Table of Contents
What does it mean to raise a number to an irrational power?
Irrational exponents are non repeating or infinite decimals while rational exponents are rational numbers. The value of an irrational exponent when calculated is approximate in nature while the value of rational exponent is exact.
Why is e an irrational number like pi?
Pi is an irrational number, i.e. it cannot be expressed as a fraction of two integers. The commonly used fraction 22/7 which is often used is only a rough approximation, although sufficient for many basic calculations where only accuracy is not required….Pi, π
| Notation | Number |
|---|---|
| Binary | 11.00100100001111110110 |
Is pi to the power of e rational?
Open Question It is not known whether π(pi) to the power of Euler’s number e: πe. is rational or irrational.
Is pi is an irrational number?
No matter how big your circle, the ratio of circumference to diameter is the value of Pi. Pi is an irrational number—you can’t write it down as a non-infinite decimal.
How do you write pi as a rational number?
Adding the next significant digit to pi can be said to involve multiplying both numerator and denominator by 10 and adding a number between between -5 and +5 (approximation) to the numerator. Since both (10^(N+1)) and (M*10+A) for A between -5 and 5 are integers, the (N+1)-digit approximation of pi is also rational.
What does raising a number to a power do?
When you “raise a number to a power,” you’re multiplying the number by itself, and the “power” represents how many times you do so. So 2 raised to the 3rd power is the same as 2 x 2 x 2, which equals 8.
How do you find E raised to an imaginary power?
Now we know what e raised to an imaginary power is. One can also show that the definition of e^x for complex numbers x still satisfies the usual properties of exponents, so we can find e to the power of any complex number b + ic as follows: e^(b+ic) = (e^b)(e^(ic)) = (e^b)((cos c) + i(sin c))
Is it easy to prove that a number is irrational?
Proving a number is irrational may or may not be easy. For example, nobody knows whether π + e is rational. On the other hand, there are properties we know rational numbers have and only rational numbers have, and properties we know irrational numbers have and only irrational numbers have.
How do you prove that $$\\pi$ is irrational?
To show that $\\pi$ is irrational is much harder—in fact so hard that it was not done until the 18th century. Another proof of irrationality begins by proving that when you divide an integer by another integer, if the decimal expansion does not terminate, then it must repeat. I posted an explanation of that here.
Are \\alpha^\\beta$ and \\beta$ irrational?
Summary of edits: If $\\alpha$ and $\\beta$ are algebraicand irrational, then $\\alpha^\\beta$ is not only irrational but transcendental. Looking at your other question, it seems worth discussing what happens with square roots, cube roots, algebraic numbers in general.