# A-level Computing 2009/CIE/Theory Fundamentals/Number representation

Binary:

• A denary value is a regular integer.
• A binary value is written as a collection of 1s and 0s.
• The first value in binary corresponds to a 1 in denary and the number to it’s left is double the previous number.
 128 64 32 16 8 4 2 1 0 1 0 1 0 0 1 0
• Using the table above we can calculate the denary value of the binary number. We can do this by adding the corresponding denary values of each column together.
• E.g. 2*1+16*1+64*1 = 82, so 01010010 is 82 in denary.
• Another way to memorize this is that each value is an increased power of 2.

• Hexadecimal is a base-16 number system which means we will have 16 different characters to represent our value.
• After 9, values are represented by letters from A to F.
• Hexadecimal is written in the same way as binary, but instead of going up in powers of 2 we go up in powers of 16.
• E.g. F1 = 16*15 + 1*1 = 241
• A quick way to convert hexadecimal to binary is converting each individual value into and binary and putting them together.
• E.g. F = 0111 and 1 = 0001, therefore F1 in binary is 01110001.

Two’s Complement:

• We can represent a negative number in binary by making the most significant bit (MSB) a sign bit, which will tell us whether the number is positive or negative.
• If the MSB is 0 then the number is positive, if 1 then the number is negative.
 Method: Converting a Negative Denary Number into Binary Twos Complement Let's say you want to convert -35 into Binary Twos Complement. First, find the binary equivalent of 35 (the positive version) ```32 16 8 4 2 1 1 0 0 0 1 1 ``` Now add an extra bit before the MSB, make it a zero, which gives you: ```64 32 16 8 4 2 1 0 1 0 0 0 1 1 ``` Now 'flip' all the bits: if it's a 0, make it a 1; if it's a 1, make it a 0: ```64 32 16 8 4 2 1 1 0 1 1 1 0 0 ``` This new bit represents -64 (minus 64). Now add 1: ```64 32 16 8 4 2 1 1 0 1 1 1 0 0 + 1 1 0 1 1 1 0 1 ``` If we perform a quick binary -> denary conversion, we have: -64 + 16 + 8 + 4 + 1 = -64 + 29 = -35
 Method 1: converting twos complement to denary To find the value of the negative number we must find and keep the right most 1 and all bits to its right, and then flip everything to its left. Here is an example: ```1111 1011 note the number is negative ``` ```1111 1011 find the right most one 1111 1011 0000 0101 flip all the bits to its left ``` We can now work out the value of this new number which is: ```128 64 32 16 8 4 2 1 0 0 0 0 0 1 0 1 4 + 1 = −5 (remember the sign you worked out earlier!) ```
 Method 2: converting twos complement to denary To find the value of the negative number we must take the MSB and apply a negative value to it. Then we can add all the heading values together ```1111 1011 note the number is negative -128 64 32 16 8 4 2 1 1 1 1 1 1 0 1 1 -128 +64 +32 +16 +8 +2 +1 = -5 ```

Image Representation:

• A bitmapped image is encoded by assigning a solid color to each pixel.
• Pixel: the smallest possible addressable area defined by a solid color, represented as binary, in an image.
• Image resolution:number of pixels an image contains per inch/cm.
• Screen resolution:the number of pixels per row by the number of pixels per column.
• Color Depth:the number of bits used to represent the color of a single pixel. An image with n bits has 2^n colors per pixel.
• File Size = Number of Pixels * Colour Depth
• Vector graphics:images defined using mathematics and geometry. Allowing for scalability.
• Drawing list: a set of commands used to define a vector image.

Sound Representation:

• Sound: vibrations that travel through a medium, they are are continuous in nature, which means there is infinite amount of detail for a sound.
• An analogue to digital converter(ADC) converts analogue sound into digital signals which can be digitally stored.
• A digital to analogue converter(DAC) converts digital signals into analogue sound that can be output.
• To convert a continuous wave signal into a digital form, the computer has to sample the sound.