Lernzettel: Advanced Complex Numbers and Roots

📋 Course Outline

1. Complex numbers: modulus, argument and conjugate
2. Cube roots and fourth roots of unity
3. Complex numbers in polar form and trigonometry
4. Functions: types, graphs and periodicity
5. Sequences and series: arithmetic and geometric
6. Factorials, permutations and combinations
7. Partial fractions and synthetic division
8. Trigonometric identities and allied angles
9. Limits and continuity with derivatives
10. Vectors: magnitude, dot and cross products

📖 1. Complex numbers: modulus, argument and conjugate

🔑 Key Concepts & Definitions

• Modulus of a complex number : Modulus is the non-negative distance of a complex number from the origin in the complex plane.
• Argument of a complex number : Argument is the angle a complex number makes with the positive real axis in the complex plane.
• Complex conjugate : Complex conjugate is formed by changing the sign of the imaginary part of a complex number.
• Polar form : Polar form expresses a complex number using its modulus and argument as $r(\cos\theta+i\sin\theta)$.

📝 Essential Points

• For $Z=a+bi$, the modulus is $|Z|=\sqrt{a^2+b^2}$ and is always non-negative.
• For $Z=a+bi$, the conjugate is $\overline{Z}=a-bi$.
• If $Z=-2+3i$, then $\overline{Z}=-2-3i$.
• For $Z=\sqrt{2}(\cos45^\circ+i\sin45^\circ)$, the modulus is $\sqrt{2}$ and the argument is $45^\circ$.
• For cube roots of unity, the conjugate of $\omega$ equals $\omega^2$ (since $\omega$ and $\omega^2$ are conjugates).
• For cube roots of unity, $\omega+\omega^2=-1$ and $\omega^2+\omega=-1$ (so their sum is real).

💡 Memory Hook

Conjugate flips the sign of $i$: $a+bi \to a-bi$; modulus is the distance $\sqrt{a^2+b^2}$; argument is the angle $\theta$.

📖 2. Cube roots and fourth roots of unity

🔑 Key Concepts & Definitions

• Fourth roots of unity : Fourth roots of unity are complex numbers $z$ satisfying $z^4=1$.
• Cube roots of unity : Cube roots of unity are complex numbers $z$ satisfying $z^3=1$.
• Equivalent powers of a root of unity : Equivalent powers of a root of unity are exponents that produce the same complex value.
• Polar form of a complex number : Polar form expresses $x+iy$ as $r(\cos\theta+i\sin\theta)$ with $r\ge 0$ and angle $\theta$.

📝 Essential Points

• The sum of the four fourth roots of unity equals 0.
• The product of all four fourth roots of unity equals 1.
• The four fourth roots of 81 are $\pm 3,\pm 3i$.
• For any $n\in\mathbb Z$, $\omega^n$ is equivalent to one of $1,\omega,\omega^2$.
• If $\omega$ is a primitive fourth root of unity, then $\omega^2=-1$ and $\omega^4=1$.
• For $x+iy=r(\cos\theta+i\sin\theta)$, the angle satisfies $\theta=\tan^{-1}\left(\frac{y}{x}\right)$.

💡 Memory Hook

Fourth roots: sum cancels to 0, product stays 1; exponents reduce mod 4 (same value for $n,n+4$).

📖 3. Complex numbers in polar form and trigonometry

📖 4. Functions: types, graphs and periodicity

🔑 Key Concepts & Definitions

• Function : A function maps each input from its domain to exactly one output in its codomain.
• Domain : The domain is the set of all allowed inputs for a function.
• Range : The range is the set of all outputs a function actually produces.
• Periodic function : A periodic function repeats its values in regular intervals along the input axis.

📝 Essential Points

• A function must assign one and only one output to each input.
• A periodic function has a positive period T such that f(x+T)=f(x) for all x in its domain.
• If f(x+T)=f(x) holds, then any multiple of T is also a period.
• The graph of a function passes the vertical line test: each x-value corresponds to exactly one y-value.
• For a periodic function, the graph repeats horizontally every period T.
• The period is the smallest positive value T that makes f(x+T)=f(x).

💡 Memory Hook

Periodicity: shift right by T and the graph lands exactly on itself (f(x+T)=f(x)).

📖 5. Sequences and series: arithmetic and geometric

🔑 Key Concepts & Definitions

• Arithmetic progression : Arithmetic progression is a sequence where the difference between consecutive terms is constant.
• Arithmetic mean : Arithmetic mean is the average of two numbers, found by adding them and dividing by 2.
• Arithmetic series : Arithmetic series is the sum of terms of an arithmetic progression.
• Geometric progression : Geometric progression is a sequence where the ratio of consecutive terms is constant.
• Geometric mean : Geometric mean is the mean of two numbers obtained as the square root of their product.

📝 Essential Points

• Arithmetic mean between x−3 and x+5 equals x+1.
• If the arithmetic mean of a and b is between an and bn, then the mean is (an+bn)/(an−1+bn−1).
• Sum of n arithmetic means between a and b equals n(a+b)/2.
• Sum of n terms of an A.P with first term a and common difference d equals n/2{2a+(n−1)d}.
• 21st term of 2+4+6+… equals 40.
• Sum of A.P −7+(−5)+(−3)+… up to 6 terms equals −12.

💡 Memory Hook

A.P: difference stays constant; G.P: ratio stays constant; A.M uses + and /2; G.M uses × and √.

📖 6. Factorials, permutations and combinations

🔑 Key Concepts & Definitions

• Factorial notation : Factorial notation $n!$ represents the product of all positive integers from 1 to $n$.
• Permutation notation : Permutation notation $nP_r$ counts ordered selections of $r$ objects from $n$ objects.
• Combination notation : Combination notation $nC_r$ counts unordered selections of $r$ objects from $n$ objects.
• Factorial form : Factorial form rewrites products like $(n-1)(n-2)\cdots(n-r+1)$ using factorials.

📝 Essential Points

• $5!=120$ and $0!=1$.
• $(n-1)(n-2)\cdots(n-r+1)=\dfrac{(n-1)!}{(n-r)!}$.
• $(n+1)(n)(n-1)\cdots 3\cdot2\cdot1=\dfrac{(n+1)!}{3!(n-2)!}$.
• $\dfrac{(n-1)!(n-3)!}{n!(n-2)!}=\dfrac{n-2}{n}$.
• $nP_0=1$ and $nP_1=n$.
• If $r=n$, then $nP_r=n!$ and $nC_r=1$.

💡 Memory Hook

Factorials are products, permutations are order, combinations are no order.

📖 7. Partial fractions and synthetic division

🔑 Key Concepts & Definitions

• Synthetic division : Synthetic division is an algorithm that evaluates a polynomial at a given number and produces the quotient using only coefficients.
• Factor theorem : The factor theorem states that a polynomial has a factor $(x-a)$ exactly when the polynomial evaluates to zero at $x=a$.
• Remainder theorem : The remainder theorem says that dividing a polynomial by $(x-a)$ leaves remainder $f(a)$.
• Allied angles : Allied angles are angle pairs that add up to $180^\circ$.

📝 Essential Points

• Synthetic division uses only addition, multiplication, subtraction, and division steps, not long polynomial division.
• If $(x-2)$ is a factor of $ax^3-12x+4$, then $a=2$.
• If $(x-2)$ is a factor of $x^3+2x^2+kx+4$, then $k=10$.
• A factor $(x+a)$ of $f(x)=x^n-a^n$ occurs when $x=-a$ makes $f(x)=0$.
• For $f(x)=x^n-a^n$ with $n$ a positive integer, a factor is $x+a$ (equivalently $x-(-a)$).
• Allied angles are formed by angles $\theta$ and $180^\circ-\theta$ (so they sum to $180^\circ$).

💡 Memory Hook

Factor check: plug in $x=a$—if $f(a)=0$, then $(x-a)$ is a factor.

📖 8. Trigonometric identities and allied angles

🔑 Key Concepts & Definitions

• Angle addition identity : Angle addition identity : It rewrites $cos(\alpha+\beta)$ and $cos(\alpha-\beta)$ in terms of $sin$ and $cos$ of $\alpha$ and $\beta$.
• Sine addition identity : Sine addition identity : It rewrites $sin(\alpha+\beta)$ using $sin$ and $cos$ of $\alpha$ and $\beta$.
• Sine and cosine sum-to-product : Sum-to-product : It converts expressions like $2\sin\left(\frac{P+Q}{2}\right)\cos\left(\frac{P-Q}{2}\right)$ into a simpler sum or difference of sines.
• Periodic function : Periodic function : A function is periodic if shifting the input by a fixed positive period leaves the output unchanged.

📝 Essential Points

• $\cos(\alpha+\beta)+\cos(\alpha-\beta)=2\cos\alpha\cos\beta$.
• $\cos(\alpha+\beta)-\cos(\alpha-\beta)=-2\sin\alpha\sin\beta$.
• $2\sin\left(\frac{P+Q}{2}\right)\cos\left(\frac{P-Q}{2}\right)=\sin P+\sin Q$.
• $2\sin\left(\frac{P+Q}{2}\right)\cos\left(\frac{P-Q}{2}\right)$ is a sum-to-product form that produces a sine sum.
• Trigonometric functions are periodic functions.
• The domain of $y=\cot x$ is $\mathbb{R}$ with $x\neq n\pi$.

💡 Memory Hook

Cosine-sum trick: add $\cos(\alpha\pm\beta)$ to get $2\cos\alpha\cos\beta$; subtract to get $-2\sin\alpha\sin\beta$.

📖 9. Limits and continuity with derivatives

🔑 Key Concepts & Definitions

• Derivative limit definition : Derivative limit definition : The derivative at a point is obtained as a limit of the difference quotient as the increment tends to 0.
• Continuity at a point : Continuity at a point : A function is continuous at $a$ when $\lim_{x\to a} f(x)=f(a)$.
• Difference quotient : Difference quotient : The expression $\frac{f(x)-f(a)}{x-a}$ measures the average rate of change between $a$ and $x$.
• Power rule : Power rule : For $f(x)=x^n$, the derivative is $f'(x)=n x^{n-1}$.
• Chain rule : Chain rule : For $f(x)=g(h(x))$, the derivative is $f'(x)=g'(h(x))\,h'(x)$.

📝 Essential Points

• $\lim_{\delta x\to 0}\frac{f(x+\delta x)-f(x)}{\delta x}=f'(x)$, and the source also shows the same idea at $x=0$ and $x=a$.
• $\lim_{x\to a}\frac{f(x)-f(a)}{x-a}=f'(a)$, matching the derivative at the point $a$.
• For $h\to 0$, $\frac{f(a+h)-f(a)}{h}$ has the derivative form shown in the source as a ratio involving $f'(a+h)$ and $f'(a)$.
• If $f(x)=x^{1/3}$, then $f'(8)=\frac{1}{12}$.
• If $f(x)=(1+x)^n$, then $f'(0)=n$.
• If $f(x)=\sqrt{a+x}$, then $f'(0)>f'(1)$ (the correct option in the source).

💡 Memory Hook

Difference quotient → derivative: $\frac{f(x)-f(a)}{x-a}$ as $x\to a$.

📖 10. Vectors: magnitude, dot and cross products

🔑 Key Concepts & Definitions

• Parallel vectors : Parallel vectors are non-zero vectors whose angle between them is 0 or π.
• Perpendicular vectors : Perpendicular vectors are non-zero vectors whose angle between them is π/2.
• Dot product : Dot product is a scalar quantity that measures how aligned two vectors are.
• Cross product : Cross product is a vector quantity perpendicular to both input vectors with magnitude tied to the sine of the angle.
• Scalar triple product : Scalar triple product is a scalar formed from three vectors using a dot and a cross product.

📝 Essential Points

• If the angle between vectors is π/2, then the vectors are perpendicular.
• If the angle between vectors is 0 or π, then the vectors are parallel.
• For unit vectors, $j\hat{}\times\u0000k\hat{}=\u0000i\hat{}$ and $\u0000k\hat{}\times\u0000j\hat{}=-\u0000i\hat{}$.
• If $\vec u\times\vec v=0$, then the vectors are parallel.
• If $\vec u\times\vec v=0$ and $\vec u\cdot\vec v=0$, then either $\vec u=0$ or $\vec v=0$.
• If $\vec a$ and $\vec b$ are non-zero and $\vec a\times\vec b=\vec 0$, then $\vec a$ and $\vec b$ are parallel (angle is 0 or π).

💡 Memory Hook

Dot = alignment (cos), Cross = perpendicular area (sin).

📊 Synthesis Tables

Cube roots and fourth roots of unity (key facts)

Object	Sum	Product
cube roots of unity	-1	1
fourth roots of unity	0	1

⚠️ Common Pitfalls & Confusions

1. Mixing up modulus and argument: modulus is |a+bi|=√(a^2+b^2) (always non-negative), while argument is the angle with the positive real axis.
2. For conjugates, forgetting it changes only the sign of the imaginary part: if Z=a+bi then Z̄=a−bi (not a+bi).
3. Using the wrong “equivalent powers” rule: for cube roots, ω^n reduces to 1, ω, or ω^2 (not mod 4).
4. Confusing “sum of cube roots of unity” with “sum of all complex roots of unity”: the source distinguishes these results.
5. In polar form, using θ=tan^{-1}(x/y) instead of θ=tan^{-1}(y/x).
6. For periodic functions, stating the period is any positive T without checking the smallest positive period.
7. For partial fractions, assuming the form is always Ax+B; the source shows different denominators (e.g., x^2−1 gives Ax+B over x^2−1, and x^2(x^2−1) needs multiple terms).

✅ Exam Checklist

1. Identify the type of number: √−1 is called a complex number.
2. Compute powers/inverses of i: evaluate (−i)^19 and the multiplicative inverse of −i.
3. Find real part and modulus from expressions like 1+3i, 3√6−√−12, and −5i.
4. Use conjugation: if Z=−2+3i then write Z̄.
5. Use cube roots of unity facts: (3+ω)(3+ω^2), ω+ω^2, and the conjugate of ω.
6. Use cube roots of unity: list cube roots of −1 and 27, and compute ω−1 and ω^29+ω^28+1.
7. Use fourth roots of unity facts: sum of the four fourth roots is 0 and product is 1; find fourth roots of 81.
8. Convert to polar form: express 1+i as √2(cos45°+i sin45°) and use θ=tan^{-1}(y/x).
9. Graph/periodicity basics: a function passes the vertical line test, and for periodic f, f(x+T)=f(x) with T the smallest positive period.
10. Arithmetic sequences: compute arithmetic mean and sums of n arithmetic means and n terms of an A.P (including the given numeric examples).
11. Geometric sequences: use common ratio constraints, geometric mean, and sums of finite/infinite geometric series (including the convergence question).
12. Factorials/permutations/combinations: use n!, nPr, nCr, factorial-form rewrites, and special cases like nP0=1 and nCr=1 when r=n.
13. Synthetic division/factor theorem: use f(a)=0 to confirm (x−a) is a factor and apply the given factor questions (e.g., x−2 factor gives a or k).
14. Trigonometric identities/allied angles: use cos(α+β)±cos(α−β) forms and allied angles θ and 180°−θ summing to 180°; apply the source’s allied-angle example choices.