Magic Number loves complex numbers.
It happily takes the square root of –4.

√ -4

It obviously knows Euler’s identity.


This is the button for imaginary unit.

Complex Imaginary Button ( Shortcut: i )

It lets you enter complex numbers in the form x + y i.
Or if you prefer, x + i y.

Complex Multiplication

Of course, you can go further than arithmetic.

Complex Logarithm

Complex numbers can be represented in a 2D complex plane.

Complex Polar Diagram

Earlier, we used the rectangular (Cartesian) form, x + y i.
You can also use the polar formr θ.

Here is the polar button.

Complex Polar Button ( Shortcut: < )

Enter 3 45, press return, and you get:

Complex Polar Entry

By default, Magic Number shows complex result in rectangular form.
To see it in polar, go to menu View ▸ Complex Result ▸ Polar.

Complex Polar Result

In practice, it’s quicker to press ⌥⌘P to switch between rectangular and polar.

Complex Polar Toggle

These functions are under Calculation ▸ Extra Functions.

In practice, it’s quicker and easier to use   F1  or  F2 .

When you click them for the first time, you are presented with a list.
(You can also press ⇧⌘F for the list.)

Complex Function List

Let’s pick re — this gives you the real part of a complex number.
You can click the Examples button to learn more.

Easier shortcuts

F1 and F2 are not easy to press. Besides, they could be used by macOS.
Here are the easier shortcuts:

F1  →  F

F2  →  G

F1 can fit 2 functions

If you look under Calculation ▸ Extra Functions ▸ Complex Numbers, notice how related functions are grouped, e.g., ‘Real Part…’ and ‘Imaginary Part…’

Complex Function Menus

If F1 is ‘Real Part’, to access ‘Imaginary Part’, you press F1 twice.

F1  →  Real Part

F1   F1  →  Imaginary Part

Or if you prefer the easy shortcut:

F  →  Real Part

F F  →  Imaginary Part

Of course the same applies to F2.

The versatile ‘part’ function

The part function gives you the real or imaginary part.
For example:

Complex Rect Part 1
Complex Rect Part 2

If your complex number is in polar form, you can get the absolute value or the argument.

Complex Polar Part 1
Complex Polar Part 2

Polar to rectangular conversions

Earlier you learned that to see the result in polar, you need to switch view by pressing ⌥⌘P.

If you like, you can do so via functions — no need to switch views.

You can convert to polar

Rectangular To Polar

and vice versa.

Polar To Rectangular

Auto bracketing for part ,▸ pol ,▸ rect

By syntax, these functions come after a complex number, therefore Magic Number can examine what you entered and insert parentheses when necessary.

For examples:

1 + 2i  part  2

Complex Auto Bracket Part

3 45  ▸ rect

Complex Auto Bracket Rect

What’s good to know:

Complex numbers have the forms a ± b i, r ∠ θ. But our workflow and habits vary. For example, a + i b and b i + a are both valid. Therefore Magic Number takes a less strict approach, it will auto bracket something ± something and something ∠ something.
Give it a try. It saves you time for most cases.

If you prefer to bracket manually, it’s also good to know these tips.

The complex pain

Take the complex number  4 + 5 i.
While it is mathematically descriptive, it can be cumbersome.

If we want to multiply by 3, we cannot just write  3 × 4 + 5 i.
Instead we have to use parentheses and write  3 × (4 + 5 i).
This is because multiplication has higher precedence than addition.

Let’s subtract. Again  3 – 4 + 5 i  is no good.
We have to write  3 – (4 + 5 i).

In fact, we need parentheses for division, square root, trigonometry, and almost anything. It’s a parenthesizing pain.

The solution

We can find inspiration nearby.

Let’s look at 4.5. It has 4 as the whole part and 5 as the fractional part.
We use the decimal separator ‘ . ’ to separate the parts. Mathematically, it means 4 + 5/10.

But we rarely write ‘4 + 5/10’. Besides being long, it’s cumbersome for math — we need parentheses for almost anything, for example:
3 × (4 + 5/10)  and  3 – (4 + 5/10).

It’s the same pain. You can see how the decimal separator helps.
For complex numbers, we need a complex separator.

The complex separator

We choose underscore ‘_’ as it is easy to type, distintive, and it doesn’t conflict with math.

Let’s compare the notations.

x + y ix_y
12 + 34 i12_34

Let’s compare the math.

3 x (4 + 5i) 3 x 4_5
Standard Simple
3 - (4 + 5i) 3 - 4_5
Standard Simple

Math is not only easier to type, it’s clearer too.

√(3 + 4i) √3_4
Standard Simple
(1.2e34 + 6πi) / (3 - 4i) 1.2e34_6πi / 3_-4
Standard Simple

In the next topic, we will exploit simple notation for 2D math.

Simple notation, x_y, makes complex numbers more approachable for 2D math. You can use it to represent width and height.

For example, let’s enlarge a 16 by 9 photo 400%.
To calculate the new dimension, we need to do the same math twice:
16 × 4 and 9 × 4.

Now we can simply use 16 _ 9 to represent 16 by 9.

Enter:  16 _ 9 × 4.

2d Calculation 1 Standard

To see the result in simple notation, go to menu
View ▸ Complex Result ▸ Notation ▸ x _ y

2d Calculation 1 Simple

2D variable

Let’s say you shoot videos in 4k (3840 by 2160) and you often need resolutions for various scales.

As we use this master resolution a lot, let’s set it to m.
Enter:  m = 3840 _ 2160

2d Calculation 2 Assign

What’s the resolution at 50%?

2d Calculation 2 Percent

Let’s make it 4 times smaller.

2d Calculation 2 Division

Distance between 2 points

What is the distance between (7, 5) and (3, 2)?
Enter:  7 _ 5 – 3 _ 2

2d Calculation 3 Rect

Not quite the result as it is the horizontal and vertical distances.
We want the direct distance as shown by the blue line.

2d Calculation 3 Graph

We can use Pythagorean theorem.
But it’s quicker to see the result in polar coordinates.
(Which in effect gives you the length and the angle of the blue line.)

To switch to polar view, press ⌥⌘P

2d Calculation 3 Polar

The distance is 5.

Scalar product

As 2D is based on complex numbers, Magic Number uses the star symbol ( * ) for scalar multiplication. To get it, press multiply twice, that is, ××.

To calculate (1, 2) • (3, 4), enter  1 _ 2 ×× 3 _ 4.

2d Calculation 4 Scalar Product

Angle between 2 vectors

This is a popular use of scalar product. It has quite a beautiful definition:

A • B  =  | A | | B | cos θ

However finding θ involves lots of ugly labor.

There is a quicker way via the polar button    ( shortcut: < ).

If you press it twice, you get the vector angle operator.

To calculate the angle between (1, 2) and (3, 4), enter  1 _ 2 << 3 _ 4.

2d Calculation 4 Vector Angle

Storing 2D numbers

2D-List is an ideal place for 2D numbers.

Here is a 2D number.

2d Number

To store it, open 2D-List (⌘5) and press

2d List

What’s good to know:

  • Notice the parts are splitted and stored under X and Y.
    This gives you more flexibility. For example, you can sort by X.
  • Right-click (secondary click) a number on the list, you get a menu
    where you can choose what to insert.
  • The function Sum XY.
    If you use X for width and Y for height, then XY is the area.
    The screenshot shows the area of a selected row.
    Another non-2D use: X as the quantity and Y as the price.