## Representing numbers as integers

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

```
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 sort that we will retailer in a given variable. For instance, we won’t retailer the worth `69420`

inside a `UInt8`

sort, as a result of there are merely not sufficient bits to signify this large quantity. ?

Let’s study our 8 bit lengthy unsigned integer sort. 8 bit signifies that we have now 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 is not that tough in any respect, however let’s come again to this matter in a while. In Swift we will examine if a sort is a signed sort and we will additionally get the size of the integer sort by way of the `bitWidth`

property.

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

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

What about signed varieties? Effectively, the trick is that 1 bit from the 8 is reserved for the optimistic / adverse image. Often the primary bit represents the signal and the remaining 7 bits can retailer the precise numeric values. For instance the `Int8`

sort can retailer numbers from -128 til 127, for the reason that **most optimistic 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 optimistic quantity and the remaining 7 bits are all ones.

So how the hack can we signify -128? Is not -127 (1111 1111) the minimal adverse worth? ?

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

## Fundamental signed quantity maths

It’s comparatively straightforward 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** then again is a bit tougher, however luckily it may be changed with an addition operation if we retailer adverse numbers in a particular approach and that is the place two’s complement is available in.

Lets say that we might like so as to add two numbers:

`0010 1010`

(+42)`0100 0101`

+(+69)`0110 1111`

=(+111)

Now let’s add a optimistic and a adverse quantity saved utilizing two’s complement, first we have to categorical -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 will merely carry out an addition operation on the optimistic and adverse numbers.

`0010 1010`

(+42)`1111 1010`

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

=(+36)

So, you may assume, what is the cope with the additional 1 to start with of the 8 bit end result? Effectively, that is referred to as a carry bit, and in our case it will not have an effect on our last end result, since we have carried out a subtraction as a substitute of an addition. As you may see the remaining 8 bit represents the optimistic quantity 36 and 42-6 is strictly 36, we will merely ignore the additional flag for now. ?

## Binary operators in Swift

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

sort. To start with, we should always speak about bitwise operators in Swift. In my earlier article we have talked about Bool operators (AND, OR, NOT) and the Boolean algebra, now we will say that these capabilities function utilizing a single bit. This time we will see how bitwise operators can carry out varied transformations utilizing a number of bits. In our pattern circumstances it is all the time 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))
```

Effectively, the issue is that we’ll preserve seeing decimal numbers on a regular basis when utilizing int varieties in Swift. We are able to print out the right 1111 1001 end result, utilizing a `String`

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

It’s because the that means of the UInt8 sort has no understanding concerning the signal bit, and the eighth bit is all the time refers back to the 2^{8} worth. Nonetheless, in some circumstances e.g. if you do low degree programming, comparable to constructing a NES emulator written in Swift, that is the fitting knowledge sort to decide on.

The Information sort from the Basis framework is taken into account to be a set of UInt8 numbers. Truly you may discover numerous use-cases for the UInt8 sort 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) - rely
guard maxLength > 0 else {
return self
}
return String(repeating: String(character), rely: 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 supply our customized logic if we wish to categorical signed numbers utilizing UInt8, however that is solely going to occur after we all know extra concerning 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 have been true, the OR operator returns a 1 if both of the bits have been true and the XOR operator solely returns a real worth if solely one of many bits have been true.

- AND
`&`

– 1 if each bits have been 1 - OR
`|`

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

– 1 if solely one of many bits have 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 may be not a lot cause 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 unique goal.

You need to use 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 sort you should 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 totally different Bool variables, however one there may be one factor that may be very inconvenient about this answer. We all the time have to make use of the fitting energy of two, after all we may use pow, however there’s a extra elegant answer for this situation.

### Bitwise left & proper shift operators

By utilizing a bitwise shift operation you may 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 fitting by one place halves its worth.” – swift.org

It is fairly easy, however let me present you a couple of sensible examples so you may 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 may see we have now to watch out with left shift operations, for the reason that end result 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 end result. Proper shifting is all the time going to finish up as a zero worth. ⚠️

Now again to our standing flag instance, we will 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 may see we have used numerous 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 check 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 may determine the final line based mostly on this data, but when not simply follow these operators, they’re very good to make use of as soon as all of the little the small print. ☺️

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