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Rational and irrational numbers. Explanation of pi being an irrational number. Thanks for the help.
Date Posted: 3 years ago
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The explanation is wrong because :
① For a/b, both a and b be have to be integers.
Basically, one of the simpler ways to think about rational numbers is that it can be expressed as a fraction where both the numerator and denominator are numbers with no fractional parts themselves. They must be exact and whole. (regardless of being negative or positive)
Irrational numbers cannot be expressed as the ratio of two integers.
If we try to write the irrational number in terms of decimal expansion, the decimal expansion does not terminate, nor end with a repeating sequence
Eg. 1/7 can be written as 0.142857142857142857....
1/7 is rational since we get repeating sequence of 142857
0.3333333.... can be rewritten as 1/3. So it is also rational.
Now something like √2 is irrational as it does not fulfill condition ①.Likewise for π.
So even if we can write π = C/d, neither C nor d can be integers.
① For a/b, both a and b be have to be integers.
Basically, one of the simpler ways to think about rational numbers is that it can be expressed as a fraction where both the numerator and denominator are numbers with no fractional parts themselves. They must be exact and whole. (regardless of being negative or positive)
Irrational numbers cannot be expressed as the ratio of two integers.
If we try to write the irrational number in terms of decimal expansion, the decimal expansion does not terminate, nor end with a repeating sequence
Eg. 1/7 can be written as 0.142857142857142857....
1/7 is rational since we get repeating sequence of 142857
0.3333333.... can be rewritten as 1/3. So it is also rational.
Now something like √2 is irrational as it does not fulfill condition ①.Likewise for π.
So even if we can write π = C/d, neither C nor d can be integers.
For the proof of √2 being irrational, see the following :
https://www.math.utah.edu/~pa/math/q1.html
https://www.math.utah.edu/~pa/math/q1.html
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