Positive Binary Numbers (OCR A Level Computer Science)

Binary

What is Binary?

• binary refers to a system of representing information using only two digits: 0 and 1.

Bits

• A bit is the smallest unit of digital information, representing either an "off" (0) or an "on" (1) state.

The status of a computer bit being on or off.

Bytes

• Bits are grouped into larger structures to form bytes (8 bits), words, and long words

• These groupings allow us to represent more complex information, like numbers, text, and instructions

Groups bits to store more complex information

What do the 0s and 1s represent?

• In binary, each level is based on powers of 2

• In the 8-bit binary number below, each of the 8 columns represents values of 2ⁿ , e.g.

• 2⁰ = 1

• 2¹ = 2

• 2² = 4

• 2³ = 8

• 2⁴ = 16

• 2⁵ = 32

• 2⁶ = 64

• 2⁷ = 128

Binary powers of two

Encoding and representation

• Various encoding schemes, like ASCII for text or JPEG for images, map these binary values to human-readable forms

• For example, the binary value 01001000 represents the letter 'H' in ASCII

• In the example below, an image is shaded black or white depending on the binary value for each pixel

• Each row in the image requires 1 byte of storage

Pixel-shading in a bitmap image

Abstraction layers

• Computers handle large volumes of basic numeric data

• To create meaningful representations of data, layers of abstraction exist so that basic data can be interpreted upwards into other forms, e.g. images, sound, video

• The same principle applies to programming languages that compile down into binary code

• At the bottom, you have binary, and each layer above it allows for more meaningful information to be represented

Abstractions of binary

Converting Between Binary & Denary

• Within computer science, two common number systems are:

  • Denary numbers – This is also known as base-10. These are used by humans and consist of 10 digits ranging from 0 to 9

  • Binary numbers - Computer systems store data using 1s and 0s. This is known as binary or base-2. Computer systems store data in binary format because computers are made up of circuits and switches that are either on (1) or off (0)

• Binary numbers can be converted into denary and vice-versa

• For example the 8-bit binary number 01101001 can be converted into denary using the following method:

Binary to decimal conversion

Therefore the 8-bit binary number 01101001 is 105 as a denary value.

Converting Between Denary & Binary

• To convert the denary number 101 to binary, we firstly write out binary number system

• Then we start at the left and look for the highest number that is less than or equal to 101 and if so, place a 1 in that column. Otherwise, place a 0 in the column

• 128 is bigger than 101 and therefore we place a 0 in that column

• 64 is smaller than 101 so we place a 1 in that column. 101 - 64 = 37. This now means we have 37 left to find

• 32 is smaller than 37 so we place a 1 in that column. 37 - 32 = 5. This now means we have 5 left to find

• 16 is bigger than 5 and therefore we place a 0 in that column

• 8 is bigger than 5 and therefore we place a 0 in that column

• 4 is smaller than 5 so we place a 1 in that column. 5 - 1 = 1. This now means we have 1 left to find

• 2 is bigger than 1 and therefore we place a 0 in that column

• 1 is equal to the number we have left so we place a 1 in that column

• 64 + 32 + 4 + 1 = 101. Therefore the denary number 101 in binary is 01100101