Representation of Irrational Numbers on Number Line
Learn to represent irrational numbers on the number line using geometric constructions, identify irrational numbers within given ranges, compare irrational values, and understand the properties of rational and irrational numbers.
Questions
Which of the following numbers lie between $1$ and $3$?
- $\dfrac13$
- $\sqrt{2}$
- $\sqrt{10}$
- $\dfrac{8}3$
Between any $2$ real numbers, __________ can always be represented on a number line.
- an integer
- an irrational number
- a natural number
- a rational number
Following are the steps to represent $\sqrt5$ on number line.
Arrange them in order.
1) Draw OC on line with $l(OC)=l(OB)$,
2) Draw $AB \perp OA\ and\ l(AB) =1$
3) Take $l(OA)=2$
4) $l(OC)=\sqrt5$, C is required point on real line.
- $1,2,3,4$
- $2,4,1,3$
- $3,2,4,1$
- $3,2,1,4$
Which of the following irrational numbers lie between $6$ and $8$?
- $\sqrt{49}$
- $\sqrt{19}$
- $\sqrt{47}$
- $\sqrt{62}$
The number $\sqrt{10}$ lies between $2$ integers $a$ and $b$ such that $b-a = 1$. Then $b+a = , ?$
- $4$
- $5$
- $7$
- None of these
Which one of the following is not true?
- $\sqrt{2}$ is an irrational number
- $\sqrt{17}$ is a irrational number
- $0.10110011100011110...$ is an irrational number
- $\sqrt[4]{16}$ is an irrational number
The greater number between $\sqrt{17}-\sqrt{12}$ and $\sqrt{11}-\sqrt{6}$ is ____.
- $\sqrt{17}-\sqrt{12}$
- $\sqrt{11}-\sqrt{6}$
- Both are equal
- Cannot comare
Which of the following is/are correct?
- Every integer is a rational number.
- The sum of a rational number and an irrational number is an irrational number.
- Every real number is rational.
- Every point on the number line is associated with a real number
To represent a rational number $\sqrt{2}$ on number line, take sides of right triangle as:
- $1$ and $1$
- $1$ and $2$
- $2$ and $0$
- $-1$ and $-1$
Use ______________ to represent an irrational number on number line.
- Isosceles-angle theorem
- Scalene angle theorem
- Right-angled theorem
- None of the above
$D$ is a real number with non terminating digits $a _1$ and $a _2$ after the decimal point. Let $D = 0, a _1 a _2 a _1 a _2 ........ $ with $a _1 & a _2$ both not zero which of the following when multiplied by $D$ will necessarily give an integer ?
- $99$
- $18$
- $125$
- $75$
The number $5\sqrt{34}$ lies between
- $29$ and $30$
- $30$ and $31$
- $31$ and $32$
- $32$ and $33$
Can $\sqrt { 3 } -3$ be represented on the number line.
- True
- False
Give an example of two irrational numbers whose difference is an irrational number.
- $\sqrt{3},-\sqrt{3}$
- $\sqrt{5,}-\sqrt{5}$
- $4\sqrt{3},-2\sqrt{3}$
- None of the above
Which is the wrong step that shows $\displaystyle 5-\sqrt{3}$ is irrational?
(I) Contradiction : Assume that $\displaystyle 5-\sqrt{3}$ is rational
(II) Find coprime a & b $\displaystyle \left ( b\neq 0 \right )$ such that $\displaystyle 5-\sqrt{3}=\frac{a}{b},\therefore 5-\frac{a}{b}=\sqrt{3}$
Rearranging above equation $\displaystyle \sqrt{3}=5-\frac{a}{b}=\frac{5b-a}{b}$
(III) Since a & b are integers we get $\displaystyle 5-\frac{a}{b}$ is irrational and so $\displaystyle \sqrt{3}$ is irrational
(IV) But this contradicts the fact that $\displaystyle \sqrt{3}$ is irrational Hence $\displaystyle 5-\sqrt{3}$ is irrational
- Both I and II
- Only III
- Only II
- Both II and III
Which of the following irrational numbers lie between $4$ and $7$?
- $\sqrt{25}$
- $\sqrt{19}$
- $\sqrt{47}$
- $\sqrt{50}$
The ascending order of the surds $\sqrt[3]{2}, \sqrt[6]{3}, \sqrt[9]{4}$ is
- $\sqrt[9]{4}, \sqrt[6]{3}, \sqrt[3]{2}$
- $\sqrt[9]{4}, \sqrt[3]{2}, \sqrt[6]{3}$
- $\sqrt[3]{2}, \sqrt[6]{3}, \sqrt[9]{4}$
- $\sqrt[6]{3}, \sqrt[9]{4}, \sqrt[3]{2}$