Irrational Numbers

The measurements used to get the calculator value are more precise, and so there are more decimal places on the calculator.

The square root I needed to calculate was not a perfect square, so finding the number that multiplies by itself to give the answer was challenging. I could have kept going forever without finding the exact right answer.

No, the decimal form of π has infinitely many places after the decimal. It never ends.

The decimal form of one third is 0.3333333333… or 0.3, where the bar above the 3 means the 3 is repeated infinitely many times.

I can calculate the square root of a non-perfect square by using the square root button on a scientific calculator.

A rational number is a number that can be written as a fraction of two integers. In decimal form, the decimals either terminate or form a repeating pattern.

An Irrational number is one whose decimal form never terminates and never repeats. They include special numbers like π, φ, or e, as well as square roots of non-perfect squares.

In practice, I can approximate an irrational number with a rational number that has enough decimal places to be useful in the problem. For example, if I need to know roughly how big √2 is, I may round to 1 decimal place to get my rational approximation. If I am an engineer and I need to be very precise, I might round to 6 decimal places. It is not practical or possible to work with infinitely many decimal places.

3.758375839203…≈3.76, √22≈4.69, [ggfrac]√20/4[/ggfrac]≈1.12, 3√2≈4.24, so the order from least to greatest is [ggfrac]√20/4[/ggfrac], 3.758375839203…, 3√2, √22.

Write the decimal as a fraction with denominator 1. Then multiply the numerator and denominator by a power of 10 until the numerator is an integer.

Example: [ggfrac]2.55676/1.00000[/ggfrac] = [ggfrac]2.55676x100,000/1.00000x100,000[/ggfrac] = [ggfrac]255,676/100,000[/ggfrac]