Approximate irrational numbers

An irrational number is understood as a real number whose decimal expansion is nonterminating and nonrepeating, and it can be located and compared by finding rational bounds or decimal approximations—for example, placing $\sqrt{2}$ between $1.41$ and $1.42$ and rounding it to a specified place value. The approximation is not the irrational number itself; using bounds, number-line position, and appropriate precision supports calculations involving radicals, measurement, and geometry, without requiring formal proofs of irrationality or advanced approximation methods.

Detailed Explanation: Approximate irrational numbers

An irrational number cannot be written as a terminating or repeating decimal. To approximate one, find two nearby rational numbers that bound it, then round to the requested place value.

Example: Approximate $\sqrt{2}$ to the nearest hundredth

We know:

1^2=1 \quad\text{and}\quad 2^2=4

so $\sqrt{2}$ is between $1$ and $2$ .

Try hundredths:

1.41^2=1.9881

and

1.42^2=2.0164

Since $1.9881<2<2.0164$ , it follows that

1.41<\sqrt{2}<1.42.

To decide whether to round to $1.41$ or $1.42$ , check the thousandths:

1.414^2=1.999396

and

1.415^2=2.002225.

Therefore,

1.414<\sqrt{2}<1.415.

The number is less than $1.415$ , so it rounds down to the nearest hundredth:

\boxed{\sqrt{2}\approx 1.41}

The symbol $\approx$ means “approximately equal to.” The decimal $1.41$ is an approximation, not the exact value of the irrational number $\sqrt{2}$ .

Learn by doing: Approximate irrational numbers

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Number Types (Irrational) - Between X and Y - Positive Square Roots, Cube Roots, Pi

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