What Are Irrational Numbers?

Irrational numbers, also known as infinite acyclic decimals, cannot be written as the ratio of two integers. If you write it as a decimal, there are an infinite number of digits after the decimal point, and it will not loop. Common irrational numbers include the square root of an incomplete square number, and e (the latter two are transcendental numbers), and so on. Another feature of irrational numbers is the infinite continuous fraction expression. Irrational numbers were first discovered by Pythagoras disciples, Heberthos. ^[1]

In mathematics, irrational numbers are all real numbers that are not rational numbers, the latter being a number consisting of a ratio (or fraction) of integers. When the length ratio of two line segments is irrational, the line segments are also described as incomparable, which means that they cannot be "measured", that is, they have no length ("metric").

Common irrational numbers are: the ratio of the circumference to its diameter, the Euler number e, the golden ratio , and so on.

It can be seen that the representation of irrational numbers in a position number system (for example, in decimal digits or any other natural basis) does not terminate, nor does it repeat, that is, it does not contain a subsequence of numbers. For example, the decimal representation of the number starts at 3.141592653589793, but numbers without a finite number can accurately represent and are not repeated. The evidence for the decimal expansion of a rational number that must be terminated or repeated is different from the evidence that the terminated or repeated decimal expansion must be a rational number. Although basic and not lengthy, both proofs require some work. Mathematicians do not usually use "termination or repetition" as a definition of the concept of rational numbers.

Irrational numbers can also be handled by unterminated continuous fractions.

Irrational numbers are numbers that cannot be expressed as a ratio of two integers in the range of real numbers. Simply put, irrational numbers are in decimal

prove

Is an irrational number (integer

)

,

Mutual prime.

Suppose

Exist

Then a is even, let

,

Substituting the above formula for a positive integer has

Then b is also an even number, and the condition (

,

)for

If the positive integer N is not a perfect square number, then

Not a rational number (is an irrational number).

Prove: if hypothesized

Is a rational number, may wish to set

, Where p and q are positive integers (not necessarily mutually prime. If p and q are assumed to be prime, the proof method is slightly changed).

Assume

Where the integer part is a, then there is an inequality

Established. Multiply both sides by q to get

Since p, q, and a are all integers, p-aq is also a positive integer.

Multiply both sides of the above inequality by

, Got

which is:

Obviously, qN-ap is also a positive integer.

So we found two new positive integers

with

And they satisfy

, which is

And have

.

Repeat the above steps to find a series of

Make

And

. Because this step can be repeated indefinitely, it means

Both can be infinitely reduced, but this contradicts a minimum of 1 for positive integers.

So the assumption is wrong,

Not a rational number.