Complex numbers

Explanation

• A quadratic with real coefficients may have two real roots, one repeated real root or a complex-conjugate pair, determined by the sign of $b^2-4ac$. Use factorisation, completing the square or $x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$, with $\sqrt{-q}=i\sqrt q$ for $q>0$.
• For a cubic or quartic, sufficient information will identify a root or factor.
• Apply the factor theorem or polynomial division, then solve every remaining quadratic factor.
• A real polynomial cannot have an isolated non-real root: its conjugate is also a root.
• A complete solution must list all roots, including repeated and non-real roots, rather than stopping after the supplied factor is used.

Explanation

• For $z=x+iy$, where $x,y\in\mathbb R$, the real part is $\operatorname{Re}(z)=x$ and the imaginary part is $\operatorname{Im}(z)=y$; the imaginary part is not $iy$. Its modulus is the non-negative length $|z|=\sqrt{x^2+y^2}$ and an argument is its directed angle from the positive real axis.
• Addition and subtraction combine corresponding parts.
• Multiplication uses ordinary expansion followed by $i^2=-1$.
• To divide by $c+di$, multiply numerator and denominator by its conjugate $c-di$ so the denominator becomes $c^2+d^2$, then collect the result into Cartesian form.
• Equality of complex numbers means equality of their real parts and equality of their imaginary parts, producing two real equations.

Explanation

• The complex conjugate of $z=x+iy$ is $\overline z=x-iy$, obtained by reflecting the point for $z$ in the real axis. Important identities are $z+\overline z=2\operatorname{Re}(z)$ and $z\overline z=x^2+y^2=|z|^2$, which is real.
• If a polynomial has real coefficients and $a+ib$ is a non-real root, conjugating the whole equation shows that $a-ib$ is also a root.
• The pair produces the real quadratic factor $[x-(a+ib)][x-(a-ib)]=(x-a)^2+b^2$.
• This rule depends on the coefficients being real.
• In polynomial questions, use the conjugate root before dividing or comparing coefficients to find the remaining real factor.

Explanation

• An Argand diagram represents $z=x+iy$ by the point $(x,y)$: the horizontal axis is the real axis and the vertical axis is the imaginary axis.
• Addition of complex numbers is vector addition, so the point representing $z_1+z_2$ is obtained by completing a parallelogram.
• The difference $z_2-z_1$ is the directed displacement from the point representing $z_1$ to the point representing $z_2$, and $|z_2-z_1|$ is the corresponding distance.
• Coordinate geometry can prove gradients, perpendicularity, lengths and areas.
• A labelled sketch should show exact complex numbers or coordinates, but conclusions such as a right angle or equal length still require algebraic justification.

Explanation

• For $z=x+iy$, the modulus is $r=|z|=\sqrt{x^2+y^2}$ and an argument $\theta$ is the directed angle from the positive real axis to the vector representing $z$. Thus $z=r(\cos\theta+i\sin\theta)$, or $z=re^{i\theta}$.
• When converting from Cartesian form, calculate $r$ and use $\tan\theta=y/x$ only as a reference angle: the signs of $x$ and $y$ determine the correct quadrant.
• The principal argument lies in $(-\pi,\pi]$.
• To return to Cartesian form, use $x=r\cos\theta$ and $y=r\sin\theta$.
• Exact special-angle values should remain exact, and radians are used throughout this topic.

Explanation

• If $z_1=r_1e^{i\theta_1}$ and $z_2=r_2e^{i\theta_2}$, then $z_1z_2=r_1r_2e^{i(\theta_1+\theta_2)}$: moduli multiply and arguments add. For division, $\dfrac{z_1}{z_2}=\dfrac{r_1}{r_2}e^{i(\theta_1-\theta_2)}$, provided $z_2\ne0$: moduli divide and arguments subtract.
• These results also follow from the compound-angle formulae in trigonometric form.
• Any argument may be adjusted by an integer multiple of $2\pi$; if a principal argument is requested, place it in $(-\pi,\pi]$.
• Geometrically, multiplication combines a scaling by the modulus with a rotation by the argument, which helps check whether the resulting modulus and quadrant are sensible.
• Final form should retain a positive modulus.

Explanation

• A locus translates complex notation into distance and direction. The equation $|z-a|=r$ is a circle of radius $r$ centred at the point representing $a$; $|z-a|\le r$ selects its closed interior and $|z-a|>r$ its exterior without the boundary.
• The equation $|z-a|=|z-b|$ is the perpendicular bisector of the segment joining the points representing $a$ and $b$, while $|z-a|\le|z-b|$ selects the side at least as close to the point representing $a$.
• The condition $\arg(z-a)=\beta$ is a ray from that point at directed angle $\beta$, excluding the endpoint.
• Double argument inequalities select a sector.
• Sketch boundaries first, then use solid or dashed curves to show inclusion correctly.

Explanation

• De Moivre's theorem states $[\cos\theta+i\sin\theta]^n=\cos(n\theta)+i\sin(n\theta)$ for every integer $n$.
• Expanding the left side and equating real or imaginary parts produces multiple-angle formulae for $\cos p\theta$ and $\sin q\theta$ in powers of $\cos\theta$ and $\sin\theta$.
• The relations $z+z^{-1}=2\cos\theta$ and $z-z^{-1}=2i\sin\theta$, with $z=e^{i\theta}$, also convert powers back into multiple angles and support formulae involving $\tan r\theta$.
• Finite trigonometric sums are handled by summing a complex geometric series and then taking its real or imaginary part.
• The chosen part and any non-zero denominator condition must be stated.

Explanation

• Euler's definition $e^{i\theta}=\cos\theta+i\sin\theta$ identifies a unit complex number with argument $\theta$. Hence $z=re^{i\theta}$ has modulus $r$ and argument $\theta$, and arguments differing by $2\pi k$ represent the same number.
• Replacing $\theta$ by $-\theta$ gives $e^{-i\theta}=\cos\theta-i\sin\theta$. Adding and subtracting these equations yields $\cos\theta=\dfrac12(e^{i\theta}+e^{-i\theta})$ and $\sin\theta=\dfrac1{2i}(e^{i\theta}-e^{-i\theta})$.
• Exponential form turns multiplication into addition of arguments and often simplifies symmetric trigonometric expressions.
• The trigonometric and exponential forms are interchangeable, so a final answer should use the form requested.
• It must not be treated as a real exponential: $|e^{i\theta}|=1$ for every real $\theta$.

Explanation

• To solve $z^n=re^{i\theta}$ with $r>0$, allow every argument of the right side: $\theta+2k\pi$. Taking nth roots gives $z=r^{1/n}e^{i(\theta+2k\pi)/n}$ for $k=0,1,\ldots,n-1$.
• These are the $n$ distinct roots; larger integer values repeat them.
• All roots have modulus $r^{1/n}$ and consecutive arguments differ by $2\pi/n$, so their points are equally spaced on a circle centred at the origin and form a regular $n$-gon.
• The polygon may be rotated, depending on $\theta$, but its size and angular spacing are fixed.
• Each proposed root can be checked by raising it to the nth power and reducing its argument modulo $2\pi$.

Explanation

• The nth roots of unity solve $z^n=1$ and are $1,\omega,\omega^2,\ldots,\omega^{n-1}$ for $\omega=e^{2\pi i/n}$. They lie on the unit circle as the vertices of a regular $n$-gon.
• For a non-real root $\omega\ne1$, factorising $\omega^n-1$ gives $1+\omega+\cdots+\omega^{n-1}=0$.
• Complex methods encode geometry efficiently: the distance between the points representing $a$ and $b$ is $|a-b|$, and multiplication by $\omega$ rotates every point representing a complex number through $2\pi/n$ without changing lengths.
• Algebraic identities among powers of $\omega$ can therefore prove equal sides, angles, parallelism and ratios.
• Any cancellation involving $\omega-1$ requires the explicit fact that $\omega\ne1$.