# 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:
* 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.