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Complex Numbers
The Foundation: Field Axioms
The set of real numbers $R$ is a Field, a mathematical structure defined by a set of rules for a Binary Operation like addition and multiplication.
$$a(b + c) = ab + ac$$
The Distributive Law links multiplication and addition across a field.
$a, b, c$=Any elements within the field(unitless)
$a = 0$
→$0(b + c) = 0$, illustrating the property of the additive identity.
Axiom Checklist: Includes Closure, Associative, Identity, Inverse, and Commutative properties.
Trichotomy Property: For any $a, b \in R$, exactly one of $a < b$, $a = b$, or $a > b$ must be true.
Additive Identity: The number 0, where $a + 0 = a$.
Introduction to Complex Numbers
Complex numbers extend the real number system by introducing the Iota ($i$), allowing for the square roots of negative numbers.
$$i = \sqrt{-1} \implies i^2 = -1$$
Definition of the imaginary unit $i$.
$i$=Imaginary unit (iota)(unitless)
$i^3$
→$-i$
$i^4$
→$1$
Standard Form: A complex number $z$ is written as $x + iy$, where $x$ is the Real Part and $y$ is the Imaginary Part.
Every Real Is Complex: Any real number $a$ is a complex number with imaginary part 0, i.e., $a = a + 0i$.
Purely Imaginary: If the real part $x = 0$, the number is called purely imaginary (e.g., $3i$, $-5i$).
Iota Cycle: Powers of $i$ repeat every four steps: $i^1 = i$, $i^2 = -1$, $i^3 = -i$, $i^4 = 1$, then back to $i$.
Power Shortcut: To find $i^n$, divide $n$ by 4 and use the remainder: $i^{n \bmod 4}$. For example, $i^{13} = i^{13 \bmod 4} = i^1 = i$.
Consecutive Sum: Any four consecutive powers of $i$ always sum to zero: $i^k + i^{k+1} + i^{k+2} + i^{k+3} = 0$.
Algebraic Operations
Operations on complex numbers involve combining like terms (real with real, imaginary with imaginary) and applying the distributive property.
$$(a + bi)(c + di) = (ac - bd) + (ad + bc)i$$
Standard multiplication formula for complex numbers.
$a, c$=Real components(unitless)
$b, d$=Imaginary components(unitless)
$(a + bi)(a - bi)$
→$a^2 + b^2$, a real non-negative number.
Equality: Two complex numbers are equal iff their real and imaginary parts match: $a + bi = c + di \Leftrightarrow a = c$ and $b = d$.
Addition: $(a + bi) + (c + di) = (a+c) + (b+d)i$ — add real parts, add imaginary parts.
Ordered Pair Form: Complex numbers can be expressed as $(a, b)$, where multiplication follows $(a,b)(c,d) = (ac-bd, ad+bc)$.
Scalar Multiplication: $k(a + bi) = ka + kbi$ for any real constant $k$.
Division Technique: To divide $\frac{z_1}{z_2}$, multiply numerator and denominator by the conjugate of $z_2$: $\frac{a+bi}{c+di} \times \frac{c-di}{c-di} = \frac{(ac+bd)+(bc-ad)i}{c^2+d^2}$.
Conjugate Properties and Identities
The Conjugate of a complex number satisfies several important algebraic identities used throughout complex number computations.
$$z\bar{z} = |z|^2 = a^2 + b^2$$
The product of a complex number with its conjugate always yields a real non-negative number equal to the square of its modulus.
$z$=Complex number $a + bi$(unitless)
$\bar{z}$=Conjugate $a - bi$(unitless)
$z$ is real ($b = 0$)
→$z\bar{z} = a^2$, and $\bar{z} = z$ (reals are self-conjugate).
Double Conjugate: $\bar{\bar{z}} = z$ — conjugating twice returns the original number.
Sum Property: $\overline{z_1 + z_2} = \bar{z_1} + \bar{z_2}$ — the conjugate distributes over addition.
Product Property: $\overline{z_1 z_2} = \bar{z_1} \cdot \bar{z_2}$ — the conjugate distributes over multiplication.
Quotient Property: $\overline{\left(\frac{z_1}{z_2}\right)} = \frac{\bar{z_1}}{\bar{z_2}}$ for $z_2 \neq 0$.
Modulus Invariance: $|-z| = |z| = |\bar{z}| = |-\bar{z}|$ — negation and conjugation do not change the modulus.
Product of Moduli: $|z_1 z_2| = |z_1| \cdot |z_2|$ — the modulus of a product equals the product of moduli.
Multiplicative Inverse: $z^{-1} = \frac{\bar{z}}{|z|^2} = \left(\frac{a}{a^2+b^2}, \frac{-b}{a^2+b^2}\right)$ for $z \neq 0$.
The Complex Plane and Modulus
The Argand Diagram visualizes complex numbers as points or vectors, where the x-axis is the real axis and the y-axis is the imaginary axis.
$$|z| = \sqrt{x^2 + y^2}$$
The modulus represents the magnitude or distance from the origin.
$|z|$=Modulus of $z$(unitless)
$x$=Real component coordinate(unitless)
$y$=Imaginary component coordinate(unitless)
$y = 0$
→$|z| = \sqrt{x^2} = |x|$, reducing to the absolute value of a real number.
Pythagorean Link: The modulus formula is derived directly from the distance formula in Cartesian coordinates.
Triangle Inequality: $|z_1 + z_2| \le |z_1| + |z_2|$ — equality holds when $z_1$ and $z_2$ point in the same direction.
Reverse Triangle Inequality: $|z_1 + z_2| \ge ||z_1| - |z_2||$ — the minimum modulus of a sum.
Real Points: Points on the x-axis represent real numbers (imaginary part 0).
Imaginary Points: Points on the y-axis represent purely imaginary numbers (real part 0).
No Ordering: Unlike $R$, complex numbers cannot be ordered — statements like $z_1 > z_2$ are meaningless for non-real complex numbers.
Polar Form and De Moivre's Theorem
The Polar Form describes a complex number using its distance $r$ from the origin and the Argument $\theta$, facilitating complex exponentiation.
$$(\cos \theta + i \sin \theta)^n = \cos n\theta + i \sin n\theta$$
De Moivre's Theorem for raising complex numbers to integer powers.
$\theta$=Argument (angle)(radians/degrees)
$n$=Integer exponent(unitless)
$n = 2$
→Double angle identities: $\cos 2\theta + i\sin 2\theta$.
Polar Form: $z = r(\cos\theta + i\sin\theta)$ where $r = \sqrt{x^2+y^2}$ is the modulus and $\theta = \tan^{-1}(y/x)$ is the argument.
Multiplication in Polar: $z_1 z_2 = r_1 r_2[\cos(\theta_1+\theta_2) + i\sin(\theta_1+\theta_2)]$ — multiply moduli, add arguments.
De Moivre's Power: $[r(\cos\theta + i\sin\theta)]^n = r^n(\cos n\theta + i\sin n\theta)$ — raise modulus to the $n$th power, multiply argument by $n$.
Application Strategy
Worked Examples
Problem Statement
Divide $(2 + 3i)$ by $(4 - i)$ and express the result in $a + bi$ form.
Conjugate multiplicationComplex division
Strategic Toolkit
$\text{Multiply by conjugate: } \frac{z_1}{z_2} \times \frac{\bar{z_2}}{\bar{z_2}}$
Step 1
Identify Denominator and Conjugate
The denominator is $4 - i$. To remove the imaginary part, find its conjugate.
$\bar{z} = 4 + i$
Step 2
Multiply Numerator and Denominator
Scale the fraction by the conjugate of the denominator.
$\frac{2+3i}{4-i} \times \frac{4+i}{4+i} = \frac{(2+3i)(4+i)}{(4-i)(4+i)}$
Step 3
Expand and Simplify
Apply the FOIL method to the numerator and the difference of squares/identity $z\bar{z} = a^2 + b^2$ to the denominator.
Numerator: $8 + 2i + 12i + 3i^2 = 8 + 14i - 3 = 5 + 14i$
Denominator: $4^2 + 1^2 = 17$
Step 4
Final Result
Divide both real and imaginary parts by the denominator.
$\frac{5}{17} + \frac{14}{17}i$
Problem Statement
Prove that the product of $z = a + bi$ and its conjugate $\bar{z}$ is always a non-negative real number.
Conjugate definitionComplex multiplication
Strategic Toolkit
$z\bar{z} = |z|^2$
Step 1
Setup the Product
Write $z$ and its conjugate $\bar{z}$ explicitly.
$z = a + bi, \quad \bar{z} = a - bi$
Step 2
Multiply and Use Identities
Use the difference of squares identity $(x+y)(x-y) = x^2 - y^2$.
$z\bar{z} = (a + bi)(a - bi) = a^2 - (bi)^2$
Step 3
Substitute i²
Since $i^2 = -1$, the imaginary term becomes positive and real.
$a^2 - (b^2 i^2) = a^2 - b^2(-1) = a^2 + b^2$
Step 4
Conclusion
Since $a$ and $b$ are real, their squares are non-negative, so their sum is non-negative and real.
$a^2 + b^2 \ge 0$
Problem Statement
Express $1 + i\sqrt{3}$ in polar form $r(\cos\theta + i\sin\theta)$.
Polar formModulusArgument
Strategic Toolkit
$r = \sqrt{x^2 + y^2}, \quad \theta = \tan^{-1}\left(\frac{y}{x}\right)$
Step 1
Identify Real and Imaginary Parts
Compare with $x + iy$ to extract the components.
$x = 1, \quad y = \sqrt{3}$
Step 2
Compute the Modulus
Use $r = \sqrt{x^2 + y^2}$.
$r = \sqrt{1^2 + (\sqrt{3})^2} = \sqrt{1 + 3} = \sqrt{4} = 2$
Step 3
Compute the Argument
Use $\theta = \tan^{-1}(y/x)$ and check the quadrant (both $x, y > 0$ so Q1).
$\theta = \tan^{-1}\left(\frac{\sqrt{3}}{1}\right) = \tan^{-1}(\sqrt{3}) = 60°$
Step 4
Write in Polar Form
Substitute $r$ and $\theta$ into $r(\cos\theta + i\sin\theta)$.
$1 + i\sqrt{3} = 2(\cos 60° + i\sin 60°)$
Problem Statement
Simplify $i^{47}$.
Powers of iCyclic pattern
Strategic Toolkit
$i^n = i^{n \bmod 4}$
Step 1
Divide the Exponent by 4
Find the remainder when 47 is divided by 4.
$47 = 4 \times 11 + 3$, so remainder $= 3$
Step 2
Use the Iota Cycle
The remainder tells us which power in the cycle $\{i^0=1, i^1=i, i^2=-1, i^3=-i\}$ to use.
$i^{47} = i^3 = -i$
Memorize formulas
Complex Modulus
Calculates the distance of a complex number from the origin on the Argand diagram.
Formula
$|z| = \sqrt{a^2 + b^2}$
Multiplicative Inverse (Pair Form)
Used to find the inverse of $z = (a, b)$ such that $z \cdot z^{-1} = (1, 0)$.
Formula
$z^{-1} = \left( \frac{a}{a^2 + b^2}, \frac{-b}{a^2 + b^2} \right)$
Complex Multiplication
Standard formula for the product of two complex numbers in ordered pair form.
Formula
(a, b)(c, d) = (ac - bd, ad + bc)
Conjugate Product
Product of a complex number with its conjugate equals the square of its modulus.
Formula
$z\bar{z} = a^2 + b^2 = |z|^2$
Polar Conversion (θ)
Finding the argument for the polar form. Always check the quadrant!
Formula
$\theta = \tan^{-1}\left(\frac{y}{x}\right)$
De Moivre's Theorem
Essential for raising complex numbers to large integer powers.
Formula
$[r(\cos \theta + i \sin \theta)]^n = r^n(\cos n\theta + i \sin n\theta)$
Review
1
Which of the following properties is NOT satisfied by the set of complex numbers?
Ans
Trichotomy Property (Order Axioms)
Review
Tap to reveal
Check which axioms complex numbers actually satisfy
The complex numbers behave like a field, so multiplication is commutative and the usual identity and distributive laws hold.
Notice the missing order structure
There is no consistent way to order all complex numbers, so the trichotomy property does not apply to \mathbb{C}.
The property not satisfied is the Trichotomy Property (Order Axioms).
2
Simplify the expression: $i^{101} + i^{102} + i^{103} + i^{104}$
Ans
$0$
Review
Tap to reveal
Use the 4-cycle of powers of i
Powers of $i$ repeat every four terms: $i, -1, -i, 1$. Reducing the exponents mod 4 gives the values of each term immediately.
$$i^{101}=i,\quad i^{102}=-1,\quad i^{103}=-i,\quad i^{104}=1$$
Add the consecutive terms
The positive and negative parts cancel in pairs, so the sum is zero.
$$i - 1 - i + 1 = 0$$
3
If the multiplicative inverse of a complex number $z$ is $\frac{3}{25} - \frac{4}{25}i$, find $|z|$.
Ans
$5$
Review
Tap to reveal
Find the modulus of the inverse
Treat the given inverse as a complex number and compute its modulus directly.
$$|z^{-1}| = \sqrt{\left(\frac{3}{25}\right)^2 + \left(\frac{4}{25}\right)^2} = \frac{1}{5}$$
Use the reciprocal modulus rule
For any nonzero complex number, $|z^{-1}| = \frac{1}{|z|}$. Therefore the modulus of $z$ is the reciprocal of the modulus of its inverse.
$$|z| = 5$$
4
If $|z_1| = 3$ and $|z_2| = 4$, what is the maximum possible value of $|z_1 + z_2|$?
Ans
$7$
Review
Tap to reveal
Apply the triangle inequality
The modulus of a sum is never larger than the sum of the moduli, so the largest possible value is obtained by aligning the two complex numbers in the same direction.
$$|z_1+z_2| \le |z_1| + |z_2| = 3 + 4$$
Take the maximum case
When the two vectors point the same way, the upper bound is achieved exactly.
$$|z_1+z_2|_{\max} = 7$$
5
Express the complex number $\sqrt{3} + i$ in polar form.
Ans
$2(\cos 30^\circ + i \sin 30^\circ)$
Review
Tap to reveal
Compute the modulus
For $\sqrt{3}+i$, the modulus is the distance from the origin.
$$r = \sqrt{(\sqrt{3})^2 + 1^2} = \sqrt{4} = 2$$
Find the argument from the right triangle
The point lies in the first quadrant and $\tan\theta = \frac{1}{\sqrt{3}}$, which gives $\theta = 30^\circ$.
$$\sqrt{3}+i = 2(\cos 30^\circ + i\sin 30^\circ)$$
6
If $z_1 = 5 + 5i$ and $z_2 = 3 - 2i$, which expression correctly represents $\overline{\left(\frac{z_1}{z_2}\right)}$?
Ans
$\frac{\bar{z_1}}{\bar{z_2}} = \frac{5-5i}{3+2i}$
Review
Tap to reveal
Apply the conjugate quotient rule
Conjugation distributes over division, so the conjugate of a quotient is the quotient of the conjugates.
$$\overline{\left(\frac{z_1}{z_2}\right)} = \frac{\bar z_1}{\bar z_2}$$
Substitute the conjugates of each number
Conjugate each given complex number by changing the sign of its imaginary part.
$$\frac{\bar z_1}{\bar z_2} = \frac{5-5i}{3+2i}$$
7
Using De Moivre's Theorem, what is $(\cos 15° + i\sin 15°)^{12}$?
Ans
$-1$
Review
Tap to reveal
Raise the argument by the exponent
De Moivre's Theorem multiplies the angle by the power.
$$\left(\cos 15^\circ + i\sin 15^\circ\right)^{12} = \cos(12 \cdot 15^\circ) + i\sin(12 \cdot 15^\circ)$$
Evaluate the resulting angle
Since $12 \cdot 15^\circ = 180^\circ$, the sine term vanishes and the cosine term is $-1$.
$$\cos 180^\circ + i\sin 180^\circ = -1$$
8
If $|z_1| = 5$ and $|z_2| = 3$, what is $|z_1 \cdot z_2 \cdot \bar{z_2}|$?
Ans
$45$
Review
Tap to reveal
Use modulus multiplicativity
The modulus of a product is the product of the moduli, and a conjugate has the same modulus as the original number.
$$|z_1 z_2 \bar{z_2}| = |z_1|\,|z_2|\,|\bar z_2| = 5\cdot 3\cdot 3$$
Evaluate the product
Now multiply the values directly.
$$5 \cdot 3 \cdot 3 = 45$$
9
Simplify $\frac{i}{1+i}$ into the form $a + bi$.
Ans
$\frac{1}{2} + \frac{1}{2}i$
Review
Tap to reveal
Rationalize the denominator
Multiply numerator and denominator by the conjugate of $1+i$, which is $1-i$.
$$\frac{i}{1+i}\cdot\frac{1-i}{1-i} = \frac{i(1-i)}{(1+i)(1-i)}$$
Simplify the fraction
Use $i^2=-1$ and $(1+i)(1-i)=2$ to rewrite the expression in $a+bi$ form.
$$\frac{i-i^2}{2} = \frac{1+i}{2} = \frac{1}{2}+\frac{1}{2}i$$
10
What are the complex roots of the equation $x^2 + 9 = 0$?
Ans
$x = \pm 3i$
Review
Tap to reveal
Isolate x squared
Move the constant term to the other side so the equation becomes a square root problem.
$$x^2 = -9$$
Take the complex square root
Since $\sqrt{-1}=i$, the square roots of $-9$ are $\pm 3i$.
$$x = \pm 3i$$
11
If $z\bar{z} = 50$ and $z + \bar{z} = 10$, what is the imaginary part of $z$?
Ans
$\pm 5$
Review
Tap to reveal
Use the sum to find the real part
Write $z=a+bi$. Then $z+\bar z=2a$, so the given sum determines the real part immediately.
$$2a = 10 \Rightarrow a = 5$$
Use the product to find the imaginary part
Since $z\bar z = a^2+b^2$, substitute $a=5$ and solve for $b$.
$$25+b^2=50 \Rightarrow b^2=25 \Rightarrow b=\pm 5$$
12
What is the value of $i^{23} + i^{46}$?
Ans
$-i - 1$
Review
Tap to reveal
Reduce each exponent modulo 4
Powers of $i$ repeat every four terms, so only the remainders matter.
$$i^{23}=i^3=-i,\quad i^{46}=i^2=-1$$
Add the two values
Combine the simplified results directly.
$$-i + (-1) = -i - 1$$
Question 1 of 12
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1
Which of the following properties is NOT satisfied by the set of complex numbers?