how to findan irrational numbr between 3.56254.......and3.56342.....

Asked by S20101980 | 13th Jun, 2018, 07:15: PM

Expert Answer:

Here are the four ways by which we can an find irrational number which is between two rational numbers:

If we consider between 3.1 and 3.2, we can write infinite such numbers.

Examples: 3.11729393223578214627482716439386424145749… (The successive dots indicate it is non-terminating).

3.1897825830294836834902937329847239230973…

3.1087654876341739263927832789201423739321153… and so on.

The second method is to square the given two rational numbers A and B, such that A2 and B2 say (A2 - 0.006, such that it isn’t a perfect square). The square root of such numbers are always irrational.

Example: Between 2.7 and 2.8, we can find the irrational number as follows,

(2.7)2 = 7.29 and (2.8)2 = 7.84.

Pick any random rational number between these two. Let it be 7.347.

The square root of 7.347 is 2.710535002541011… It is irrational. Similarly,

Sq.rt of 7.562 is 2.74990908941…

The third method is by multiplying the lowest number by an irrational number, such that the product won’t alter the value of that number significantly.

Here I can specifically tell that the number for a two digit number is almost between 1.001 and 1.0001(such that it isn’t a perfect square). It goes on higher as the number of digits before the decimal place increases.

Example: Between the numbers 75.6 and 89.2, an irrational number can be found as, 75.6 x square root of (1.0002) is an irrational.

Like that we can multiply infinite irrational numbers with that rational number such that its value doesn’t exceed the greater number.

The fourth method is by dividing the greater number by an irrational number such that the resulting number isn’t lower than the other number.

If the numbers are 27.54 and 34.82, then, 34.82/(square root of 1.0033) is an irrational number between those two. (It is more apt to consider numbers like 1.0000283, 1.000726 etc.., as its effect on the value is insignificant).

Answered by Arun | 18th Jun, 2018, 10:21: AM

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