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An irrational number is expressed as a non-repeating decimal that goes on forever. Write out the enough of the decimal expansion of each number to find the first digit where the two numbers disagree. Truncate the larger number at that digit, and the result is a rational number (terminating decimal) that is between the two.

Q: How do you find rational numbers between two irrational numbers?

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All fractions are rational numbers because irrational numbers can't be expressed as fractions

Let your sum be a + b = c, where "a" is irrational, "b" is rational, and "c" may be either (that's what we want to find out). In this case, c - b = a. If we assume that c is rational, you would have: a rational number minus a rational number is an irrational number, which can't be true (both addition and subtraction are closed in the set of rational numbers). Therefore, we have a contradiction with the assumption that "c" (the sum in the original equation) is rational.

The idea is to look for a rational number that is close to the desired irrational number. You can find rational numbers that are as close as you want - for example, by calculating more decimal digits.

There are an infinite number of integers that meet this criteria.Ans 2Root 2 and root 3 are both irrational, but there is no integer between them.Did you mean to say 'an infinite number of pairs of integers" ?

root 2 * root 2 = 2

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A rational number is one that is the ratio of two integers, like 3/4 or 355/113. An irrational number can't be expressed as the ratio of any two integers, and examples are the square root of 2, and pi. Between any two rational numbers there is an irrational number, and between any two irrational numbers there is a rational number.

All fractions are rational numbers because irrational numbers can't be expressed as fractions

See lemma 1.2 from the cut-the-knot link. Yes, you can.

The one thing they have in common is that they are both so-called "real numbers". You can think of them as points on the "real number line".Both are infinitely dense, in the sense that between any two rational numbers, you can find another rational number. The same applies to the irrational numbers. Thus, there are infinitely many of each. However, the infinity of irrational numbers is a larger infinity than that of the rational numbers.

Find the arithmetic average of the two rational numbers. It will be a rational number and will be between the two numbers.

Irrational numbers are not closed under any of the fundamental operations. You can always find cases where you add two irrational numbers (for example), and get a rational result. On the other hand, the set of real numbers (which includes both rational and irrational numbers) is closed under addition, subtraction, and multiplication - and if you exclude the zero, under division.

Let your sum be a + b = c, where "a" is irrational, "b" is rational, and "c" may be either (that's what we want to find out). In this case, c - b = a. If we assume that c is rational, you would have: a rational number minus a rational number is an irrational number, which can't be true (both addition and subtraction are closed in the set of rational numbers). Therefore, we have a contradiction with the assumption that "c" (the sum in the original equation) is rational.

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.

There are infinitely many of them. In fact there are more of them in that interval than there are rational numbers in total.

Irrational numbers are infinitely dense. That is to say, between any two irrational (or rational) numbers there is an infinite number of irrational numbers. So, for any irrational number close to 6 it is always possible to find another that is closer; and then another that is even closer; and then another that is even closer that that, ...

The idea is to look for a rational number that is close to the desired irrational number. You can find rational numbers that are as close as you want - for example, by calculating more decimal digits.

For two rational numbers select any terminating or repeating decimal number which starts with 2.10 and for irrational numbers you require a non-terminating, non-repeating decimal which also starts with 2.10.