Unit 1 - Lesson 1: Introduction to Complex Numbers
Master the extended numerical system: The imaginary unit i, cyclic powers of i, the standard complex form z = a + bi, complex conjugates, and solving quadratic equations over C.
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The Origin of the Imaginary Unit
Over the set of Real Numbers $\mathbb{R}$, equations such as $x^2 + 1 = 0$ have no solution because $x^2 = -1$ has no real square root. Mathematicians defined the imaginary unit $i$ such that:
Cyclic Powers of i
Integer powers of $i$ repeat in a 4-step cycle:
Definition of a Complex Number
A complex number $z$ is any number written in the standard form:
Here, $a$ is called the Real Part, and $b$ is called the Imaginary Part. The set of all complex numbers is denoted by $\mathbb{C}$.
Complex Conjugates
The complex conjugate of $z = a + bi$ is $\bar{z} = a - bi$.
$(a + bi)(a - bi) = a^2 - (bi)^2 = a^2 - b^2(-1) = a^2 + b^2$ (a non-negative real number).
Step-by-Step Solved Examples
Example 1 Simplify: i⁴³
Solution:
Divide 43 by 4: $43 = 4 \times 10 + 3$ (remainder 3).
Therefore, $i^{43} = i^3 = -i$.
Example 2 Compute: (3 + 2i) + (4 - 5i)
Solution:
Combine real and imaginary parts:
Real: $3 + 4 = 7$
Imaginary: $2i - 5i = -3i$
Result = $7 - 3i$.
Example 3 Simplify the fraction: (5 - i) / (2 + i)
Solution:
Multiply numerator and denominator by the conjugate $(2 - i)$:
Denominator: $(2 + i)(2 - i) = 2^2 + 1^2 = 5$
Numerator: $(5 - i)(2 - i) = 10 - 5i - 2i + i^2 = 10 - 7i - 1 = 9 - 7i$
Result = $\frac{9 - 7i}{5} = \frac{9}{5} - \frac{7}{5}i$.
Practice Assessment
Select the correct option for each problem and click submit to check your score.
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