Definition
Irrational number
An irrational number is a real number that no fraction of two integers can equal exactly.
Continue from this idea into connected AXIO math.
Boundary check: A repeating decimal such as 0.333… is rational; √2 does not eventually repeat.
What it means
Fractions made from integers produce decimals that end or eventually repeat. Irrational numbers do neither, so no such fraction names them exactly. You can still use a nearby decimal when measuring or calculating.
Remember it
The decimal keeps going and never settles into a repeating block.
Example
√2 is irrational. Since 1.414² = 1.999396 and 1.415² = 2.002225, √2 lies between 1.414 and 1.415. The decimal 1.414 is an approximation, not the exact value.
Watch for
The decimal 0.333… is not irrational. Its repeating 3s show that it equals the fraction 1 ÷ 3.
Common mix-up
A decimal can continue forever and still be rational if a block eventually repeats. Irrational decimals never eventually repeat a fixed block.
Exact meaning
An irrational number is a real number that cannot be written as a ÷ b for integers a and b with b ≠ 0. Its decimal expansion does not terminate and does not eventually repeat a fixed block.