What are they?

Let’s say you want to calculate

It’s natural to type  ‘11 + 33 / 2’.

But doing so gives you

As division has higher precedence than addition, it acts like glue —
it bonds 33 and 2.

As a workaround, we use parentheses,  ( 11 + 33 ) / 2.
While workable, it’s also unnatural and slows us down.

Ideally we want ‘+’ to act upon 11 and 33.
We want it to bond 11 and 33.
We want it to be sticky.

How?

⌥ (option) is the key

Or rather, ⌥ is the sticky key.

To enter a sticky ‘+’,  press ⌥ + 

So to calculate

just enter  11   ⌥ +  33  /  2

Subtraction

As you have guessed, the shortcut is ⌥ – 

For example:

Enter:  33   ⌥ –  11  /  2

(Don’t go away yet. There is a little twist in our story.)

Multiplication

To avoid confusion with the standard multiplication sign ‘×’, the lower-case x,
and the upper-case X, Magic Number uses the dot ( • ) notation from math.

Shortcut: ⌥ . 

Example:  To convert 3600 minutes into days, we want

For that, we enter  3600  /  60   ⌥ .  24

Division

No drama here, the shortcut is ⌥ / 

Previously, to calculate

you had to use parentheses, 8 ^ (2 / 3).

And now,  8 ^ 2   ⌥ /  3.

What’s next?

You are almost an expert.
With the next 2 tips, you can handle any sticky situation.

Sensing sticky

With great power comes great responsibility…

When you have  2 × 3 + 4 = 14, and without knowing ‘+’ is sticky, you might think something is wrong. How can you tell something is sticky?

Let’s look at standard and sticky operators side by side.

Addition

Standard

Sticky

Subtraction

Standard

Sticky

Multiplication

Standard

Sticky

Division

Standard

Sticky

See the bond

You can also tell by double-clicking the operator.
This selects the operator and its operands as a group.

Standard

Left operand	2 × 3  =  6
Right operand	4
Left + Right	10

Sticky

Left operand	3
Right operand	4
Left + Right	7

Avoid misunderstanding

Whether it is for someone or for your future self, parentheses are still essential to make your expressions understandable.

Here is a quick way to bracket a sticky for clarity:

1. Double-click to select it as a group.

2. Press ⌘ )  or  ⌘ ]  to bracket the group.

Semi-sticky

There are times you need a less sticky operator.
To see why, let’s start with the standard ‘+’.

Enter  √ 3 + 4  and you get

For comparison, let’s try the sticky ‘+’.
Enter  √ 3 ⌥+ 4

You can see sticky ‘+’ bonds directly to 3.
That is, it bonds directly to whatever comes before it.

To get a better insight, enter  8 ^ 2 ⌥+ 3

As expected sticky ‘+’ bonds directly to 2.
This is better than entering  8 ^ (2 + 3).

However when dealing with expressions like

Using a sticky ‘+’ for  √ 3 ^ 2 ⌥+ … yields

And using a standard ‘+’ doesn’t help either.

You are stuck.
Sticky ‘+’ won’t do as it bonds with 2 — it’s too sticky.
Standard ‘+’ won’t do as it bonds with √ 3 ² — it’s not sticky.

You need something in between.
You need a semi-sticky ‘+’ that bonds with 3 ².

You can achieve that by pressing ⌥+ twice.
That is, by entering  √ 3 ^ 2  ⌥+ ⌥+  4 ^ 2.

Your turn

Well done for getting this far.
Let’s put what you’ve learned to the test.

Here’s a quadratic equation

And here’s the quadratic formula.

Your task:  Solve x by entering

Your extra challenge:  Do it with Apple’s calculator. Do it in RPN mode.