles, e.g. 3, 6, ...
5. Some numbers can be shown by two rectangles, for example,
Number 12 may be represented as,
●●●●
●●●●
●●●●
3 rows with 4 dots.

Or by
●●●
●●●
●●●
●●●
4 rows with 3 dots each.

Give at least five other such examples.

Patterns Observation

Observation of patterns can guide you in simplifying processes. Study the following:
(a) 117 + 9
= 117 + 10 − 1
= 127 − 1
= 126

(b) 117 − 9
= 117 − 10 + 1
= 107 + 1 = 108

(c) 117 + 99
= 117 + 100 − 1
= 217 − 1 = 216

(d) 117 − 99
= 117 − 100 + 1
= 17 + 1 = 18

Does this pattern help you to add or subtract numbers of the form 9, 99, 999,…?

Here is one more pattern :

(a) 84 × 9
= 84 × (10 − 1)

(b) 84 × 99
= 84 × (100 − 1)

(c) 84 × 999
= 84 × (1000 − 1)

Do you find a shortcut to multiply a number by numbers of the form 9, 99, 999,…?

Such shortcuts enable you to do sums verbally.

The following pattern suggests a way of multiplying a number by 5 or 25 or 125. (You can think of extending it further).

(i) 96 × 5
= 96 × 10/2
= 960/2= 480

(ii) 96 × 25
= 96 × 100/4
= 9600/4
= 2400

(iii) 96 × 125
= 96 × 1000/8
= 96000/8
= 12000...

What does the pattern that follows suggest?

(i) 64 × 5
= 64 × 10/2
= 32 × 10
= 320 × 1

(ii) 64 × 15
= 64 × 30/2
= 32 × 30
= 320 × 3

(iii) 64 × 25
= 64 × 50/2
= 32 × 50
= 320 × 5

(iv) 64 × 35
= 64 × 70/2
= 32 × 70
= 320 × 7.......

EXERCISE 2.3

Q1. Which of the following will not represent zero:
(a) 1 + 0
(b) 0 × 0
(c) 0/2
(d) (10 − 10)/2

A1. (a)

Q2. If the product of two whole numbers is zero, can we say that one or both of them will be zero? Justify through examples.

A2. Yes

Q3. If the product of two whole numbers is 1, can we say that one or both of them will be 1? Justify through examples.

A3. Both of them will be ‘1’

Q4. Find using distributive property :
(a) 728 × 101
(b) 5437 × 1001
(c) 824 × 25
(d) 4275 × 125
(e) 504 × 35

A4.
(a) 73,528
(b) 54,42,437
(c) 20,600
(d) 5,34,375
(e) 17,640

Q5. Study the pattern :
1 × 8 + 1 = 9
12 × 8 + 2 = 98
123 × 8 + 3 = 987
1234 × 8 + 4 = 9876
12345 × 8 + 5 = 98765

Write the next two steps. Can you say how the pattern works?
(Hint: 12345 = 11111 + 1111 + 111 + 11 + 1).

A5. 123456 × 8 + 6 = 987654
1234567 × 8 + 7 = 9876543

What have we discussed?

1. The numbers 1, 2, 3, 4,... which we use for counting are known as natural numbers.

2. If we add 1 to a natural number, we get its successor. Whereas, if we subtract 1 from a natural number, we get its predecessor.

3. Every natural number has a successor. Every natural number, except 1, has a predecessor.

4. If we add the number zero to the collection of natural numbers, we get the collection of whole numbers. Thus, the numbers 0, 1, 2, 3,... form the collection of whole numbers.

5. Every whole number has a successor. Every whole number, except zero, has a predecessor.

6. All natural numbers are whole numbers, but all whole numbers are not natural numbers.

7. We take a line, mark a point on it, and label it as 0. We then mark out points to the right of 0, at equal distances. Label them as 1, 2, 3,.... Thus, we have a number line with the whole numbers represented on it. We can easily perform the number operations of addition, subtraction and multiplication on the number line.

8. Addition corresponds to moving to the right on the number line, whereas subtraction corresponds to moving to the left. Multiplication corresponds to making jumps of equal distances starting from zero.

9. Adding two whole numbers always gives a whole number. Similarly, multiplying two whole numbers always gives a whole number. We say that whole numbers are closed under addition and also under multiplication. However, whole numbers are not closed under subtraction and under division.

10. Division by zero is not defined.

11. Zero is the identity for addition of whole numbers. The whole number 1 is the identity for multiplication of whole numbers.

12. We can add two whole numbers in any order. We can multiply two whole numbers in any order. We say that addition and multiplication are commutative for whole numbers.

13. Addition and multiplication, both, are associative for whole numbers.

14. Multiplication is distributive over addition for whole numbers.

15. Commutativity, associativity and distributivity properties of whole numbers are useful in simplifying calculations and we use them without being aware of them.

16. Patterns with numbers are not only interesting, but are useful especially for verbal calculations and help us to understand properties of numbers better.

Congratulations! You have completed this chapter. I hope you enjoyed studying this chapter. In case you found any difficulties in this chapter or have any suggestions to improve it, please write to us at ‘blind2Visionary@gmail.com’.

End of Chapter 2: Whole Numbers Class 6 Mathematics for blind and visually impaired students