which of the following are irrational number root 2. Pi. 14.879924752494242. 3.625
Answers
Answer:
√2 and π are irrational numbers.
14.879924752494242 and 3.625 are rational numbers
Step-by-step explanation:
Prove that √2 is irrational.
Assume $\sqrt{2}$ is rational, i.e. it can be expressed as a rational fraction of the form $\frac{b}{a}$, where $a$ and $b$ are two relatively prime integers. Now, since $\sqrt{2}=\frac{b}{a}$, we have $2=\frac{b^2}{a^2}$, or $b^2=2a^2$. Since $2a^2$ is even, $b^2$ must be even, and since $b^2$ is even, so is $b$. Let $b=2c$. We have $4c^2=2a^2$ and thus $a^2=2c^2$. Since $2c^2$ is even, $a^2$ is even, and since $a^2$ is even, so is a. However, two even numbers cannot be relatively prime, so $\sqrt{2}$ cannot be expressed as a rational fraction;
hence √2 is irrational.
Prove that π is irrational
1. Assume π is rational, π = a/b for a and b relatively prime.
2. Create a function f(x) that depends on constants a and b
3. After much work, prove that integral of f(x) sin(x) evaluated from 0 to π must be an integer, if π is rational.
4. Simultaneously show that integral of f(x) sin(x) evaluated from 0 to π will be positive but tend to 0 as the value of n gets arbitrarily large. This is the required contradiction: if the integral evaluates to an integer, it cannot also be equal to a value between 0 and 1.
5. Conclude π is irrational.
prove that 14.879924752494242 is rational
It can be written in the form of p/q where q≠0 so it is rational.
Example : 14.879924752494242/1
14.879924752494242/2
14.879924752494242/3
And so on....
prove that 3.625 is rational
Similarly
It can Also be written in the form of p/q where q≠0 so it is rational.
Example : 3.625/1,3.625/2,3.625/3 and so on....
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