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

inside a `UInt8`

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

Let’s look at our 8 bit lengthy unsigned integer kind. 8 bit implies that we’ve 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 afterward. In Swift we will examine if a sort is a signed kind and we will additionally get the size of the integer kind by means 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 simple 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? Nicely, the trick is that 1 bit from the 8 is reserved for the optimistic / destructive image. Often 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, because the **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 characterize -128? Is not -127 (1111 1111) the minimal destructive worth? 😅

Nope, that is not how **destructive binary numbers** work. In an effort to perceive destructive integer illustration utilizing binary numbers, first we’ve to introduce a brand new time period known as two’s complement, which is a straightforward methodology 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 thankfully it may be changed with an addition operation if we retailer destructive numbers in a particular method 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 destructive 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 destructive numbers.

`0010 1010`

(+42)`1111 1010`

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

=(+36)

So, you would possibly suppose, what is the cope with the additional 1 to start with of the 8 bit outcome? Nicely, that is known 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 possibly can 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 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 will say that these capabilities function utilizing a single bit. This time we’ll see how bitwise operators can carry out numerous transformations utilizing a number of bits. In our pattern instances 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))
```

Nicely, 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 cause the inverted quantity represents 249 based on our debug console. 🙃

It’s because the that means of the UInt8 kind has no understanding concerning the signal bit, and the eighth bit is at all times refers back to the 2^{8} worth. Nonetheless, in some instances e.g. if you do low degree programming, resembling 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 group of UInt8 numbers. Truly you may discover various use-cases for the UInt8 kind if you happen to take a deeper have a look at the prevailing frameworks & libraries. Cryptography, knowledge transfers, and so forth.

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 offer 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 count on 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 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 should use a bitmask to extract & set given elements 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 if you happen to do not wish to fiddle with 8 totally different Bool variables, however one there may be 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

Through the use of a bitwise shift operation you possibly can transfer a bit in a given quantity to left or proper. Left shift is actually a multiplication operation and proper shift is equivalent 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 number 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 possibly can see we’ve to watch out with left shift operations, because the 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 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 possibly can 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 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’ll set the worth utilizing a bitwise or operate. I suppose you possibly can determine 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 already know all of the little the small print. ☺️

I believe I will reduce it right here, and I am going to make simply one other submit about overflows, carry bits and numerous transformations, perhaps we’ll contain hex numbers as effectively, 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. 👾

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