If you can find a number whose decimals run on for ever without ever repeating, that number cannot be written as a fraction. Such numbers are called irrational.
An irrational number cannot be written as p over q. Its decimals go on for ever and never fall into a repeating block.
Where you meet them
The square root of 2 is the diagonal of a square with sides of 1 unit, so it is a length you can draw. Pi is another. The roots of 4, 9, 16 and 25 are not irrational, because those are perfect squares.
22/7 agrees with pi to two decimal places, but 22/7 is 3.142857 repeating while pi is 3.14159265 with no repeat anywhere. Using it in a calculation is fine; saying pi equals it is not true.
Why the square root of 2 cannot be a fraction
Suppose it could be, written in lowest terms. Squaring gives top squared equals twice bottom squared, so the top is even.
Substituting back shows the bottom is even too. But we said the fraction was in lowest terms. That is a contradiction, so no such fraction exists.
Assume the opposite, follow the consequences honestly, and arrive somewhere impossible. So the starting point must have been false. This proof is around 2500 years old.
Real numbers
The rationals and irrationals together make the real numbers. The square root of 2 lies between 1.4 and 1.5, then between 1.41 and 1.42, narrowing for ever without landing exactly.
One seventeenth has a repeating block sixteen digits long, so on a short display it looks patternless, but it is rational. A calculator screen is far too short to settle the question.
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