## Representing numbers as integers

Now that we all know what sort of integers can be found in Swift, it is time to speak a bit about what sort of numbers can we symbolize utilizing these knowledge sorts.

```
print(Int.min)
print(Int.max)
print(UInt.min)
print(UInt.max)
print(UInt8.min)
print(UInt8.max)
print(UInt16.min)
print(UInt16.max)
print(UInt32.min)
print(UInt32.max)
print(UInt64.min)
print(UInt64.max)
print(Int8.min)
print(Int8.max)
print(Int16.min)
print(Int16.max)
print(Int32.min)
print(Int32.max)
print(Int64.min)
print(Int64.max)
```

So there’s a minimal and most worth for every integer kind that we are able to retailer in a given variable. For instance, we will not retailer the worth `69420`

inside a `UInt8`

kind, as a result of there are merely not sufficient bits to symbolize this large quantity. 🤓

Let’s look at our 8 bit lengthy unsigned integer kind. 8 bit signifies that we now have actually 8 locations to retailer boolean values (ones and zeros) utilizing the binary quantity illustration. 0101 0110 in binary is 86 utilizing the “common” decimal quantity format. This binary quantity is a base-2 numerical system (a positional notation) with a radix of two. The quantity 86 might be interpreted as:

**0***2^{8}+**1***2^{7}+**0***2^{6}+**1***2^{5}+**0***2^{4}+**1***2^{3}+**1***2^{2}+**0***2^{1}+**0***2^{0}**0***128+**1***64+**0***32+**1***16 +**0***8+**1***4+**1***2+**0***1- 64+16+4+2
- 86

We are able to convert forwards and backwards between decimal and binary numbers, it isn’t that arduous in any respect, however let’s come again to this matter in a while. In Swift we are able to examine if a kind is a signed kind and we are able to additionally get the size of the integer kind by the `bitWidth`

property.

```
print(Int.isSigned)
print(UInt.isSigned)
print(Int.bitWidth)
print(UInt8.bitWidth)
```

Primarily based on this logic, now it is fairly easy that an 8 bit lengthy unsigned kind can solely retailer 255 as the utmost worth (1111 1111), since that is 128+64+32+16+8+4+2+1.

What about signed sorts? Properly, the trick is that 1 bit from the 8 is reserved for the constructive / damaging image. Normally the primary bit represents the signal and the remaining 7 bits can retailer the precise numeric values. For instance the `Int8`

kind can retailer numbers from -128 til 127, for the reason that **most constructive worth** is represented as **0111 1111**, 64+32+16+8+4+2+1, the place the main zero signifies that we’re speaking a couple of constructive quantity and the remaining 7 bits are all ones.

So how the hack can we symbolize -128? Is not -127 (1111 1111) the minimal damaging worth? 😅

Nope, that is not how **damaging binary numbers** work. With the intention to perceive damaging integer illustration utilizing binary numbers, first we now have to introduce a brand new time period referred to as two’s complement, which is an easy technique of signed quantity illustration.

## Primary signed quantity maths

It’s comparatively simple so as to add two binary numbers, you simply add the bits so as with a carry, identical to you’d do **addition** utilizing decimal numbers. **Subtraction** however is a bit more durable, however luckily it may be changed with an addition operation if we retailer damaging numbers in a particular method and that is the place two’s complement is available in.

We could say that we would like so as to add two numbers:

`0010 1010`

(+42)`0100 0101`

+(+69)`0110 1111`

=(+111)

Now let’s add a constructive and a damaging quantity saved utilizing two’s complement, first we have to specific -6 utilizing a signed 8 bit binary quantity format:

`0000 0110`

(+6)`1111 1001`

(one’s complement = inverted bits)`1111 1010`

(two’s complenet = add +1 (0000 0001) to 1’s complement)

Now we are able to merely carry out an addition operation on the constructive and damaging numbers.

`0010 1010`

(+42)`1111 1010`

+(-6)`(1) 0010 0100`

=(+36)

So, you would possibly assume, what is the cope with the additional 1 at first of the 8 bit outcome? Properly, that is referred to as a carry bit, and in our case it will not have an effect on our last outcome, since we have carried out a subtraction as a substitute of an addition. As you’ll be able to see the remaining 8 bit represents the constructive quantity 36 and 42-6 is precisely 36, we are able to merely ignore the additional flag for now. 😅

## Binary operators in Swift

Sufficient from the speculation, let’s dive in with some actual world examples utilizing the `UInt8`

kind. To start with, we should always discuss bitwise operators in Swift. In my earlier article we have talked about Bool operators (AND, OR, NOT) and the Boolean algebra, now we are able to say that these capabilities function utilizing a single bit. This time we will see how bitwise operators can carry out numerous transformations utilizing a number of bits. In our pattern circumstances it is at all times going to be 8 bit. 🤓

### Bitwise NOT operator

This operator (~) inverts all bits in a quantity. We are able to use it to create one’s complement values.

```
let x: UInt8 = 0b00000110
let res = ~x
print(res)
print(String(res, radix: 2))
```

Properly, the issue is that we’ll hold seeing decimal numbers on a regular basis when utilizing int sorts in Swift. We are able to print out the right 1111 1001 outcome, utilizing a `String`

worth with the bottom of two, however for some motive the inverted quantity represents 249 in response to our debug console. 🙃

It’s because the that means of the UInt8 kind has no understanding in regards to the signal bit, and the eighth bit is at all times refers back to the 2^{8} worth. Nonetheless, in some circumstances e.g. whenever you do low stage programming, similar to constructing a NES emulator written in Swift, that is the correct knowledge kind to decide on.

The Knowledge kind from the Basis framework is taken into account to be a set of UInt8 numbers. Truly you will discover various use-cases for the UInt8 kind when you take a deeper take a look at the present frameworks & libraries. Cryptography, knowledge transfers, and many others.

Anyway, you can also make an extension to simply print out the binary illustration for any unsigned 8 bit quantity with main zeros if wanted. 0️⃣0️⃣0️⃣0️⃣ 0️⃣1️⃣1️⃣0️⃣

```
import Basis
fileprivate extension String {
func leftPad(with character: Character, size: UInt) -> String {
let maxLength = Int(size) - depend
guard maxLength > 0 else {
return self
}
return String(repeating: String(character), depend: maxLength) + self
}
}
extension UInt8 {
var bin: String {
String(self, radix: 2).leftPad(with: "0", size: 8)
}
}
let x: UInt8 = 0b00000110
print(String(x, radix: 2))
print(x.bin)
print((~x).bin)
let res = (~x) + 1
print(res.bin)
```

We nonetheless have to offer our customized logic if we wish to specific signed numbers utilizing UInt8, however that is solely going to occur after we all know extra in regards to the different bitwise operators.

### Bitwise AND, OR, XOR operators

These operators works identical to you’d anticipate it from the reality tables. The AND operator returns a one if each the bits had been true, the OR operator returns a 1 if both of the bits had been true and the XOR operator solely returns a real worth if solely one of many bits had been true.

- AND
`&`

– 1 if each bits had been 1 - OR
`|`

– 1 if both of the bits had been 1 - XOR
`^`

– 1 if solely one of many bits had been 1

Let me present you a fast instance for every operator in Swift.

```
let x: UInt8 = 42
let y: UInt8 = 28
print((x & y).bin)
print((x | y).bin)
print((x ^ y).bin)
```

Mathematically talking, there’s not a lot motive to carry out these operations, it will not provide you with a sum of the numbers or different primary calculation outcomes, however they’ve a special goal.

You should utilize the bitwise AND operator to extract bits from a given quantity. For instance if you wish to retailer 8 (or much less) particular person true or false values utilizing a single UInt8 kind you need to use a bitmask to extract & set given components of the quantity. 😷

```
var statusFlags: UInt8 = 0b00000100
print(statusFlags & 0b00000100 == 4)
print(statusFlags & 0b00010000 == 16)
statusFlags = statusFlags & 0b11101111 | 16
print(statusFlags.bin)
statusFlags = statusFlags & 0b11111011 | 0
print(statusFlags.bin)
statusFlags = statusFlags & 0b11101111 | 0
print(statusFlags.bin)
statusFlags = statusFlags & 0b11101011 | 4
print(statusFlags.bin)
```

That is good, particularly when you do not wish to fiddle with 8 completely different Bool variables, however one there’s one factor that could be very inconvenient about this answer. We at all times have to make use of the correct energy of two, after all we might use pow, however there’s a extra elegant answer for this problem.

### Bitwise left & proper shift operators

By utilizing a bitwise shift operation you’ll be able to transfer a bit in a given quantity to left or proper. Left shift is actually a multiplication operation and proper shift is an identical with a division by an element of two.

“Shifting an integer’s bits to the left by one place doubles its worth, whereas shifting it to the correct by one place halves its worth.” – swift.org

It is fairly easy, however let me present you a couple of sensible examples so you will perceive it in a bit. 😅

```
let meaningOfLife: UInt8 = 42
print(meaningOfLife << 1)
print(meaningOfLife << 2)
print(meaningOfLife << 3)
print(meaningOfLife >> 1)
print(meaningOfLife >> 2)
print(meaningOfLife >> 3)
print(meaningOfLife >> 4)
print(meaningOfLife >> 5)
print(meaningOfLife >> 6)
print(meaningOfLife >> 7)
```

As you’ll be able to see we now have to watch out with left shift operations, for the reason that outcome can overflow the 8 bit vary. If this occurs, the additional bit will simply go away and the remaining bits are going for use as a last outcome. Proper shifting is at all times going to finish up as a zero worth. ⚠️

Now again to our standing flag instance, we are able to use bit shifts, to make it extra easy.

```
var statusFlags: UInt8 = 0b00000100
print(statusFlags & 1 << 2 == 1 << 2)
statusFlags = statusFlags & ~(1 << 2) | 0
print(statusFlags.bin)
statusFlags = statusFlags & ~(1 << 2) | 1 << 2
print(statusFlags.bin)
```

As you’ll be able to see we have used various bitwise operations right here. For the primary examine we use left shift to create our masks, bitwise and to extract the worth utilizing the masks and at last left shift once more to match it with the underlying worth. Contained in the second set operation we use left shift to create a masks then we use the not operator to invert the bits, since we will set the worth utilizing a bitwise or operate. I suppose you’ll be able to work out the final line primarily based on this data, but when not simply observe these operators, they’re very good to make use of as soon as you realize all of the little the small print. ☺️

I feel I’ll minimize it right here, and I will make simply one other put up about overflows, carry bits and numerous transformations, perhaps we’ll contain hex numbers as nicely, anyway do not wish to promise something particular. Bitwise operations are usueful and enjoyable, simply observe & do not be afraid of a little bit of math. 👾