Proving 2√+5√ is Irrational

Study Notes

Proving 2√+5√ is Irrational

Understanding irrational numbers and their properties is a fundamental aspect of algebra and real numbers. To establish that a particular expression is irrational, we must explore its structure and the underlying definitions. In this case, we're looking at the expression 2√+5√.

Irrational Numbers

By definition, an irrational number is a real number that cannot be expressed as a fraction (rational number) with a finite or repeating decimal representation. This includes numbers like π (pi), √2, and √7, which cannot be expressed as fractions with a finite number of digits.

Proving Irrationality

To prove that 2√+5√ is irrational, we'll show that it cannot be expressed as a fraction with a finite or repeating decimal representation. We can start by expanding the expression:

2√ + 5√ = (2 + 5)√ = 7√

Now, we'll assume that 7√ is a rational number, which means it can be expressed as a fraction with a finite or repeating decimal representation. If 7√ is rational, then we can write it as:

7√ = a/b

where a and b are positive integers and b is not divisible by 7 (to avoid 7 as a common factor).

Next, we'll square both sides:

(7√)^2 = (a/b)^2

Now, expand the left side:

7^2 * √^2 = a^2/b^2

Since 7^2 = 49, we have:

49 * √ = a^2/b^2

Now, simplify the expression:

49√ = a^2/b^2

Now, square both sides again to eliminate the square root:

(49√)^2 = (a^2/b^2)^2

49^2 * √^2 = a^4/b^4

Since 49^2 = 2401, we have:

2401 * 1 = a^4/b^4

2401 = a^4/b^4

Now, we'll use the fact that a and b are positive integers and not divisible by 7. Since 2401 cannot be expressed as a perfect square of an integer, the equation is not satisfied. Therefore, our initial assumption that 7√ is a rational number is false.

Conclusion

We've shown that 7√ cannot be expressed as a rational number, and by extension, neither can 2√+5√, since it is equal to 7√. This demonstrates that 2√+5√ is irrational.

This process of contradiction, expanding an expression, and reaching a conclusion is a common technique for proving irrationality. It's a powerful tool to understand and prove properties of real numbers.