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Thought-Process to Discover Knowledge

Welcome to nubtrek.

Books and other education websites provide "matter-of-fact" knowledge. Instead, nubtrek provides a thought-process to discover knowledge.

In each of the topic, the outline of the thought-process, for that topic, is provided for learners and educators.

Read in the blogs more about the unique learning experience at nubtrek.
mathsDecimalsConversion of Decimals

### Decimals: Not repeating and not ending

This page provides an overview of numbers with digits in decimal place not repeating and not ending. These are called irrational numbers.

This introduction on irrational numbers is amazingly simple and revolutionary. All students should go through this once to understand irrational numbers.

click on the content to continue..

Consider a pencil. How to measure the length of a pencil?

• mark the ends of the pencil into a line-segment and measure the length of the line-segment
• measure the distance-span between the two ends of the pencil
• either one of the above
• either one of the above

The answer is "either one of the above".

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 bar(AB)?

• length cannot be found
• 2.7cm
• 2.7cm

The answer is "2.7cm".

The figure is magnified. Note the legend showing the modified 1cm length. The point A is not shown. As per the figure, Which of the following is correct?

• the length of bar(AB) is accurately 2.7cm
• the length of bar(AB) is approximately 2.7cm
• the length of bar(AB) is approximately 2.7cm

The answer is "the length of bar(AB) is approximately 2.7cm"

The figure is magnified even larger. Note the legend showing the modified 1mm length. As per the figure, which of the following is more accurate?

• the length of bar(AB) is still 2.7cm
• the length of bar(AB) is better approximated to 2.68cm
• the length of bar(AB) is better approximated to 2.68cm

The answer is "the length of bar(AB) is better approximated to 2.68cm"

The line-segment bar(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 2.68892cdots 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 2cm is possible. That object, when magnified and measured with high accuracy, will be of length 2cm 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 3.68023cdotscm length. The decimals in the number does not end or repeat. It is an interesting observation, that will help to understand some concepts later.

Which of the following is a meaning for the word "accuracy"?

• state of being exactly correct
• state of being exactly correct
• gather together in large numbers

The answer is "state of being exactly correct".

What is the term used to refer "state of being exactly correct"?

• Pronunciation : Say the answer once
Spelling: Write the answer once

The answer is "accuracy".

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 are called "irrational numbers".

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 1cm. 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.

Which of the following is a meaning for the word "irrational"?

• that cannot be given as a ratio
• that cannot be given as a ratio
• not given in fixed small amounts

The answer is "that cannot be given as a ratio".

What is the term used to refer "that cannot be given as a ratio"?

• Pronunciation : Say the answer once
Spelling: Write the answer once

The answer is "irrational". The word can be understood as "ir (not) + rational (represented 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.

slide-show version coming soon