1.12 Complex numbers in Cartesian form questions for IB Maths AA

Express $z$ in the form $a+b\text{i}$, where $a,b\in \mathbb{Q}$.

Find the exact value of $\mathrm{\mid }z\mathrm{\mid }$.

Find $\arg z$, where $-\pi <\arg z\le \pi$.

Show that $N(a+b\text{i})=a^2+b^2$.

Let $z=a+b\text{i}$ and $w=c+d\text{i}$, where $a,b,c,d\in \mathbb{R}$. Show that $(zw{)}^{\ast }=z^{\ast }{w}^{\ast }$.

Hence show that $N(zw)=N(z)N(w)$.

Let $z=1+2\text{i}$ and $w=2+3\text{i}$.

Find $zw$ and $z{w}^{\ast }$.

Hence write $65$ as the sum of the squares of two positive integers in two different ways.

Use part (c) to show that if $m$ and $n$ can each be written as the sum of two squares of integers, then so can $mn$.

It is given that $1105=5\times 13\times 17$ and $17=1^2+4^2$.

By using your answers to part (d)(i), find four different ways of writing $1105$ in the form $p^2+q^2$, where $p,q\in {\mathbb{Z}}^{+}$ and $p>q$.

Show that, for any $a\in \mathbb{Z}$, the remainder when $a^2$ is divided by $4$ is $0$ or $1$.

Hence show that no integer of the form $4m+3$, where $m\in \mathbb{N}$, is the norm of a Gaussian integer.

By considering $(1+\text{i})(a+b\text{i})$, show that if $n=a^2+b^2$ then $2n=(a-b{)}^2+(a+b{)}^2$. Hence write $2210$ as the sum of the squares of two positive integers.

Let $z=a+b\text{i}$, where $a,b\in {\mathbb{Z}}^{+}$ and $a>b$. By considering $N\left(z^2\right)$, show that ${(a^2-b^2)}^2+(2ab{)}^2={(a^2+b^2)}^2$.

Hence, using $z=4+\text{i}$, find three positive integers that are the side lengths of a right-angled triangle.

Find $z$, giving your answer in the form $x+y\text{i}$, where $x,y\in \mathbb{R}$.

Hence find the exact value of $\mid z^2-z^{\ast }\mid$.

Find the two square roots of $u$, giving your answers in the form $a+b\text{i}$, where $a,b\in \mathbb{R}$.

Hence solve the equation $z^2+3z+6-2\text{i}=0$, where $z\in \mathbb{C}$.