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Find the difference between the two numbers, then add an irrational number between zero and one, divided by this difference, to the lower number. Such an irrational number might be pi/10, (square root of 2) / 2, etc.

Q: How do you find an irrational no between 0.365789 and 0.365478?

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The set of irrational numbers is infinitely dense. As a result there are infinitely many irrational numbers between any two numbers. So, if any irrational number, x, laid claim to be the closest irrational number to 3, it is possible to find infinitely many irrational numbers between x and 3. Consequently, the claim cannot be valid.

Any number that can't be expessed as a fraction is an irrational number as for example the square root of 4.5

72 = 49 and 82 = 64. So, the square root of any integer between these two numbers, for example, sqrt(56), is irrational.

There may be many easier and better ways, but here's how I would do it: -- Square the first given irrational number. -- Square the second irrational number. -- Pick a nice ugly complicated decimal between the two squares. -- Take the square root of the number you picked. It's definitely between the two given numbers, and it would be a miracle if it's not irrational.

Irrational numbers are infinitely dense. Between any two numbers, there are infinitely many irrational numbers. So if it was claimed that some irrational, x, was the closest irrational to 6, it is possible to find an infinite number of irrationals between 6 and x. Each one of these infinite number of irrationals would be closer to 6 than x. So the search for the nearest irrational must fail.

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It is proven that between two irrational numbers there's an irrational number. There's no method, you just know you can find the number.

863

the numbers between 0 and 1 is 0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,0.10.

I can't tell you the irrational number between 0.2 and 0.3; there an infinite number of irrationals in this range.For an example - root(2) / 7 is slightly more than 0.202, and is irrational.

The set of irrational numbers is infinitely dense. As a result there are infinitely many irrational numbers between any two numbers. So, if any irrational number, x, laid claim to be the closest irrational number to 3, it is possible to find infinitely many irrational numbers between x and 3. Consequently, the claim cannot be valid.

Any number that can't be expessed as a fraction is an irrational number as for example the square root of 4.5

There may be many easier and better ways, but here's how I would do it: -- Square the first given irrational number. -- Square the second irrational number. -- Pick a nice ugly complicated decimal between the two squares. -- Take the square root of the number you picked. It's definitely between the two given numbers, and it would be a miracle if it's not irrational.

72 = 49 and 82 = 64. So, the square root of any integer between these two numbers, for example, sqrt(56), is irrational.

0.12=1.1201001000100001 0.13=1.12101001000100001

sqrt(5) and sqrt(6)

Irrational numbers are infinitely dense. Between any two numbers, there are infinitely many irrational numbers. So if it was claimed that some irrational, x, was the closest irrational to 6, it is possible to find an infinite number of irrationals between 6 and x. Each one of these infinite number of irrationals would be closer to 6 than x. So the search for the nearest irrational must fail.

Any number that can't be expressed as a fraction is irrational