Math101 · Independent Ontario learning library
Math101learn.math101.caComplex Numbers
A complex number has form $a+bi$, where $a,b$ are real and $i^2=-1$. The real part is $a$ and imaginary part is $b$.
Complex numbers make algebraically missing roots available and encode planar rotation, oscillation, waves, and electrical systems in a compact form.
Intuition and core definition
A complex number has form $a+bi$, where $a,b$ are real and $i^2=-1$. The real part is $a$ and imaginary part is $b$. Complex numbers extend the real numbers so every nonconstant degree-$n$ polynomial has exactly $n$ complex roots when multiplicity is counted, including equations such as $x^2+1=0$.
Notation, language, and conditions
$\mathbb C$ denotes the complex numbers. $\operatorname{Re}(z)=a$, $\operatorname{Im}(z)=b$, and the conjugate of $a+bi$ is $a-bi$. Two complex numbers are equal exactly when their real parts and imaginary parts match. Powers of $i$ repeat with period four: $i,i^2=-1,i^3=-i,i^4=1$.
Why this idea matters
Complex numbers extend algebra to equations with negative discriminants and encode two-dimensional magnitude and direction in a single number.
A dependable method
- For addition or subtraction, combine real parts and imaginary parts separately.
- For multiplication, distribute and replace every $i^2$ by $-1$.
- Simplify powers of $i$ using exponent modulo $4$.
- For division, multiply numerator and denominator by the denominator’s conjugate.
- Write the result in standard $a+bi$ form and check by reversing the operation.
Worked example
Representations and interpretation
The complex plane plots $a+bi$ as $(a,b)$. Addition becomes vector addition, conjugation reflects across the real axis, and magnitude $|a+bi|=\sqrt{a^2+b^2}$ is distance from the origin.
Reasoning about variations
The symbol $|z|$ for a complex number means modulus, not applying absolute value independently to components. Also, $\sqrt{-9}=3i$ under the principal convention, while solving $z^2=-9$ gives $z=\pm3i$.
Common mistakes
How to check your work
- Separate real and imaginary components and recompute them independently.
- Plot addition geometrically as a vector sum.
- Multiply a quotient by its divisor to recover the dividend.
Practice
- Simplify $(2+3i)+(5-7i)$.
- Simplify $i^{23}$.
- Find the conjugate of $4-9i$.
Answers and brief solutions
Show answers
- $7-4i$ Real parts give $7$ and imaginary coefficients give $-4$.
- $-i$ $23$ leaves remainder $3$ modulo $4$, so $i^{23}=i^3=-i$.
- $4+9i$ The sign of the imaginary part changes.
Synthesis and transfer
Multiplication by $i$ rotates a point ninety degrees in the complex plane; applying it four times returns the original point and explains the power cycle geometrically.
For $z=2+i$, multiplication by $i$ gives $iz=-1+2i$, sending the point $(2,1)$ to $(-1,2)$. The distance from the origin is unchanged, and the directed angle increases by $90^\circ$. A second multiplication gives $-2-i$, and four applications return $2+i$, matching the four-step cycle of powers of $i$. Conjugation behaves differently: it reflects $(2,1)$ across the real axis to $(2,-1)$ rather than rotating it. These geometric actions provide checks on sign-heavy symbolic products and prepare the transition to polar form, where multiplication combines scale and rotation directly.
Related topics
Teaching and accessibility note
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
Simplify $(2+3i)+(5-7i)$.
- Real parts give $7$ and imaginary coefficients give $-4$.
End of lesson
Nice work making it this far.
Understanding grows through return visits. Save this lesson, try the practice, or continue when you are ready.
