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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.continue

User Guide   

Welcome to nubtrek.

The content is presented in small-focused learning units to enable you to
  think,
  figure-out, &
  learn.

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User Guide   

To make best use of nubtrek, understand what is available.

nubtrek is designed to explain mathematics and science for young readers. Every topic consists of four sections.

  nub,

  trek,

  jogger,

  exercise.

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User Guide    

nub is the simple explanation of the concept.

This captures the small-core of concept in simple-plain English. The objective is to make the learner to think about. continue

User Guide    

trek is the step by step exploration of the concept.

Trekking is bit hard, requiring one to sweat and exert. The benefits of taking the steps are awesome. In the trek, concepts are explained with exploratory questions and your thinking process is honed step by step. continue

User Guide    

jogger provides the complete mathematical definition of the concepts.

This captures the essence of learning and helps one to review at a later point. The reference is available in pdf document too. This is designed to be viewed in a smart-phone screen. continue

User Guide    

exercise provides practice problems to become fluent in the concepts.

This part does not have much content as of now. Over time, when resources are available, this section will have curated and exam-prep focused questions to test your knowledge. continue

summary of this topic

Introduction to Complex Numbers

Introduction to Complex Numbers

Voice  

Voice  



Home



Number System Hierarchy


 »  Natural or counting numbers
    →  Counting with hands 1, 2, 3, ...


 »  whole numbers
    →  0 is included


 »  integers
    →  negatives are included. in-teger meaning not-touched (not fractions)


 »  rational numbers
    →  fractions and decimals are included. ratio-nal : numbers in the p/q ratio


 »  real numbers
    →  irrational numbers included. numbers represented as expressions


 »  complex number : `a+ib` form
    →  `a` is the real part
    →  `b` is the imaginary part
    →  `i = sqrt(-1)`
    →  Complex means "consisting of many different parts"

Number System Quick Revision

plain and simple summary

nub

plain and simple summary

nub

dummy

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

simple steps to build the foundation

trek

simple steps to build the foundation

trek

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


Keep tapping on the content to continue learning.
Starting on learning "Number System Hierarchy". ;; A quick revision of number system hierarchy and introduction to complex numbers as an extension of Real Numbers.

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

  • `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

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, ...`

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
  • 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`

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

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

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

Rational and irrational numbers together is called ...

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

The answer is 'real numbers'

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

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

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

comprehensive information for quick review

Jogger

comprehensive information for quick review

Jogger

dummy

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)



           

practice questions to master the knowledge

Exercise

practice questions to master the knowledge

Exercise

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

  • integers
  • natural numbers
  • negative whole numbers

The answer is 'Integers'

your progress details

Progress

About you

Progress

Consider "whole numbers". Which of the following represents the whole numbers?
0;1;2
0 , 1, 2, 3, et cetera
minus
minus 3 minus 2 minus 1 et cetera
The answer is 0, 1, 2, 3, et cetera
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?
equation;no;solution
the equation has no solution
extended;number;system;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
0 , 1, 2, 3, et cetera
minus
minus 3 minus 2 minus 1 et cetera
The answer is minus 3 minus 2 minus 1 et cetera
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?
equation;no;solution
the equation has no solution
number;system;extended;rational
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
0 , 1, 2, 3, et cetera
minus
minus 3 minus 2 minus 1 et cetera
numbers;represented;p;q;queue;integers
numbers that can be represented as p by q where p,q are integers, and q not equals 0
The answer is numbers that can be represented as p by q where p, q are integers, and q not equals 0
When using the rational numbers, we come across problems that are mathematically modeled to x squared equals 2. There is no solution to this in rational number system. So what is done?
equation;has;no;solution
the equation has no solution
extended;irrational;number;system
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
0 , 1, 2, 3, et cetera
minus
minus 3 minus 2 minus 1 et cetera
represented;p;q;queue;integers
numbers that can be represented as p by q where p,q are integers, and q not equals 0
line;on
numbers that can not be represented as p by q and are on the number line
The answer is "numbers that cannot be represented as p by q and are on the number line"
Rational and irrational numbers together is called ...
real
real numbers
not;combined;into
they are not combined into a number system
The answer is 'real numbers'
When using the real numbers, we come across problems that are mathematically modeled to x squared = minus 1. There is no solution to this in real number system. So what is done?
equation;solution;has;no
the equation has no solution
number;system;extended;beyond
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)
To accommodate negative numbers, whole number system is extended into ...
integers;integer
integers
natural
natural numbers
whole;negative
negative whole numbers
The answer is 'Integers'

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