Which number is rational?
StartRoot 2 EndRoot
Pi
StartRoot 10 EndRoot
StartRoot 16 EndRoot
starRoot 16 end root
A rational number is a number that can be expressed as a fraction of two integers. Let's go through each of the given numbers and determine if they are rational or not.
1. √2 (Square Root of 2):
To check if √2 is rational or not, we can assume that √2 is rational and express it as a fraction in simplest form, like (√2 = a/b).
Squaring both sides, we get 2 = (a^2) / (b^2).
This implies that 2b^2 = a^2.
From this equation, we can conclude that 'a' must be an even number, let's say a = 2k, where k is an integer.
Substituting 'a' in the equation, we get 2b^2 = (2k)^2 = 4k^2.
Dividing both sides by 2, we obtain b^2 = 2k^2.
This means 'b' must also be an even number, which contradicts our assumption.
Hence, √2 is not a rational number.
2. π (Pi):
Pi is an irrational number, which means it cannot be expressed as a finite or repeating decimal or as a fraction of two integers. Its decimal representation is non-terminating and non-repeating.
3. √10 (Square Root of 10):
Similar to how we checked for √2, let's assume √10 is rational and express it as (√10 = a/b).
Squaring both sides, we get 10 = (a^2) / (b^2).
Given that 10 is even, its prime factorization is 2 * 5.
Since (a^2) / (b^2) must equal 10, both 'a' and 'b' need to include 2 and/or 5 in their prime factorization.
However, this contradicts the assumption that a/b is in simplest form, as there will be a common factor between 'a' and 'b'.
Hence, √10 is not rational.
4. √16 (Square Root of 16):
Again, let's assume √16 is rational and express it as (√16 = a/b).
Squaring both sides, we get 16 = (a^2) / (b^2).
This equation simplifies to 16b^2 = a^2.
Since 16 is a perfect square, we can rewrite it as 16 = 4 * 4.
With this, we can express a^2 as a = 4k, where k is an integer.
Substituting 'a' back into the equation, we get 16b^2 = (4k)^2 = 16k^2.
Dividing both sides by 16, we obtain b^2 = k^2.
Therefore, 'b' must be an integer. Thus, √16 is a rational number.
In summary, out of the given numbers, only √16 is rational.