How to tell between irrational numbers and rational numbers, and how to open the number under the ro

An irrational number is an infinite non-cyclic decimals, for example, a rational number is an integer (positive integer, negative integer, 0), fraction (positive fraction, negative fraction).

Open Root: Let's give you an example! The root number is 3364, open square, first look at the following 4, 1 10 in which number is squared and finally four? The answer is 8,2

Looking at the previous 33, what about the square of the number in 1 10 that is closest to 33 (the one in front of 33)? The answer is 5

Then it could be 58 or 52! (Correct answer: 58) Finally, do the math one by one to find out the correct answer!

If you don't believe me, go home and give it a try, practice makes perfect!!

Infinite non-cyclic real numbers are irrational numbers.

Infinite looping decimals are rational numbers because they can be reduced to fractions.

Finite decimals are also rational numbers.

Will the short division be? It is similar to finding the product of several prime numbers of a number in elementary school, except that it is replaced by the product of several square numbers.

eg： √1056

Can be turned into 4 * 4 * 66 4 * 66

It's as simple as that.

It's about the subordination until it becomes a prime number, and then change the same divisor to one and mention it outside the root number

For example, 100 is about 2*5*2*5, and putting 2 and 5 together is 10.

Of course, the obvious can be mentioned directly, no prime numbers are needed, such as 1230000, 123*10000, 123*100*100, just put 100 directly, and then see if 123 can be contracted

Real numbers that can be represented by p q, (where p and q are integers) are rational numbers, otherwise they are irrational numbers.

Under the root number, you can just use the computer to calculate, and you can see at a glance that it is really sweaty.

Let the root number 7 be a rational number.

Then there must be a root number 7 = p q

pq is an integer and coprimous).

then there is p 2 q 2 = 7

i.e. p qp q=7

Because the numerator and denominator are mutual.

So there is no common prime factor.

It won't be about 7i.e. the equation does not hold. So the root number 7 can only be an irrational number.

Suppose 6 is not an irrational number, but a rational number.

Since 6 is a rational number, it must be written in the form of a ratio of two integers:

6=p q and since p and q have no common factor to be reduced, p q can be considered to be a reduced fraction, that is, the simplest fractional form.

Square 6=p q on both sides.

6q 2 = p 2

Since 6q 2 must be even, p is even.

Let p=2m, and substitute to get 6q 2=4m 2, that is, 3q 2=2m 2, since 2m 2 must be even, 3q 2 is also even.

That is, q 2 is an even number, and q is also an even number.

Since p and q are both even, they must have a common factor of 2, which contradicts the previous assumption that p q is a reduced fraction. This contradiction is caused by the assumption that 6 is a rational number.

Hence 6 is an irrational number.

The root number 5 is an irrational number, and there are 2 commonly used methods to calculate it:

1) Series method. Utilize the Taylor formula under the root number (1+x).

2) Iterative algorithms. Use the iterative formula: x0=a 2, x(n+1)=(xn+a xn) 2.

Proof process

1. Let 5 under the root number not be an irrational number but a rational number, then let 5 = p q under the root number (p, q are positive integers and are prime numbers of each other, that is, the greatest common divisor is 1).

2. Square both sides, 5=p 2 q 2, p 2=5q 2(*).

3. p 2 contains a factor of 5, let p = 5m, substitute (*) 25m 2 = 5q 2, q 2 = 5m 2, q 2 contains a factor of 5, that is, q has a factor of 5.

4. So that p,q have a common factor of 5, which contradicts the assumption that the greatest common divisor of p,q is 1.

5. 5=p q under the root number (p, q are positive integers and are prime numbers of each other, i.e., the greatest common divisor is 1) is not true, therefore, 5 under the root number is not a rational number but an irrational number.

Categories: Education Academic Exams >> Study Help.

Problem description: Given the square of a number? How can you tell if an open root number is rational or irrational?

Analysis: It depends on whether the number under the root number is perfectly squared, i.e. it can be written as the square of another number. If it is a perfectly square number, it is a rational number after the root number; Otherwise, it is an irrational number.

A perfectly squared number is a number that can be written as a square of a positive integer. For example, 36 is 6 6 and 49 is 7 7.

The sum of n odd numbers starting from 1 is a perfectly square number, n 2 i.e. 1 3 5 7 ....2n-1) n 2, e.g. 1 3 5 7 9 25 5 2. The last digit of each perfect square is 0, 1, 4, 5, 6, or 9

Each perfect square is divisible by 3 at the end, and divisible by 3 by subtracting 1 from the end. Each perfect square number is divisible by 4 at the end of the tassel, and subtracted by the macro type 1 at the end is divisible by 4.

Each perfect square is divisible by 5 at the end, and divisible by 5 by adding 1 or subtracting 1 from the end.

Additional note: If there is a fraction under the root number, the numerator and denominator must be distinguished separately. If there is a decimal place under the root number, it is converted into a fraction and then identified using the above method.

The arithmetic square root of any perfectly squared number is a rational number, and the arithmetic square root of any other natural number is an irrational number. For example, 4, 9, etc. are rational numbers. And 3 and 5 are both irrational numbers.

Irrational numbers should meet three conditions: first, decimals; the second is an infinite decimal; The third is not circulating.

In addition to some root numbers that are irrational, some constants or fractions are also irrational numbers, such as the constant e, etc., are also irrational numbers.

It depends on whether the number under the root sign is perfectly squared, i.e. it can be written as the square of another number. If it is a perfectly square number, it is a rational number after the root number; On the contrary, the coarse trembling celebration is an irrational number.

Mathematically, a rational number is the ratio of an integer a to a positive integer b, e.g. 3 8, and a is also a rational number. A rational number is a set of integers and fractions, and an integer can also be thought of as a fraction with a denominator of one. The decimal part of a rational number is a finite or infinitely looping number.

Real numbers that are not rational numbers are called irrational numbers, i.e., the fractional part of an irrational number is an infinite number that is not cyclical.