What are the irrational numbers between 2 and 3
Answers
Step-by-step explanation:
Hence √7, 3√17, 4√54 and 5√178 are all irrational numbers between 2 and 3, as 4<7<9; 8<17<27; 16<54<81 and 32<178<243.
Answer:
Irrational numbers are numbers that can not be represented in the form of pq
. The square roots and cube roots of most numbers are irrational. Use this to find the irrational numbers between 2 and 3.
Complete step-by-step answer:
Irrational numbers and rational numbers belong to the bigger domain of real numbers.
Rational numbers are numbers that can be represented in the form of pq
where q≠0
.
Irrational numbers are numbers that can not be represented in the form of pq
. Irrational numbers are non-recurring and non-terminating decimal numbers. The square roots of non-perfect square numbers are all irrational and the cube roots of non-perfect cubes are also irrational and so on.
Few examples of irrational numbers are π
, 2–√
, 7–√3
and so on.
In this question, we need to find at least two irrational numbers between 2 and 3. We use the fact that the square root of non-perfect squares is irrational to solve it.
We know that the value of the square of 2 is 4 and the value of the square of 3 is 9.
We choose two numbers between 4 and 9, let's say 6 and 7. Then, we have the following:
4<6<9
4<7<9
Taking square root on all the three terms in each inequality, we get:
4–√<6–√<9–√
4–√<7–√<9–√
Simplifying the above expressions, we get:
2<6–√<3
2<7–√<3
Hence, the two irrational numbers between 2 and 3 are 6–√
and 7–√
.
Note: There are infinite irrational numbers between two rational numbers. Hence, you can obtain as many irrational numbers as you want between the numbers 2 and 3. The answer is just one among them.
So any five irrational numbers between 2√5 and 3√3 are : √21, √22, √23, √24, and √26.