Representing Numbers

Applications of Hexadecimal

You might be wondering why we would want to use hexadecimal when we have binary and denary, and when computers store and calculate everything in binary. The answer is that it is entirely for human ease. Consider the following example:

• EFFE11₁₆ = 15728145₁₀ = 111011111111111000010001₂

All the digit strings are the same number and the easiest version to remember/understand for humans is the base-16, especially if it represents a sequence of 3 bytes as in colour codes. Hexadecimal is used in computers for representing numbers for human consumption, having uses for things such as:

• memory addresses
• error messages / memory dumps
• colour codes.

NOTE: Hexadecimal is used as it is shorthand for binary and easier for people to remember (than binary). It DOES NOT take up less space in computer memory, only on paper or in your head! Computers still have to store everything as binary even if it appears as hexadecimal on the screen.

In exam questions for why hexadecimal is used, you usually need to choose from the following

• Easier to write / Easier to remember / Quicker to transcribe than binary
• Less likely to make a human error writing / copying (compared to binary)
• Less digits to use / more compact (than binary or denary)
• Is closer to the underlying binary than denary is, so better for representing bytes of data

Hex Color Codes

"True color" in computers is where each pixel has 24 bits of color depth divided amongst the 3 colour channels Red, Green, and Blue - the three additive primary colours. That is, there is 1 Byte of information for each of RGB for each pixel. This is often written as 6 hex digits: #RRGGBB. For example. #FF0000 is bright red, #FFFF00 is bright yellow, #FF00FF is bright pink/magenta, #A8A8A8 is a light grey.

To approximately figure out what colour is represented by a RGB mix, place them on the rainbow / colour wheel. (Note that the concept of Blue in the 7 ROYGBIV colours due to Newton has shifted over the years)

Magenta R O Y G B I V Magenta

        R     G   B

• R+G = Orange to Yellow to Lime Green depending on the ratio
• G+B = Cyan / Aqua
• B+R = Hot Pink / Magenta

Note that the additive primary colours RGB used for mixing light are the opposite of the subtractive primary colours used for mixing paint and inks. Printers use CMYK, which stands for Cyan, Magenta, Yellow, Key/blacK.

Resources

• There is no Pink Light (Minute Physics), There is no White Light (The Science Asylum)
• Hex Color Game - Really Good practice!
• C0FFEE is the Color, Color Name and Hue.
• Colours & Maths Understanding the formulas of colour conversion

MAC Addresses, IPv4 and IPv6

Media Access Control addresses identify networking devices at the hardware level. It is a 48 bit address usually written as 6 pairs of hexadecimal digits separated by hyphens or colons. The first 24 bits correspond to the vendor of the hardware (ethernet, WiFi, bluetooth, etc...) device, the organizationally unique identifier (OUI).

IP addresses are used for identifying devices communicating with the Internet Protocol.

• IPv4 is a 4 byte (32 bit) address written using Denary, e.g., 192.168.0.5 might be the IP address of your laptop on your home network.
• IPv6 is 128 bits usually written as 8 groups of 4 hex digits, e.g., 2001:0db8:0000:0042:0000:8a2e:0370:7334
• IP addresses are hierarchal. From left-to-right representing the network, then subnetwork, then device.

You can find your MAC addresses and IP addresses by using the terminal command ifconfig (MacOS or Linux) or ipconfig in Windows.

You can find your external facing IP address (controlled by your ISP / school / etc...) using whatsmyip.org, and you can lookup the vendors associated with your MAC addresses using whatsmyip.org/mac-address-lookup