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Thought-Process to Discover Knowledge

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mathsComplex NumbersAlgebra of Complex Numbers

Understanding Complex Arithmetic

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

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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
• All the above

The answer is 'All the above'

For the complex number the following definitions and properties
•  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.
• 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
• 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.

Complex arithmetic is the extension of Real numbers arithmetic.

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.

Solved Exercise Problem:

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

• Integers
• Rational numbers
• Real numbers
• Real numbers