The Argand plane

Numbers as points

A complex number $a + ib$ is fixed by two real numbers, $a$ and $b$. A point of the plane is also fixed by two numbers. So we draw $a + ib$ as the point $(a, b)$.

The plane used this way is the Argand plane, after Jean-Robert Argand, who wrote about it in 1806.

• The horizontal axis is the real axis. The number $a + 0i = a$ lies on it.
• The vertical axis is the imaginary axis. The number $0 + ib = ib$ lies on it.

The modulus is a distance

The point $(a, b)$ is at distance $\sqrt{a^2 + b^2}$ from the origin $O$, by Pythagoras. That is exactly $|z|$.

So the modulus is the distance of $z$ from $0$. The numbers $3 + 4i$, $-5$, $4 - 3i$ and $5i$ all have modulus $5$. They all lie on the circle of radius $5$ around $O$.

The conjugate is a reflection

The conjugate of $a + ib$ is the point $(a, -b)$. It is the mirror image of $z$ in the real axis. Two more mirror images are worth knowing.

• $-z$ is the point $(-a, -b)$, opposite $z$ through the origin.
• $-\bar z$ is the point $(-a, b)$, the mirror image of $z$ in the imaginary axis.

All four points are the same distance from $O$, so $|z| = |\bar z| = |-z| = |-\bar z|$. A real number is its own mirror image, so it is its own conjugate.

The distance between two numbers

Take $z_1 = a_1 + ib_1$ and $z_2 = a_2 + ib_2$. Their difference is

Its modulus is $\sqrt{(a_1 - a_2)^2 + (b_1 - b_2)^2}$. This is the distance formula for the two points. So the distance between $z_1$ and $z_2$ is

For $z_1 = -2 + 3i$ and $z_2 = 4 - 5i$, we get $z_1 - z_2 = -6 + 8i$. The distance is $\sqrt{36 + 64} = 10$.

The point halfway between them is $\dfrac{z_1 + z_2}{2}$. Here it is $\dfrac{2 - 2i}{2} = 1 - i$.

Put both points on the real axis. The distance is then just the size of the gap between two real numbers.

Adding moves points

Adding $2 + i$ to every number moves every point $2$ to the right and $1$ up. The shape of a figure does not change. So distances, and the triangles they make, are kept.

A worked example

Are the points $1 + i$, $4 + 5i$ and $7 + i$ the corners of an isosceles triangle?

Step 1. Name them. Let $A = 1 + i$, $B = 4 + 5i$ and $C = 7 + i$.

Step 2. Side AB. $B - A = 3 + 4i$, so $AB = \sqrt{9 + 16} = 5$.

Step 3. Side BC. $C - B = 3 - 4i$, so $BC = 5$.

Step 4. Side AC. $C - A = 6$, so $AC = 6$.

Answer: two sides are equal, so the triangle is isosceles.

Practise with new numbers

Try it in Python

The program puts three numbers on the Argand plane. It prints each point and its distance from $0$, then the distance between two of them.

Common mistakes

• Swapping the coordinates. $a + ib$ is the point $(a, b)$, with the real part across. It is not $(b, a)$.
• Putting $i$ on the real axis. The number $i$ is the point $(0, 1)$, on the imaginary axis.
• Reflecting in the wrong axis. The conjugate is the mirror image in the real axis, so only the height changes sign.
• Subtracting moduli. The distance between $z_1$ and $z_2$ is $|z_1 - z_2|$, not $|z_1| - |z_2|$.

Remember these

• $a + ib$ is the point $(a, b)$ of the Argand plane.
• $|z|$ is the distance of $z$ from the origin.
• $\bar z$ is the mirror image of $z$ in the real axis.
• The distance between $z_1$ and $z_2$ is $|z_1 - z_2|$.
• The midpoint of $z_1$ and $z_2$ is $\dfrac{z_1 + z_2}{2}$.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. In which quadrants do $-4 - 7i$ and its conjugate lie?

2. How far from the origin is the point $6 - 8i$?

3. What is the distance between $1 + i$ and $4 + 5i$?

4. Which complex numbers lie on the imaginary axis?

5. A number $z$ is at the same point as $\bar z$. Where on the plane is it?