Complex numbers in Cartesian form

For $z=a+bi$, the real part is $\operatorname{Re}(z)=a$ and the imaginary part is $\operatorname{Im}(z)=b$. Both parts are real numbers.

The imaginary part is the coefficient $b$, not the term $bi$. This distinction is frequently tested.

A complex number is real exactly when $\operatorname{Im}(z)=0$, and pure imaginary exactly when $\operatorname{Re}(z)=0$ and $z\ne0$.

The zero complex number is $0+0i$; it is real but is not normally described as pure imaginary under the non-zero convention.

Two complex numbers are equal when their real parts are equal and their imaginary parts are equal. Thus $a+bi=c+di$ implies $a=c$ and $b=d$.

Add and subtract complex numbers componentwise. Real parts combine with real parts, and imaginary parts combine with imaginary parts.

When subtracting, distribute the negative sign across both components of the second complex number before collecting terms.

Equating components turns one complex equation into two simultaneous real equations, which is especially useful when unknown parameters are real.

Successive powers of $i$ repeat every four exponents: $1,i,-1,-i$. This follows because $i^4=(i^2)^2=1$.

To simplify $i^n$ for an integer $n$, divide $n$ by 4 and use the remainder. A remainder of 0, 1, 2, or 3 gives $1,i,-1,-i$, respectively.

Negative powers are reciprocals. Since $i^{-1}=\frac{1}{i}=-i$, the same four-cycle continues for every integer exponent.

When powers of $i$ occur inside a longer expression, reduce them before collecting real and imaginary parts.

Multiply complex numbers using the distributive law exactly as for binomials, then replace $i^2$ by $-1$.

For $(a+bi)(c+di)$, the real part is (ac-bd) and the imaginary part is $ad+bc$.

A square such as $(a+bi)^2$ contains a middle term $2abi$. Missing this term is a common algebra error.

Products may become purely real or purely imaginary when one component cancels. Always finish by writing the result in Cartesian form.

The conjugate of $z=a+bi$ is $\overline z=a-bi$. Conjugation keeps the real part and reverses the sign of the imaginary part.

A complex number and its conjugate have the same modulus. On an Argand diagram, their points are reflections in the real axis.

Multiplying conjugates removes the imaginary part: $z\overline z=a^2+b^2$. This non-negative real result is central to division and modulus calculations.

Conjugation respects addition and multiplication. If a polynomial has real coefficients, every non-real root occurs with its conjugate.

A quotient is in Cartesian form only when its denominator is real. To divide by $c+di$, multiply numerator and denominator by the conjugate (c-di).

The denominator becomes $(c+di)(c-di)=c^2+d^2$, which is positive whenever $c+di\ne0$.

After expanding the numerator, divide both its real and imaginary components by the real denominator.

Do not divide real parts and imaginary parts separately. Complex division follows ordinary fraction algebra with conjugates.

The modulus of $z=a+bi$ is $|z|=\sqrt{a^2+b^2}$. It is the Euclidean distance from the origin to the point ($a,b$) in the Argand plane.

A modulus is always a non-negative real number. Moreover, $|z|=0$ exactly when $z=0$.

The distance between complex numbers $z_1$ and $z_2$ is $|z_1-z_2|$. This is the usual distance formula expressed compactly.

Modulus behaves like length: $|zw|=|z||w|$, and for $w\ne0$, $\left|\frac zw\right|=\frac{|z|}{|w|}$. The triangle inequality states $|z+w|\le|z|+|w|$.

An Argand diagram represents $z=a+bi$ by the point ($a,b$), or by the position vector from the origin to that point. The horizontal axis is the real axis and the vertical axis is the imaginary axis.

Addition of complex numbers corresponds to vector addition. Subtraction $z_1-z_2$ is the displacement vector from $z_2$ to $z_1$.

Conjugation reflects a point across the real axis. For example, $z=3+2i$ and $\overline z=3-2i$ lie the same distance $\sqrt{13}$ from the origin.

Conjugation reflects $z=3+2i$ to $\overline z=3-2i$ across the real axis while preserving the modulus.

Label axes and scale clearly in an exam sketch. A correct algebraic value plotted with the real and imaginary coordinates reversed is still an incorrect Argand point.

Linear equations in $z$ use the same inverse operations as real equations, followed by conjugate division if the coefficient is complex.

When an equation contains $z$ and $\overline z$, set $z=a+bi$, replace $\overline z$ by (a-bi), and equate real and imaginary parts.

Quadratic equations may have non-real solutions. The quadratic formula remains valid, with $\sqrt{-k}=i\sqrt{k}$ for $k>0$.

To find a Cartesian square root, write $(a+bi)^2=(a^2-b^2)+2abi$ and equate components. Remember that a non-zero complex number has two square roots differing by sign.

If a polynomial has real coefficients and $a+bi$ is a non-real root, then (a-bi) is also a root. This is the conjugate-root theorem.

The factor pair associated with $a\pm bi$ has real coefficients: $(x-a-bi)(x-a+bi)=(x-a)^2+b^2$.

Use the conjugate pair to construct the least-degree real polynomial with a specified non-real root. Additional real or conjugate roots contribute additional factors.

The theorem depends on the coefficients being real. A polynomial with complex coefficients does not have to contain conjugate pairs.

Advanced IB questions often combine several skills: component equations, conjugates, moduli, and polynomial structure. Translate each condition into its simplest Cartesian statement before calculating.

For $z=a+bi$, the relations $z+\overline z=2a$, $z-\overline z=2bi$, and $z\overline z=a^2+b^2$ can turn a complex problem into familiar real algebra.

A condition such as $|z-c|=r$ describes a circle in the Argand plane centered at the point represented by $c$, with radius $r$. A condition $|z-z_1|=|z-z_2|$ describes the perpendicular bisector of the segment joining $z_1$ and $z_2$.

Keep exact values such as radicals and fractions until the question requests an approximation.

In calculator-permitted work, select complex mode and rectangular output $a+bi$. Enter the imaginary unit using the calculator's dedicated $i$ key, not a letter variable.

A GDC can evaluate complex arithmetic, conjugates, moduli, and polynomial roots. Menu names vary, so practise the exact sequence on the approved model before the examination.

Use the calculator to verify arithmetic, not to replace required reasoning. For an exact question, show the conjugate multiplication, component equations, or factor construction that leads to the answer.

If the GDC returns decimals, decide whether an exact Cartesian value is expected. Convert recognizable decimals to fractions only when justified, and retain full internal precision until the final line.

A strong solution states restrictions, includes both roots where necessary, writes final answers in $a+bi$ form, and checks the result by substitution or multiplication.

Attempt each question before opening the worked steps. These problems progress from basic Cartesian fluency to multi-step HL reasoning.

For non-calculator practice, keep every value exact. For calculator practice, use the GDC only after setting up the mathematics clearly.

Complex-number work becomes reliable when every expression is returned to Cartesian form, every component equation is stated explicitly, and every division uses a conjugate.

Use the summary below as a final checklist, then practise mixed questions without referring to it.