“Settle into a pattern” is a vague term. If you refer to the decimal fractions (e.g. 3.14…), then if the fraction is finite or repeating the number is rational (basically follows from the definitions of rational numbers and positional fractions). We know that numbers like √2, π and e are irrational, and hence, their decimal fraction representations are infinite and non-repeating.
However, some irrational numbers can be expressed as periodic continued fractions. For example √2=1+1/(2+1/(2+1/(…
So what you’re asking about is probably more about the properties of decimal fractions, and not irrational numbers.
If a number has some repeating pattern, then we call that rational. Rational numbers can all be expressed by a fraction of two integers.
You can prove that a number is irrational by showing that it's not possible for a certain number to be expressed as such a fraction. A common proof technique for achieving this is a proof by contradiction, where you assume that a number can be expressed by M/N and those two integers M and N are the smallest integers with this property, then showing that you can find an even smaller pair of integers with the same property, hence a contradiction.
Suppose the decimal repeats after n places. For example, .252525... repeats after 2 digits. You can find the fraction that represents this number by dividing the n repeating digits by n 9s. .252525... = 25/99.
Since this number can be represented by a ratio of integers, it must be rational.
You can use similar tricks for numbers that repeat after an initial nonrepeating string.
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How do we know that irrational numbers never settle into a pattern? Do we just assume it or is there any kind of proof? | Spyke
By definition. If they did, they’d be rational.
And we also know that irrational numbers exist, and we have proven that certain numbers are irrational.
“Settle into a pattern” is a vague term. If you refer to the decimal fractions (e.g. 3.14…), then if the fraction is finite or repeating the number is rational (basically follows from the definitions of rational numbers and positional fractions). We know that numbers like √2, π and e are irrational, and hence, their decimal fraction representations are infinite and non-repeating.
However, some irrational numbers can be expressed as periodic continued fractions. For example √2=1+1/(2+1/(2+1/(…
So what you’re asking about is probably more about the properties of decimal fractions, and not irrational numbers.
If a number has some repeating pattern, then we call that rational. Rational numbers can all be expressed by a fraction of two integers.
You can prove that a number is irrational by showing that it's not possible for a certain number to be expressed as such a fraction. A common proof technique for achieving this is a proof by contradiction, where you assume that a number can be expressed by M/N and those two integers M and N are the smallest integers with this property, then showing that you can find an even smaller pair of integers with the same property, hence a contradiction.
https://en.wikipedia.org/wiki/Proof_that_pi_is_irrational
https://en.wikipedia.org/wiki/Proof_that_e_is_irrational
You can easily find the proofs for the other irrational numbers with a quick web search.
Suppose the decimal repeats after n places. For example, .252525... repeats after 2 digits. You can find the fraction that represents this number by dividing the n repeating digits by n 9s. .252525... = 25/99.
Since this number can be represented by a ratio of integers, it must be rational.
You can use similar tricks for numbers that repeat after an initial nonrepeating string.