Questions and Answers
What is a complex number?
An ordered pair (x, y) with x ∈ R and y ∈ R.
The imaginary unit is denoted by ____.
i
How is addition of complex numbers defined?
(a, b) + (c, d) = (a + c, b + d)
How is multiplication of complex numbers defined?
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The equation $z^2 + 1 = 0$ has roots in complex numbers.
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What does $i^2$ equal?
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Study Notes
The Complex Numbers
- A complex number ( z ) is represented as an ordered pair ( (x, y) ) where ( x ) and ( y ) are real numbers.
- The real part of ( z ) is denoted as ( \text{Re} z = x ) and the imaginary part as ( \text{Im} z = y ).
- The set of complex numbers is denoted by ( \mathbb{C} ).
- The number ( x ) corresponds to the complex form ( (x, 0) ) and ( i ) represents ( (0, 1) ).
- A field isomorphism exists between the real numbers ( \mathbb{R} ) and a subset of ( \mathbb{C} ).
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Addition of complex numbers is defined as:
( (a, b) + (c, d) = (a + c, b + d) ) -
Multiplication of complex numbers is defined as:
( (a, b)(c, d) = (ac - bd, bc + ad) ) -
Commutative Properties:
- ( z_1 + z_2 = z_2 + z_1 )
- ( z_1 z_2 = z_2 z_1 )
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Associative Property:
( z_1 (z_2 + z_3) = z_1 z_2 + z_1 z_3 ) - Complex numbers can be expressed in the form ( z = x + iy ) by associating ( (x, 0) ) with ( x ) and ( (0, y) ) with ( iy ).
- The imaginary unit ( i ) satisfies the equation ( i^2 = -1 ).
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Division of complex numbers, where ( z \neq 0 ), is defined as:
( \frac{1}{z} = \frac{1}{x + iy} = \frac{x - iy}{x^2 + y^2} ) - This operation leads to results of the form:
( \frac{x_1 + iy_1}{x_2 + iy_2} = \frac{x_1 x_2 + y_1 y_2}{x_2^2 + y_2^2} + i\frac{y_1 x_2 - x_1 y_2}{x_2^2 + y_2^2} )
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Description
This quiz explores the fundamentals of complex numbers, including their representation as ordered pairs and the operations of addition and multiplication. Understand the properties such as commutativity and associativity, as well as the significance of real and imaginary parts. Test your knowledge of complex number theory with this engaging quiz.
