Magic Number

What’s New

2.8.25 Fixes ’24.01

Improved compatibility with older versions of macOS

♥︎ If you love Magic Number, please rate it. It helps a lot. Thanks!

2.8.24 Sonoma

Updated for macOS Sonoma. Apologize for the later than usual release — we discovered some issues in the late betas.

Menubar App Mode is now functional in macOS Sonoma

Fixed many cosmetic issues in macOS Sonoma

2.8.23 Fixes ’23.02

Minor bug fixes

Improved robustness for older versions of macOS

2.8.22 Fixes ’22.10

Made the undo more robust

Fixed some drawer issues from the last update

Advance notice: In the next update, macOS 10.13 or above is required

2.8.21 Undo

Finally a long overdue enhancement. Some of you ask for Undo a while back.
My apologies for taking this long.

Ability to undo while you are typing an expression

Improved copy and paste of expressions

Improved compatibility with newer versions of macOS

Fixed the issue that some buttons in Settings didn’t work

Fixed the cosmetic issues with drawer buttons

Fixed an editing bug

2.8.20 Fixes ’22.08

Fixed an equation solver issue involving the n-th root

Fixed cosmetic issues in Touch Bar

Fixed minor bugs in Help

2.8.18 Silicon

New update for a new chip. Enjoy!

Fully optimized for Macs with Apple M1 chip

Fixed cosmetic issues in macOS Big Sur

Fixed a bug related to the # operator

2.8.16 Fix & Polish

Improved clarity of some UI elements in macOS Big Sur

Further improvement for macOS Big Sur compatibility

Fixed an issue with inserting current time due to daylight saving

Fixed the single quote shortcut for dates

2.8.15 Big Sur Ready

Improved compatibility with macOS Big Sur

You can view numbers as time in the List drawer

You can paste text with time such as 12:30

Fixed an issue with complex polar format

Minor bug fixes

2.8.14 A Bit Wiser

Better bitwise

We have added lots of enhancements to make bitwise calculations easier.
It’s time to give programmers more love. Without further ado, here we go.

Format Options Improvements

You can quickly switch between bit lengths without opening the drawer (sidebar).

A You can automatically hide the options after you switched base or bit length.
B You can set the separator distance individually for binary, hexadecimal etc.

Shortcuts

⌃⌘0	Set bit length to none
⌥⌘<	Increase bit length
⌥⌘>	Decrease bit length

New Bitwise Functions

You now have the negated alternatives of AND, OR, XOR:
NAND, NOR, XNOR

When using a bit length like 8-bit, you also get 1’s and 2’s complements.
They are prefix functions just like NOT.

For computing purposes, you have integer division div in place of floating point division ÷.

Easier Shortcuts

When you know one, you know the other.

For instance, AND has the shortcut &.
Its alternative NAND has the shortcut &&.

Basic functions
AND	&
OR	\|
XOR	^
NOT / 1’s Complement	!
div (Integer)	/

Alternative functions
NAND	&&
NOR	\| \|
XNOR	^^
2’s Complement	! !
÷ (Floating point)	/ /

This shortcut scheme is based on the Double Key idea.

New Bitwise Actions

This is the bitwise action button.

When clicked, you get a set of actions.
These actions directly affect the numbers you typed on the display.

The rotate and complement actions are meant for numbers with bit length.

Shortcuts

Tip

If you have a floating point number, you can use decrement or increment to convert it into an integer.

Smarter paste

Imperial math

Maths with imperial units is tedious.
For example, 5 feet 11 inches + 4 inches.

As you know 1 foot = 12 inches
Handling base 12 in a decimal world hurts the brain.

Unlike metric, which is decimal, we cannot type the above as “5 . 11 + 0 . 4”.
Yet similar to metric, we can type “5 .. 11 + 0 .. 4” and get the result.

The answer is 6 feet 3 inches. Or simply 6 ft 3 in.

By default, Magic Number uses feet & inches as the unit & subunit scheme.
We will show you how to use a different scheme later.

We refer the two dots (··) as the subunit separator.

Here’s a quick tour of what you can do.

Multiply — No need to enter 4 × (5..1).

Divide — You can see 23..5 is treated as a single entity.

Result as fraction — Shortcut: ⌘/.

Mixing decimals and fractions.

Percentage — 80% of 5 ft 8 in.

Tip — Convert 4.75 feet to feet and inches.

Using a different subunit scheme

Hold down the decimal button to reveal the menu.

For more options, choose More… from the menu.

We could modify existing schemes, or we can create a custom one.

We will create one for stone (st) and pound (lb).
Knowing that 1 st = 14 lb, our configuration will look like:

Click Apply and we are are ready to calculate.

Example: 10 st 8 lb – 4 st 12 lb

Result: 5 st 10 lb

Improved compatibility with macOS Catalina

Various bug fixes

2.8.12 Quadratic, Cubic, Quartic

Quadratic equations

Let’s solve a simple quadratic equation
x² + 2x – 3 = 0

For that you enter
?² + 2? – 3 = 0

And press ↩ for the results.

So x = -3 or x = 1.

Magic Number is flexible.
You can express the equation differently.

Let’s try a complex example:
(2x – 5) (3x + 4) = 7x² – 6

Usually you have to expand the terms to get x² + 7x + 14 = 0.
Not anymore — Magic Number can do that for you.

Notice how the results -3.5+1.3229i and -3.5–1.3229i are presented as
-3.5 ± 1.3229i.

Tip

In algebra, a a = a · a = a × a = a²
In Magic Number, you can press ?? for ?²

? ? → ?²

Likewise,

? ? ? → ?³

You will find this easier especially for cubic and quartic equations.

Cubic and quartic equations

Easy function for polynomial equations

Tip: Reusing result from polynomial equations

2.8.11 Extending !

For the love of combinatorics, the factorial button ! is now multifunctional.

Multifactorial

Subfactorial

Fixes for date calculation

Improved compatibility with macOS Mojave

2.8.10 Let’s Make a Date

3 things you can do with dates.

Day of the week

Date after some days

Days between 2 dates

Tips

Year shorthand

You can abbreviate the year with the single quote '

For example, '18 . 12 . 25 is equivalent to 2018 . 12 . 25.

The single quote represents the ‘20’ in ‘2018’— you can think of it as the current century. But there’s more.

The single quote shortcut

The single quote is actually the current shortcut — it gives you the current century, year, month, or day.

To illustrate, suppose the current date is 2019.2.14.

To begin, let’s clear the display by pressing AC

Enter ' for the current century abbreviation ( ’ ).

Enter ' again; this time you get the current year.

And again for the current month and day. In other words,

Other shortcuts

You can insert the current date by pressing ⌃ '
( ⌃ = control )
You can omit the day.
For example, ’19.7 is treated as the start of the month, that is, ’19.7.1.
You can omit the day and month.
’19 is the start of the year 2019 or the New Year, that is, ’19.1.1.
Tip for European users
If your Mac uses the comma ( , ) as a decimal mark, then the dot ( . ) is solely for dates. Here you can omit the current year. You can enter .12.25 instead of 2019.12.25.

2.8.9 Overcoming Overflow

Some math involves gigantic numbers — numbers so large that they can easily overflow calculators and spreadsheets.

Previously Magic Number could stumble. Now it just marches on.

Smarter modulo for modular exponentiation

Useful in computer science and cryptography, modular exponentiation is the calculation of aᵇ mod m, where a, b, and m are positive integers.

For example, 5² mod 7 = 25 mod 7 = 4.
The modulo (mod) is the remainder when 5² is divided by 7.

Problem arises when a and b are fairly big, resulting a huge number for aᵇ.
For example, 1013⁵⁹ has 178 digits; it would overflow most calculators.
Magic Number does better but still cannot escape the fate of 1013⁵⁹⁰⁰⁰.

Once overflowed, further calculation was not possible.

Magic Number is smarter now — it uses fast modular exponentiation.

In case you are wondering, a, b, m can be 14 digits long.

Smarter permutation and combination

Website update

Fixes

2.8.8 二八八

Happy Chinese New Year!
新年快乐！

Or Happy Valentine’s Day.
Whatever the case, it’s time for more love.

万事如意 = 😀，千事如意 = 🙁

To Russia with love

Equation of a line

Given 2 points (x₁, y₁) and (x₂, y₂), what is the equation of the line in the form of y = mx + b ?

This involves finding the values for m, the slope, and b, the y-intercept.

The adjacent function has a second role.
It doubles as a slope/intercept function. (Shortcut: ⇧H)

It’s helpful to see its icon this way:

Let’s do some math.
What is the equation of the line that passes through (2, 2) and (6, 5)?

Enter (2, 2)(6, 5) and you get 2 results.

This means our equation is y = 0.75 x + 0.5

Tip

Press R to access the first result, 0.75. And twice for the second result, 0.5.

R → 0.75

RR → 0.5

Vector product

Refinement and fixes

2.8.5 Higher

In the spirit of macOS High Sierra, we spent some time to improve Magic Number’s foundation.

The journey actually began in the last update when we introduced an additional form for logarithm, log(a , b). While not groundbreaking, it is the groundwork for functions with multiple inputs or arguments.

This means functions in Magic Number can be as sophisticated as those in spreadsheets and programming languages. And yet they are simpler to use.

Before we begin…

You can use comma or semicolon for function arguments. For examples:

Comma	log (3 , 9)
Semicolon	log (3 ; 9)

Geometric functions

Despite the lack of true vector support, Magic Number’s new functions can support multiple arguments which work like multi-dimensional vectors. For brevity, we use 2D as examples.

Scalar product

Also known as the dot product. Hence you use the dot ( • ) operator.

Shortcut: ⌥ .

To calculate enter (1, 2) • (3, 4)

Here is a 3D example:

Angle between 2 vectors

The polar operator is now multi-talented.

∠ ( shortcut: < )

Generally, it is used to specify complex number in polar format, in the form of
radius ∠ angle.*

If you have multiple arguments on either side, they will be treated as vectors and Magic Number will calculate the angle between them.
Let’s find the angle between and
Enter (1, 2) ∠ (3, 4)

* When the radius and angle are complex, they will be also treated as vectors.

Magnitude of vector

Also known as the absolute value.
You find it under Menu ▸ Calculation ▸ Extra Functions ▸ Complex Numbers.

Let’s do a 4D example:

Distance between 2 points

Before we show you the new way, let’s look at the math.

The distance d is the hypotenuse. And according to Pythagorean theorem

Typing the formula is tedious, it’s easier to use the hypotenuse operator.

⊿ h ( shortcut: H )

This simplifies to

Of course we still need to calculate a and b where
a = x₂ – x₁ and b = y₂ – y₁

Well not anymore. The hypotenuse operator can now accept your coordinates and do the rest. That is (x₁, y₁) ⊿ (x₂, y₂)

For example

Just enter (3, 2) ⊿ (7, 5)

Logarithm perfected

Mixed base calculation

Bitwise improvements

Bug fixes

Compatibility

Magic Number supports the latest 3 major releases of macOS —
High Sierra (10.13), Sierra (10.12), El Capitan (10.11)

2.8.3 A Touch of Magic

Happy New Year!

If you have a MacBook Pro with Touch Bar, this is a lovely gift for you.
If you don’t, think of this as your extra Christmas present.

A feature for billionaires — Making sense of big numbers

Here’s a big number: 1 000 000 000.
Scientists think of it as 10⁹. And technical people think of it as 1 G.
That’s 1 Giga as in 1 gigabyte.

But billionaires don’t say I have 1 giga dollars in the bank.
And we certainly don’t say there are over 7 giga people on this planet.

Time to let Magic Number do the talking.

This is the new Quantifier format.

Menu: View ▸ Number Format ▸ Quantifier
Shortcut: ⌥⌘K

By default, this format uses the metric system.
Let’s have it in plain, simple language.

Go to menu View ▸ Number Options (⌥⌘ , )
Select Linguistic and enable Use word.

Changing your perspective

Not everyone’s a billionaire. Maybe you are a millionaire.
To see 1 billion in your units, press ⌥⌘< to lower the ‘scale’.

Likewise to see 1 billion in terms of trillion, let’s go up a ‘scale’ by pressing ⌥⌘>.

As you might have noticed, changing the scale is really shifting the exponent by 3.

Abbreviations

By turning off Use word you can see 1 Billion abbreviated as 1 B.

To avoid confusion, here are the conventions we used.

Copying for purists

A more versatile log

Gamma function

% as modulo

Extra Functions — UI refinements

Automatic bracketing for certain functions

Parentheses will be inserted automatically for absolute value, argument, and many complex functions.

Improved keyboard support

This is particularly useful if you use many different functions.
You can insert a function via keyboard.

Quick illustration:

Press ⇧⌘F to open the Function Browser.
Type the first few letters of the function you want.
For example, type “flo” to select the floor function.
Press return to insert the function.

The floor function becomes your favorite function. To use it again, press ctrl-F.

Previously if you select a function via ⇧⌘F, it will override your chosen function for F1. Now your choice is respected.

Touch Bar support — This is a big one

Touch Bar is a wonderful feature on the new MacBook Pro. Something we’d love to have on future Macs.

Here is Magic Number’s Touch Bar. (Full image here.)

It is fully customizable.
Menu ▸ View ▸ Customize Touch Bar…

Instead of menu, you can also tap

Within minutes, you have a business calculator:

Or a specialized time calculator:

You get the idea.

Let’s look at the special buttons.

Trigonometry

You may be wondering where are the inverse and hyperbolic functions.
When you tap sin for example, all is revealed:

If you don’t use hyperbolic functions, there is also a quicker way.
Check out Menu ▸ View ▸ Touch Bar Options ▸ Use Double Key

Angle Unit

On the surface, this button switches between degrees and radians.

When you tap and hold, you get contextual options.

In radians, you can see your result in terms of pi or tau.

Memories

Initially you have 2 memories for the Touch Bar. You can get more:

Format

This expands into 2 parts. Part B is quite smart — it examines the context and brings you the relevant options.

A Number formats

They are general, decimal places, scientific, and fraction.
For engineering, quantifier, and significant digits, tap

B Contextual control

Sometimes, depending on the context of your calculation and format, you want to look at results differently. Previously you had to dig deep, now it’s one tap away.

Here are the scenarios:

Transform

This one is good to know. It transforms a number from one form to another.

Sign Transform
e.g. 2 → -2
Inverse Transform
e.g. 2 → 2⁻¹ = ½ = 0.5
Also: cosine → secant
Percent Transform
e.g. 0.5 → 50%
Time Transform
e.g. 2.5 → 2:30

For your reference, their shortcuts are ⌘–, ⌘', ⌘%, and ⌘:

🎄 Bonus

Themes

🎁
Menu ▸ View ▸ Touch Bar Options

😊
We hope this brightens your Mac
and your New Year.

❤️
If you want to show your love,
a nice rating will warm our hearts.

Goodbye.

2.8 Anytime, Anywhere

Menubar app

If you are familiar with Spotlight, you know you can access it anytime and anywhere — by clicking its menubar icon or by pressing its shortcut. And when you switch apps via ⌘ tab, it doesn’t get in the way. What’s more, it coexists nicely with full screen apps.

You can make Magic Number behave this way.

In Magic Number ▸ Preferences, click the arrow for more options.

When Menubar App Mode is enabled, Magic Number resides on the menubar instead of the Dock.

Its menubar icon, by default, gives you the full menu. (If you want Magic Number to be available all the time, make sure Open Magic Number at login is checked.)

Tip #1 — Menu within reach

A quicker way to access the menu is to right-click (secondary click) the arrow button on the calculator.

Tip #2 — Always on top

When working with a full screen app, you might want to keep Magic Number’s window on top. You do so by right-clicking its title and choose Window Mode ▸ Float Above Others.

Tip #3 — Prevent accidental quitting

If you find yourself pressing ⌘Q when you just want to hide Magic Number, there is an option for you. Under Preferences > Shortcuts, click the arrow for more.

Significant digits

Future proofing

2.7.10 Love For Film-makers

And love for the time-obsessed.

Timecode and frame rates

Video is a series of still images which, when viewed in rapid succession, creates an illusion of motion.

How rapid? For films, it’s 24 images or frames per second or 24 fps.
This is the frame rate of film.

Suppose we are adding music to a film. The music starts at ½ second. As time format, this is 0 : 0 : 0.5. But our video app prefers timecode instead, in the form of hours : minutes : seconds : frames.

We will use Magic Number to find the timecode.

Type 0:0:0.5 and press ↩

To see the result as timecode, select View ▸ Timecode ▸ 24 fps.

As expected, ½ second equates to 12 frames.

Our music is 3 minutes and 7.25 seconds long.
Let’s see exactly where it ends in the film.

Type + 0:3:7.25

You can think of timecode as time with the extra frame component.
Dealing with 4 components can be messy. The following tips will help.

Tip #1 — Type less zeros

To enter 6 minutes, just type 0 : 6 — no need to type 0 : 6 : 0 : 0.
Also, no need to type the leading zero, just type : 6.

Likewise for 6 frames, you can type : : : 6 for 0 : 0 : 0: 6.

Example:	0 : 03 : 07 : 18 + 6 frames
Enter:	: 3 : 7 : 18 + : : : 6

Tip #2 — Know your frame rate

When you have 0 : 0 :0 :12, its actual time depends on the frame rate;
it can be 0.5 second for 24 fps or 0.4 second for 30 fps.

To see what frame rate you’re using, press ⌥⌘. (option-command dot).

To see the frame rate permanently, right-click (secondary-click) the window’s title.

Then choose Timecode.

Tip #3 — Toggle timecode on or off

The menu None ↔︎ Recent fps (⌥⌘:) switches between no frame rates and your most recently used frame rate. It’s a quick way to turn timecode on or off.

Custom frame rate

Slow motion requires shooting at high frame rates. The iPhone can do 240 fps. While impressive, some do it 10 times better. And few do it insanely well.

It’s good to know that there is a customizable frame rate.

Example

Suppose we have 1 minute 25 seconds of video at 240 fps.
When exporting it as images, how many images do we get?

Before we do any calculation, let’s establish the frame rate.
Go to View ▸ Timecode ▸ Customize* and enter 240 for frame rate.

Normally you would calculate how many seconds in 0:1:25.
Then you multiply the result by 240.

There is a better way. How about a Time to Frames function?
Better still, how about a new set of time conversion functions?
This leads us to the next topic.

Finer time conversions

How many minutes in 1:20:30?
And how many seconds are there?

Time to Minutes and Time to Seconds can give you the answers.
They are parts of the time conversion family.

You can find them under Calculation ▸ Extra Functions ▸ Time.

Time to Minutes ( t ▸ m )

How many minutes in 1:20:30?

Enter 1 : 20 : 30 t ▸ m

Notice Magic Number automatically encloses 1 : 20 : 30 in parentheses to avoid ambiguity.

Time to Seconds ( t ▸ s )

How many seconds in 1:20:30?

Enter 1 : 20 : 30 t ▸ s

Time to Hours ( t ▸ h )

Sometimes it’s easier to think of time as duration.
For 180 minutes, it’s perfectly fine to type 0 : 180.

What is 36 million seconds (36×10⁶) in hours?

Automatic time conversion

You may be wondering is there a Minutes to Time function.
There isn’t one. Because Magic Number can do it automatically.

To convert 80.5 minutes into time, type 0 : 80.5 and press ↩

For 4830 seconds, type 0 : 0 : 4830.
(You can also type ::4830.)

For 2.5 hours, type 2.5 : 0.

When a number has colon ‘ : ’ as in ‘2.5 : 0’, you are letting Magic Number know you want the result as time.

Tips

As usual, you can assign these functions to F1 and F2.
You can fit 4 functions under 1 button.
Let’s say F1 is assigned to Time to Frames.
By pressing F1 repeatedly you cycle through its related functions.

In other words:

F1 → t ▸ f

F1F1 → t ▸ s

F1F1F1 → t ▸ m
If F1 and F2 are occupied, the following recent function will help.

Recent function

When you used a function from Menu ▸ Calculation ▸ Extra Functions,
you can use it again by pressing control-F (⌃F).

Example + Tips

The screenshot shows Modulo is the recently used function.
To use Modulo again, press ⌃F.
Quotient is related to Modulo (as indicated by the menu).
To use Quotient, press ⌃F ⌃F.
You can use the same technique for F1 and F2.
That is, press F1 twice for its related function. Learn more here.
Idea: You can use F1 and F2 for your favorite functions and set aside ⌃F
for the occasional ones.

There are other small yet nice refinements to Extra Functions.
Check out the next topic.

Refinements and fixes

2.7.8 Sticky But Smooth

Sticky operators

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.
But 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 surprises here, the shortcut is ⌥ /

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

And now, 8 ^ 2 ⌥ / 3.

What’s next?

You are 5 minutes away from being an expert.
These 2 tips will give you the edge.

Refinements and fixes

2.7.6 R is for…

Results

Previously when you press R, you get the most recent result.
Now you can press R repeatedly for 5 most recent results.

Here is a quick illustration.

The most recent result is 10. You press R to insert it.
If you press R again, you get the next result, 20.

In other words:

R → 10

RR → 20

RRR → 30

And so on… until you reach the 5th one, 50. Another press will take you back to the first result.

The result button

It now has a new menu — click and hold to reveal it.

It lets you see and insert results without opening History.

To see the expression, just hover your mouse over it.

Be informed

You may have noticed, as you press R successively, Magic Number flashes the name of the result.

Besides name, you can see the expression.

Choose “Flash Expression” and you get:

You can turn off flashing with “Do Not Flash”.
You can still see the detail by pressing ⌘I (that’s the shortcut for Inspect).

Robustness

2.7.5 Complex Made Simple

Simple notation for complex numbers — A little story

CXXIII

This is the Roman numeral for one hundred and twenty three. It’s shorter than words, but not as easy as 123. What’s more, it’s awkward for math.

Our numeral system began with whole numbers. Fractions were added later. The number 123.4 has 123 as the whole part and 4 as the fractional part; both separated by the decimal separator. The notation means 123 + 4/10.

Mathematicians could have invented a different notation like 123 + 4 f.
Here f is 1/10. Overall it’s shorter than 123 + 4/10. It’s a step forward.

For fun, let's do some math.
Let's calculate 100 × 123.4… oh I mean 100 × 123 + 4f.

But wait… multiplication has higher precedence than addition.
We really have to bracket 123 + 4 f.

100 × ( 123 + 4 f )

Let’s subtract, 200 – 123.4.
Again we need those parentheses.

200 – ( 123 + 4 f )

In fact, we need parentheses for division, square root, trigonometry…
The list goes on. It’s a parenthesizing nightmare.

Does this look familiar?

100 × ( 123 + 4 i )
200 – ( 123 + 4 i )

Complex number also has 2 parts, just like 123.4.
Its format, x + y i, while informative, is also cumbersome.

Just as we have decimal separator, we can devise a complex separator for complex numbers. The question is, what symbol should we use?

The symbol for complex separator

In the spirit of decimal separator, the symbol has to be easy to type and easy on the eyes (compare 8.88 with 8&88). More importantly, it has to be distinctive from other math symbols.

In writing, we use hyphens to join related words, for example, two-dimensional, full-length. Unfortunately hyphens look too similar to subtraction.

In computing, underscores are commonly used for the same job, for example, full_length. Furthermore, they don’t conflict with math.

There you have it. Underscore is it.

x + y i	x_y
Standard	Simple
123 + 4 i	123_4
Standard	Simple

Besides easier to type, it’s clearer too:

100 × 123_4
200 – 123_4

Here are some comparisons:


Standard		Simple

Standard		Simple

What’s next?

Simple notation makes complex numbers more approachable as 2D numbers.
Whether it is x and y or width and height, anyone can benefit. Read on.

2D for everyone

Let’s enlarge a 16 by 9 photo 400%.
To calculate the new dimension, you do the same math twice: 16 × 4 and 9 × 4.

As 16 by 9 is two-dimensional, let’s use a ‘2D’ number, 16 _ 9. This number is actually a complex number, usually written as 16 + 9 i, but for convenience, we use Magic Number’s simple notation, x _ y.

If math is not your subject, don’t let the word complex deter you.

Remember our math evolves over time. First we had whole numbers (1, 2, 3) and later fractions. Early humans probably think 1 ½ is complex.

Numbers can have two parts. That is, they can be two-dimensional. Everyday numbers have one part — the real part, represented horizontally. The vertical or the imaginary part emerges under special circumstances.

Note also negative numbers were once thought as ‘imaginary’ as people could not imagine having less than nothing. But if the subject is money, negative would mean debt. Likewise, the imaginary part has significance in many subjects.

The nice thing is, for 2D arithmetic and geometry, you don’t have to understand the imaginary part. Just use it as the vertical dimension.

If you want to learn more, go to BetterExplained for the wonderful articles on imaginary and complex numbers. We’ve learned a lot there. Last but not least, check out Wikipedia.

Enter: 16 _ 9 × 4.

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

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

What’s the resolution at 50%?

Let’s make it 4 times smaller.

Distance between 2 points

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

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

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.)

Go to View ▸ Complex Result ▸ Polar
Shortcut: ⌥⌘P

The distance is 5.

2D-List

2D-List is now easier for storing and reusing complex numbers.

Here is a complex number.

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

3 subtle things

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.
New 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.

What’s next?

2D-List can split a complex number. The new ‘Part’ function can do this too.
It does so without populating your list. It’s easy and versatile. Check it out.

Easier complex functions

These functions are under Calculation ▸ Extra Functions, towards the end.
( Shortcut: ⇧⌘F )

Part

The part function lets you pick the real and imaginary part.
(In 2D speaking, that’s x and y.)

If the complex number is in polar, you can pick the absolute value (magnitude) and the argument (angle).

What’s good to know — Auto bracketing

As the part function is used for complex numbers, for convenience and clarity, Magic Number inserts parentheses if needed.

Complex numbers have the forms a ± b i, a_b, r ∠ θ. But our needs 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, something_something, and something ∠ something. Give it a try. It works well for most cases.

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

Polar & Rectangular

Previously if you want to see the result in polar, you need to switch to polar view by pressing ⌥⌘P.

You now have dedicated functions. There’s no need to switch views.

You can convert to polar

and vice versa.

These functions also have auto bracketing.

Modulo & quotient

For mathematical correctness, Magic Number now uses Euclidean division for modulo and quotient. This only affects negative numbers, for example:

Before	Now
7 mod 3 = 1	7 mod 3 = 1
-7 mod 3 = -1	-7 mod 3 = 2
7 mod -3 = 1	7 mod -3 = -2
-7 mod -3 = -1	-7 mod -3 = -1

Here is the full story behind the change.

Let’s look at simple division like 7 ÷ 3. If we want integer result, the answer (quotient) is 2 and the remainder (modulo) is 1.

We can see

dividend = divisor × quotient + modulo 7 = 3 × 2 + 1

and crucially

| modulo | < | divisor |

In Magic Number, you can calculate modulo and quotient with mod and quo.

7 mod 3 = 1 7 quo 3 = 2

But when we have negative number, as in –7 mod 3, there are two possibilities.

We are dividing by a positive 3.
The modulo should be positive too.

This is only possible when

modulo = 2 quotient = -3

-7 = 3 × quotient + modulo -7 = 3 × -3 + 2

This means

quotient = floor (-7 ÷ 3)

The modulo should have the same sign as –7. It should be negative.

This is only possible when

modulo = –1 quotient = –2

-7 = 3 × quotient + modulo -7 = 3 × -2 + -1

This means

quotient = integer of (-7 ÷ 3)

B is used in popular programming languages like C, Java, and PHP. As early users of modulo were mostly programmers, Magic Number opted for B.

A, being based on Euclidean division, is a better suited for math. It is used in spreadsheets, Mathematica, and languages like Haskell and R.

For math and new users, Magic Number now uses A.

If you prefer the old way…

You are taken care of. You get the ‘new’ functions Remainder and Integer Division. They use B for modulo and quotient.

Summary

A mod	B rem
7 mod 3 = 1	7 rem 3 = 1
-7 mod 3 = 2	-7 rem 3 = -1
7 mod -3 = -2	7 rem -3 = 1
-7 mod -3 = -1	-7 rem -3 = -1

A quo	B div
7 quo 3 = 2	7 div 3 = 2
-7 quo 3 = -3	-7 div 3 = -2
7 quo -3 = -3	7 div -3 = -2
-7 quo -3 = 2	-7 div -3 = 2

Optimization & bug fixes

2.7.2 Healthier

Improved compatibility with OS X El Capitan.

Engineering-wise, Magic Number is more streamlined and optimized.

More functions for history. (Less taxing for tax.)

How the tax functions can help you:

We start with the U.S. where prices are exclusive of tax.
‘399 + 749’ is actually the sum of an Apple Watch and an iPhone.

If you want to know the sum with tax, you have to do another
calculation like ‘(399 + 749) + tax’.

Now you can do it with one click.

(For simplicity, we use a tax rate of 10%.)

In Europe, prices are usually inclusive of tax. Let’s say for bookkeeping purposes, we need to find the price without tax. Again it’s just one click away:

What this means:

You can use History to quickly look at the effect of tax. Our examples show tax for one calculation, but you are not limited to one.

Another nice touch:

When you use a tax function, your result is automatically shown in currency format (2 decimal places).

A sweeter list

2.7 Obsession

We obsess every detail — from design to workflow.
And you get a better app in every possible way.

Refined math typesetting

Subscripts for everyone

In math, we often use subscripts for variables, for example, a₁ , a₂ , a₃ …
We can use them in everyday life too.
They make memory and tax names more compact, yet informative.

( For memory name, go to menu Calculation ▸ Memory Options
and for tax name, go to Calculation ▸ Tax & Rates )

Automatic subscripts

If the name ends with a number, you get subscript automatically.

Examples:

Namea2

a₂

NameiMac27

iMac₂₇

Nametax2015

tax₂₀₁₅

Manual subscripts

You can also specify in the form of ‘name_subscript’.
The underscore ‘_’ acts as a separator.

Examples:

Namea_b

a_b

NameiMac_retina

iMac_retina

Nametax_USA

tax_USA

Note:

If you want ‘tax_USA’ instead of ‘tax_USA’,
prefix it with ‘_’ so that it becomes ‘_tax_USA’.

More love for scientists

Working with scientific constants is easier with the new Constants drawer.

Click to sort constants by name. Click again to sort by symbols.
If you want see the constant’s value quickly, press ⌘i.
Press ⌃K (ctrl-K) to insert the most recent constant (Standard gravity).
Press again to insert the next recent one (Electron volt).
You can look up constants on Wikipedia.
You can see constant names with units.

Other improvements:

To cycle through the recent constant in reverse, press ⇧⌃K.
(Useful when you want Magic Number to remember 10 recent constants.)
Constants are also listed under the new Insert menu.
The sorting and content are also dictated by 1 and 5.
You can search constants by pressing ⇧⌘/.
(This is really harnessing OS X’s power to search the menus.)
Right-click a constant in the drawer, you get a contextual menu
with detailed actions.
Last but not least, the constants are updated with the latest CODATA recommended values.

Here are some tips if you want to learn more.

Global shortcut + Custom keys

(Menu ▸ Magic Number ▸ Preferences)

Custom keys

In brief, custom key let you use an ‘alpha’ key as a shortcut for a button.
The first example uses ‘p’ for the ‘%’ button.*

Technically you can use any ‘character’ based key. The above uses ‘\’ for ‘÷’. Another example, ⇧2 for x². If you like, you can also use ⌥2.

‘Esc’ is the only non-character key you can use. ‘Esc’ for ‘clear’ is a shortcut on Apple’s calculator. If you are a Spotlight fan, you can use ‘esc’ for ‘hide’.

* Note: it’s the lower case ‘p’. If you use capital ‘P’, you need to press ⇧.

Global shortcut

Global shortcut lets you switch to Magic Number from any app.
The default shortcut is control-space. You can easily change this.

When you press again, it hides Magic Number.
You can customize this ‘press again’ behavior too.

Here are some useful tips on customizations.

Smart select — Clever double-clicking

In text, when you double-click, you select a word.
What about double-clicking an expression?

You can say a word is a just group of related letters.
An expression is similar; it’s made of groups of related elements.

The most obvious kind of group is the bracket (parenthesis).
It literally groups elements together.

To select it as a group, double-click either ( or ).

You can also press ⌘G.

Like editing text, it’s easier to double-click to select a word.
In Magic Number, double-clicking is faster and more intelligent.

Selective bracketing

Once you have your selection, you can bracket it.

Shortcut: ⌘ ] or ⌘ )
(Whatever is easier for you.)

Notice the cursor ( ) is at the right of the bracket.
This makes it handy to insert more at the right, e.g.
( 8² + 15² ) / 2

What if you want to make it √ ( 8² + 15² ), i.e. you want to insert at the left.
As you might have guessed, there is a left-biased shortcut.

Obsessive detail

Whether you use the left or right-biased shortcut, the cursor is always outside the bracket. This makes sense because you want to do something to the group, such as dividing by 2 or taking the square root.

When your selection is just a number, the cursor will be inside. The top reason for bracketing a number is to combine it with other element to form a group. Here is an example.

More nice touches

A bigger status

The status is now clearer and easier to read.
If you like it bigger, go to menu View ▸ Status Size

A bigger history too

To have it bigger, open the drawer and click:
▸ Options ▸ Text Size: Large

Export multiple PDFs in history

Speaking of history, the days of exporting one PDF at a time is over.

Simply select multiple histories and click:
▸ Export # items as PDF

Extra drawer shortcuts

`⌘ ▲` command-up	Select first item
`⌘ ▼` command-down	Select last item
`⌃ ↩` control-return	Insert an item to the display.
`⌃ ⇥` control-tab	Switch the focus between the calculator and drawer