As I was revising once again The chapter number system,I couldn't understand the concept of Irrationality of square roots and non-perfect numbers and I did not get any idea of how to do it. Please could you explain it.
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By definition, an irrational number is a real number that cannot be expressed as a ratio of integers
Example of a Rational Number
0.75 can be expressed as 3/4 therefore making it a rational number.
Similarly Pi, which is denoted as 22/7 when printed in decimal form is unending (starting from 3.142857 and goes on) thus making it an irrational number.
Now let us come to the topic of square roots. Whenever you try to find out the square root of a number and the answer is an integer or a rational number (as explained above), you will say the the square root is a rational number.
Lets take a few examples.
Find the square root of 1. The answer is 1 (rational)
Find the square root of 2. The answer is 1.414...... (unending and hence irrational)
Find the square root of 3. The answer is 1.732...... (unending and hence irrational)
Find the square root of 4. The answer is 2 (rational)
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Now lets talk about non perfect numbers. Generally speaking, any number that is equal to half the sum of all its divisors is called a perfect number. For instance 6 = (1+2+3+6)/2. The next perfect number is 28 which is equal to (1+2+4+7+14+28)/2.
But I think here you are talking about perfect squares and non perfect squares. Any number whose square root is a whole number is called a perfect square. For instance the square root of 25 is 5. Hence 25 is a perfect square. Similarly 100 is a perfect square because 10 is the square root of 100.
And any number whose square root is not a perfect number is called a non perfect square. For instance, 10 is a non perfect square because its square root is 3.16 and so on.
In summary, it will be safe to say that irrational square roots and non perfect squares are talking about the same thing.
Example of a Rational Number
0.75 can be expressed as 3/4 therefore making it a rational number.
Similarly Pi, which is denoted as 22/7 when printed in decimal form is unending (starting from 3.142857 and goes on) thus making it an irrational number.
Now let us come to the topic of square roots. Whenever you try to find out the square root of a number and the answer is an integer or a rational number (as explained above), you will say the the square root is a rational number.
Lets take a few examples.
Find the square root of 1. The answer is 1 (rational)
Find the square root of 2. The answer is 1.414...... (unending and hence irrational)
Find the square root of 3. The answer is 1.732...... (unending and hence irrational)
Find the square root of 4. The answer is 2 (rational)
==========================================
Now lets talk about non perfect numbers. Generally speaking, any number that is equal to half the sum of all its divisors is called a perfect number. For instance 6 = (1+2+3+6)/2. The next perfect number is 28 which is equal to (1+2+4+7+14+28)/2.
But I think here you are talking about perfect squares and non perfect squares. Any number whose square root is a whole number is called a perfect square. For instance the square root of 25 is 5. Hence 25 is a perfect square. Similarly 100 is a perfect square because 10 is the square root of 100.
And any number whose square root is not a perfect number is called a non perfect square. For instance, 10 is a non perfect square because its square root is 3.16 and so on.
In summary, it will be safe to say that irrational square roots and non perfect squares are talking about the same thing.
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