__maths__>__Complex Numbers__>__Properties of Complex Number Arithmetic__### Understanding Properties of Complex Arithmetic

Properties of complex number arithmetic is almost same as real number arithmetic as Complex number arithmetic is nothing but extension of real number arithmetic.

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What is 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'

To understand properties of the complex numbers, the following are to be learned

• Properties of addition and subtraction

• Properties of multiplication and division

• Properties of 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'

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

Complex arithmetic is the extension of Real numbers arithmetic.

**Complex Arithmetic Fundamentals: ** The 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
- None of the above

The answer is 'Real numbers'

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