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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.
mathsComplex NumbersIntroduction to Complex Numbers

### Number System Quick Revision

This page starts with a quick revision of number system hierarchy. Complex numbers are introduced as an extension of Real Numbers.

click on the content to continue..

Consider "whole numbers". Which of the following represents the whole numbers?

• 0, 1, 2, 3, ...
• 0, 1, 2, 3, ...
• ..., -3, -2, -1, 0, 1, 2, 3, ...

The answer is '0, 1, 2, 3, ...'

When using the whole numbers, we come across problems that are mathematically modeled to x+1=0. There is no solution to this in whole number system. So what is done?

• the equation has no solution
• number system is extended to integers
• number system is extended to integers

The answer is 'number system is extended to integers'. Whole numbers are extended to integers by including negative numbers.

Which of the following represents the integers?

• 0, 1, 2, 3, ...
• ..., -3, -2, -1, 0, 1, 2, 3, ...
• ..., -3, -2, -1, 0, 1, 2, 3, ...

The answer is '..., -3, -2, -1, 0, 1, 2, 3, ...'.

When using the integers, we come across problems that are mathematically modeled to 2x=1. There is no solution to this in integer number system. So what is done?

• the equation has no solution
• the equation has no solution
• number system is extended to rational numbers

The answer is 'number system is extended to rational numbers '

Which of the following represents the rational numbers?

• 0, 1, 2, 3, ...
• ..., -3, -2, -1, 0, 1, 2, 3, ...
• numbers that can be represented as p/q where p,q are integers, and q !=0
• numbers that can be represented as p/q where p,q are integers, and q !=0

The answer is 'numbers that can be represented as p/q where q !=0'

When using the rational numbers, we come across problems that are mathematically modeled to x^2=2. There is no solution to this in rational number system. So what is done?

• the equation has no solution
• number system is extended to irrational numbers
• number system is extended to irrational numbers

The answer is 'number system is extended to irrational numbers '

Which of the following represents irrational numbers?

• 0, 1, 2, 3, ...
• ..., -3, -2, -1, 0, 1, 2, 3, ...
• numbers that can be represented as p/q where p,q are integers, and q !=0
• numbers that cannot be represented as p/q and are on the number line
• numbers that cannot be represented as p/q and are on the number line

The answer is 'numbers that cannot be represented as p/q and are on the number line'

What does the combination of Rational and irrational numbers called?

• real numbers
• real numbers
• they are not combined into a number system

When using the real numbers, we come across problems that are mathematically modeled to x^2=-1. There is no solution to this in real number system. So what is done?

• the equation has no solution
• number system is extended beyond real numbers
• number system is extended beyond real numbers

The answer is 'number system is extended beyond real numbers'.

Learning about the extended numbers is the objective of this topic.

Number system is in a hierarchy that extends one to another to include additional mathematical models and solutions.

Number System :
•  Whole numbers
•  Integers (Whole numbers extended for negative numbers)
•  Rational Numbers (Integers extended for fractions)
•  Real numbers (Rational numbers extended for irrational numbers)
•  Complex numbers (Real numbers extended to include solutions to polynomials)

Solved Exercise Problem:

To accommodate negative numbers, whole number system is extended into ...

• integers
• integers
• natural numbers
• negative whole numbers