Irrational values such as square roots of positive non-perfect-square integers are understood as numbers that cannot be expressed exactly as terminating or repeating decimals, but can be located and approximated between rational benchmarks on a number line. By comparing nearby perfect squares and refining bounds to tenths or hundredths, learners estimate and round radical values and use those approximations to compare quantities or evaluate simple expressions; general methods for higher roots or advanced numerical approximation are beyond this scope.
To estimate an irrational value such as , find nearby numbers whose squares are easier to calculate. Since squaring positive numbers preserves their order, comparing squares lets us locate the square root.
Example: Estimate to the nearest hundredth.
Find nearby perfect squares.
Since ,
Refine the estimate to tenths.
Because ,
Refine the estimate to hundredths.
Test hundredths near :
Therefore,
so
Round to the nearest hundredth.
The halfway point between and is . Check its square:
Thus is less than , so it rounds down to .
The symbol means “approximately equal to,” because is irrational and cannot be written as an exact terminating or repeating decimal.
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