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Algebra of Complex Numbers

Algebra of Complex Numbers

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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

Understanding Complex Arithmetic

plain and simple summary

nub

plain and simple summary

nub

dummy

Complex arithmetic is the extension of Real numbers arithmetic.

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trek

simple steps to build the foundation

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


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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'

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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'

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