overview

On measuring a real-life object, it is found that a number can have decimal digits that do not repeat and do not end. Such decimal numbers -- with decimal digits that do not repeat and do not terminate -- are called "irrational numbers".

numbers in practice

Consider length of a pencil. One of the ways to measure the length of a pencil, is to mark the ends of the pencil into a line-segment and measure the length of the line-segment. This essentially measures the "distance-span between the two ends of the pencil".

To measure the length of a pencil, an equivalent line segment is taken. The line segment and the graduated scale are shown in the figure. Note the legend on the right-top of the figure showing $1$ centimeter. What is the length of the line segment $\overline{AB}$?

The figure is magnified. Note the legend showing the modified $1cm$ length. The point $A$ is not shown. "the length of $\overline{AB}$ is approximately $2.7cm$"

The figure is magnified even larger. Note the legend showing the modified $1mm$ length. The length of $\overline{AB}$ is better approximated to $2.68cm$

at what accuracy?

The line-segment $\overline{AB}$ can be measured at different accuracies.

• The length is $2.6cm$ when measured with a graduated-scale.

• The length is $2.68cm$ when magnified and measured with a better instrument than a scale.

• The length is $2.68892\cdots$ when measured at much higher accuracy.
For most practical applications, the length of $2.6cm$ is good enough.

Note 1: The line represents a real object. It can be a leaf, or a rod. The important point being, the measurements in real-life are approximated.

Note 2: The length of a real object can be exactly measured too. For example, a pencil of length exactly $2$cm is possible. That object, when magnified and measured with high accuracy, will be of length $2$cm only.

Note 3: The length of a real object can be a decimal number, that does not end or does not repeat. For example, a pencil is of length $3.68023\cdots$cm length. The decimals in the number does not end or repeat. It is an interesting observation, that will help to understand some concepts later.

The word "accuracy" means: state of being exactly correct.

defining

On measuring a real-life object, it is found that a number can have decimal digits that do not repeat and do not end. Such decimal numbers -- with decimal digits that do not repeat and do not terminate -- are called "irrational numbers".

The word "irrational" means: that cannot be given as a ratio.

**Irrational Numbers** : Decimal numbers that has decimal digits that do not end and do not have a pattern of digits that repeats.
eg: accurate length of a pencil is $2.68892\cdots$.

advanced concepts

**advanced concepts (not for regular readers)**

There are two types of decimal numbers with decimal digits that do not end and do not have a pattern that repeats.

• Algebraic irrational numbers: Irrational numbers that are solutions to algebraic equations. eg: The square root of $2$ which is a solution to ${x}^{2}=2$.

• Transcendental irrational numbers: Numbers that are not solutions to algebraic equations. The ratio of circumference to diameter of a circle is an irrational numbers. The diameter is chosen to be in the given standard $1$cm. Circle is a regular curve and the length of the curve is measured accurately and found to be an irrational number.

In transcendental irrational numbers, the following is included.

Accurate measurement of real-life object in a unrelated standard.

eg: $1$ pound is approximated to $453.5924277$ grams. Pound is a standard defined based on mass of $7000$ grains (wheat or barley). Kilogram is a standard defined based on mass of $1$ liter water at the temperature of melting point. This irrational number is similar to the example illustrated above -- accurately measuring a real-life object.

Similarly, curve-length of a circle is a measurement in an unrelated standard and so is a irrational number.

summary

**Irrational Numbers** : Decimal numbers that has decimal digits that do not end and do not have a pattern of digits that repeats.
eg: accurate length of a pencil is $2.68892\cdots$.

Outline

The outline of material to learn "decimals" is as follows.

Note: *goto detailed outline of Decimals *

• Decimals - Introduction

→ __Decimals as Standard form of Fractions__

→ __Expanded form of Decimals__

• Decimals - Conversion

→ __Conversion between decimals and fractions__

→ __Repeating decimals__

→ __Irrational Numbers__

• Decimals - Arithmetics

→ __Comparing decimals__

→ __Addition & Subtraction__

→ __Multiplication__

→ __Division__

• Decimals - Expressions

→ __Expression Simplification__

→ __PEMA / BOMA__