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Irrational Numbers
An irrational number is a number that can't be expressed by a fraction having integers in both its numerator and denominator. While their existence was once kept secret from the public for philosophical reasons, they are now well accepted, yet still surprisingly hard to prove on an individual basis. Please post all questions about irrational numbers, including the famous examples of π, e, and √2, into this category.
3,962 Questions
Are non-repeating decimals irrational numbers?
No because non-repeating decimals may be terminating.
But suppose you consider terminating decimals as consisting of repeating 0s. That is, 1/8 = 0.125 = 0.12500....
Then all non-repeating decimals are irrational.
What is a good definition of irrational numbers?
A real number that can not be expressed as a rational number.
What are some rational numbers?
All integers and fractions are rational numbers whereas an irrational number can't be expressed as an integer or a fraction.
Which set does 6 belongs to isn't rational numbers or what?
6 is a rational number and it belongs to any set that contains it!
It belongs to {6},
or {45, sqrt(2), 6, pi, -3/7},
or all whole numbers between -43 and 53,
or multiples of 2,
or composite numbers,
or counting numbers,
or integers,
or rational numbers,
or real numbers,
or complex numbers,
etc.
Rational numbers are numbers that can be written as a fraction. Irrational numbers cannot be expressed as a fraction.
How do you find the square root of 2?
With great difficulty because the square root of 2 is an irrational number
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One way to find the square root of a number is an iterative method. This entails making a guess at the answer and then improving on it. Repeating the procedure should lead to a better estimate at each stage. One such is the Newton-Raphson method.
If you want to find the square root of 2, define f(x) = x2 - 2.
Then finding the square root of 2 is equivalent to solving f(x) = 0.
Let f'(x) = 2x. This is the derivative of f(x) but you do not need to know that to use the N-R method.
Start with x0 as the first guess.
Then let xn+1 = xn - f(xn)/f'(xn) for n = 0, 1, 2, …
Provided you made a reasonable choice for the starting point, the iteration will very quickly converge to the true answer. Even if your first guess is not so good:
Suppose you start with x0 = 2 (a pretty poor choice since 22 is 4, which is nowhere near 2).
Even so, x3 = 1.414215686, which is less than 1-in-a billion from the true value. Finally, remember that the negative value is also a square root.
Can you add two rational number to get irrational number?
Yes, we can get a rational number on the addition of two irrational numbers.
e.g. Let us consider two irrational numbers: 3 + √2 and 4 - √2.
Addition yields:
(3 + √2)+ (4 - √2) = 3 + 4 = 7(a rational number).
Another example is:
Addition of √2 and -√2.
√2+ (-√2) = 0(a rational number).
Explanation of example 1:
Irrational numbers in the form of of p + q are are the irrational numbers which are obtained on addition of two terms: one is rational(p) and another is irrational(q).
And on taking the conjugate of p + q we get p - q, which is an another irrational number. And the addition of these two yields a rational number.
When are irrational numbers used in mathematics?
Two of the most important numbers in advanced mathematics are pi and e and both are irrational.
What is an irrirational number?
A irrational number is a number that can't be written in the form of a fraction. Like 3.14.......
What is a subsets of the irrational numbers?
One possible set, out of infinitely many, is positive irrational numbers.
Why are some numbers irrational?
There are numbers which cannot be expressed as ratios of two integers. These are called irrational numbers.
Is the perfect square of 64 a rational or irrational number?
It is a rational number - as are ALL perfect squares.
Definition of irrational number?
A rational number is a number that can be expressed as a ratio of two integers, the second of them being non-zero. An irrational number is one that cannot be so expressed.
