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Thought-Process to Discover Knowledge

Welcome to **nub****trek**.

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

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

Welcome to **nub****trek**.

The content is presented in small-focused learning units to enable you to

think,

figure-out, &

learn.

**Just keep tapping** (or clicking) on the content to continue in the trail and learn. continue

The content is presented in small-focused learning units to enable you to

think,

figure-out, &

learn.

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.

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

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

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

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

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Complex Arithmetic as Numerical Expression

» Real Number arithmetic

→ Addition with closure, commutative, associative, identity, inverse properties

→ Multiplication with closure, commutative, associative, distributive, identity, inverse properties

→ subtraction and division as inverse of addition and multiplication respectively

→ PEMDAS / BODMAS Precedence rule

» Complex number arithmetic

→ extension of real arithmetic with additional number `i`

→ `i` is handled like a variable

→ `i^2 = -1` maps `i` to a real number

*plain and simple summary*

nub

*plain and simple summary*

nub

dummy

Complex arithmetic is the extension of Real numbers arithmetic.

*simple steps to build the foundation*

trek

*simple steps to build the foundation*

trek

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Complex number arithmetic like addition, multiplication, etc are nothing but extension of real number arithmetic.

Starting on learning "Understanding Complex Arithmetic". ;; Complex number arithmetic like addition, multiplication, etc are nothing but extension of real number arithmetic.

What is the complex number system?

- Real number system extended to include solutions to polynomials
- Real number system added with an additional number `sqrt(-1)`
- Model of amplitude and phase of sine waves and elements interacting with sine waves
- All the above

The answer is 'All the above'

For the complex number the following definitions and properties

• addition and subtraction

• multiplication and division

• exponents and roots

• properties like commutative, distributive, associative, etc.

Should one learn each of these afresh?

- No, just extend the definitions and properties of real numbers as though complex number is a numerical expression.
- Yes, `sqrt(-1)` is not a real number so it will not adhere to the definitions and properties of real numbers.

The answer is 'No, Just extend the definitions and properties of real numbers'

One person decides to write all real numbers in the form `3.1+7xx2.4` - that is, two real numbers with `7` appearing as in the expression. How would the representation affect the arithmetic operations, like addition, subtraction, associative, etc?

- All definitions and properties will hold true for the numerical expressions too
- The definitions and properties need to be figured out afresh.

The answer is 'All the definitions and properties will hold true' for the numerical expressions too, as long as the expression is kept in tact.

Consider the example of writing real numbers in the form `a+7b`:

• Two identical numerical expressions may not be evident from the numbers in the expressions. eg: `3.1+7xx2.4 = 4.5+7xx2.2`, though the expression looks different, they evaluate to the same.

• There can be additional properties that is a combination of already existing properties. eg: To add `3.1+7xx2.4 + 1.1+7xx0.2`, add the coefficients `3.1+1.1` and `2.4+0.2`. This looks to be a new definitions for addition. But it is derived from the distributive and associative properties of addition and subtraction.

Complex number is a numerical expression with `sqrt(-1)` as an element. When learning rules and properties of complex numbers, visualize the complex number as a numerical expression of real numbers.

*comprehensive information for quick review*

Jogger

*comprehensive information for quick review*

Jogger

dummy

**Complex Arithmetic Fundamentals: ** The definitions and properties of real number arithmetic is extended to include `sqrt(-1)` as a number that cannot be added or multiplied to other numbers.

*practice questions to master the knowledge*

Exercise

*practice questions to master the knowledge*

Exercise

To understand complex numbers arithmetic operations and properties, which of the following number system is used?

- Integers
- Rational numbers
- Real numbers

The answer is 'Real numbers'

*your progress details*

Progress

*About you*

Progress

What is the complex number system?

polynomials

Real number system extended to include solutions to polynomials

minus;1;square;root

Real number system added with an additional number square root minus 1

model;amplitude;phase

Model of amplitude and phase of sine waves and elements interacting with sine waves

all;above

All the above

The answer is 'All the above'

For the complex number the following definitions and properties ;; addition and subtraction ;; multiplication and division ;; exponents and roots ;; properties like commutative, distributive, associative, etc. ;; Should one learn each of these afresh?

No;just;extend

No, just extend the definitions and properties of real numbers as though complex number is a numerical expression.

yes;square;root;minus

Yes, square root of minus 1 is not a real number so it will not adhere to the definitions and properties of real numbers.

The answer is 'No, Just extend the definitions and properties of real numbers'

One person decides to write all real numbers in the form 3 point 1+7 multiplied by 2 point 4 ; - that is, two real numbers with 7 appearing as in the expression. How would the representation affect the arithmetic operations, like addition, subtraction, associative, etc?

will;hold;true

All definitions and properties will hold true for the numerical expressions too

need;figured;out;afresh

The definitions and properties need to be figured out afresh.

The answer is 'All the definitions and properties will hold true' for the numerical expressions too, as long as the expression is kept in tact.

Consider the example of writing real numbers in the form a+7b: ;; Two identical numerical expressions may not be evident from the numbers in the expressions. for example 3 point 1+7 into 2 point 4 = 4 point 5+7 into 2 point 2, though the expression looks different, they evaluate to the same.;; There can be additional properties that is a combination of already existing properties. eg: To add 3 point 1+7 into 2 point 4; + 1 point 1+7 into 0 point 2, add the coefficients 3 point 1+1 point 1 and 2 point 4+0 point 2. ;; This looks to be a new definitions for addition. But it is derived from the distributive and associative properties of addition and subtraction.

Complex number is a numerical expression with square root minus 1 as an element. When learning rules and properties of complex numbers, visualize the complex number as a numerical expression of real numbers.

Complex arithmetic is the extension of Real numbers arithmetic.

Complex Arithmetic Fundamentals: The definitions and properties of real number arithmetic is extended to include square root minus 1 as a number that cannot be added or multiplied to other numbers.

To understand complex numbers arithmetic operations and properties, which of the following number system is used?

integer;integers

Integers

rational

Rational numbers

real

Real numbers

The answer is 'Real numbers'