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Solutions for Mathematics, Class 9, CBSE
(i) True
Reason — Whole numbers are = 0, 1, 2, 3, 4, 5,.......∞
Natural numbers are = 1, 2, 3, 4, 5,......∞
From above we see all natural numbers are whole numbers.
(ii) False
Reason — Integers can be positive as well as negative i.e., integers are = 0, +(-1), +(-2), +(-3), +(-4)......∞ whereas whole number are 0 or positive only i.e., whole numbers are = 0, 1, 2, 3, 4,.......∞
(iii) False,
Reason — Rational numbers are = , , , ,.....
Whole numbers are = 0, 1, 2, 3, 4,.......∞
(i) True
Reason — Every irrational number is a real number, because rational number, natural number, whole number all the numbers comes into real number.
(ii) False
Reason — Every point on the number line represents a unique real number.
(iii) False
Reason — For example is real but not irrational.
Representing as the sum of squares of two natural numbers:
5 = 4 + 1 = (2)2 + (1)2
Let l be the number line. If point O represents number 0 and point A represents number 2 then draw a line segment OA = 2 units.
At A, draw AC ⊥ OA. From AC, cut off AB = 1 unit.
We observe that OAB is a right angled triangle at A. By Pythagoras theorem, we get:
(OB)2 = (OA)2 + (AB)2
(OB)2 = (2)2 + (1)2
(OB)2 = 4 + 1
(OB)2 = 5
OB = units
With O as centre and radius = OB, we draw an arc of a circle to meet the number line l at point P.
As, OP = OB = units, the point P will represent the number on the number line as shown in the figure below:
(i) = 0.36, terminating decimal expansion
(ii)
The remainder 1 keeps repeating.
Hence, = . This is a non-terminating repeating decimal expansion
(iii)
Hence, = 4.125. This is a terminating decimal expansion.
(iv)
Hence, = . This is a non-terminating repeating decimal expansion.
(v)
Hence, . This is a non-terminating repeating decimal expansion.
(vi) = 0.8225, terminating decimal expansion
Hence, = 0.8225. This is a terminating decimal expansion.
(i)
Let x =
x = 0.666666........... (1)
Here one digit 6 is repeated so we multiply both side by 10 in equation (1)
10x = 6.6666.......
we can write
10x = 6.66666.......
10x = 6 + 0.666666...........
From equation (1)
10x = 6 + x
10x - x = 6
9x = 6
x = =
Hence, =
(ii)
Let x = 0.47777777........ (1)
Here one digit 7 is repeated so we multiply both side by 10 in equation (1)
We can write
10x = 4.77777..... (2)
On subtracting equation (2) from equation (1)
10x - x = 4.777777........ - 0.477777.........
9x = 4.300000......
x =
Multiplying numerator and denominator by 10 we get,
x = =
Hence, =
(iii)
Let x = 0.001001001...... (1)
Here three digit 001 are repeating so we multiply both side by 1000 in equation (1)
1000x = 0001.001001........ (2)
On subtracting equation (2) from equation (1)
1000x - x = 0001.001001...... - 0.001001......
999x = 1.0000000.....
x =
Hence, =
Let x = 0.99999 ........ (1)
Here one digit 9 is repeating so we multiply both side by 10 in equation (1)
10 x = 9.99999.......... (2)
By subtracting equation (2) - equation (1)
10 x -x = 9.99999........ - 0.99999........
9x = 9
x =
x = 1
Therefore, (0.99999...) is equivalent to (1), which might seem surprising at first glance, but it makes sense mathematically. This result can be demonstrated by the fact that any number infinitesimally close to 1 but less than 1, when infinitely added to itself, equals 1.
For example, if we take (0.9) and add (0.09), we get (0.99), and if we add (0.009) to that, we get (0.999), and so on. As we keep adding these infinitesimal increments, we approach (1). So, in a way, (0.99999...) is just another way to represent the number (1).
Suppose, = = 0.4
or
= 0.9
or
= 1.5
q must contain factors of 2 or 5 or both.
For example,
In , 8 = 2 x 2 x 2
and
in , 10 = 2 x 5
(i) is a prime number. It is irrational because does not have a complete root.
(ii) is not a prime number. 15 is root of so it is rational.
(iii) 0.3796 is terminating decimal expansion. Hence, it is rational.
(iv) 7.478478... = , it is non-terminating but repeating decimal expansion. Hence, it is rational.
(v) 1.101001000100001... is non-terminating and non-repeating decimal expansion. Hence, it is irrational.
(i) 2 −
If in any rational number we add, subtract, multiply or divide any irrational number it becomes irrational number.
Hence, 2 − ( is an irrational number.
(ii) 3 + −
= 3 + −
= 3
Hence, (3 + ) − is a rational number
(iii)
=
It is in where q ≠ 0.
Hence, is a rational number
(iv)
= irrational number
If in any rational number we add, subtract, multiply or divide any irrational number it becomes irrational
Hence, is an irrational number
(v) 2π
2 x π = irrational number [∵ rational x irrational number = irrational number]
Hence, 2π is an irrational number
In the question given π is irrational, π = There is no contradiction. When we measure a length with a scale or any other device we only get an approximate rational value. So, we may not realize that either c or d is irrational.
Steps:
Hence, point E represents on the number line.
(i)
Multiply by in numerator and denominator
= x
=
Hence, =
(ii)
Multiply by in numerator and denominator
= x
=
=
=
=
=Hence, =
(iii)
Multiply by in numerator and denominator
= x
=
=
=
=Hence, =
(iv)
Multiply by in numerator and denominator
= X
=
=
=
= Hence, =