Data Representation

The binary system is the way in which a computer represents positive and negative whole numbers.

Binary is a two state system which means numbers are represented using a series of 1s and 0s

1 meaning a voltage is present and a 0 meaning a voltage is not present - specifically transistors turn on at around 0.7 volts

At National 5 level, we only had to deal with positive numbers, but at higher we need to know how to convert positive and negative numbers

Let’s figure out what the binary number 1001 means in denary:

To do this, add each number with a 1 under it together, so 8 + 1. Total is 9. So 1001 = 9.

Convert:

1000 1000

To denary.

Convert:

1100 1111

To denary.

Convert:

0010 0100

To denary.

Now let’s calculate 35:

Double check it by adding all the numbers with a 1 under them together.​32 + 2 + 1 = 35

Convert:

28

To 8-bit binary

Convert:

32

To 8-bit binary

Convert:

254

To 8-bit binary

Fewer arithmetic operations

Any slight voltage drop does not change the data.
    Imagine trying to represent 0 to 9 with discrete voltages!
    Just use zero volts = 0,   any volts = 1

Two state storage is easy in storage devices such as CDs

Large number of digits required for even moderately large numbers 

E.g. 1000 0000 0000     =     4096
       (12 digits)                       (4 digits)

Previously, we’ve only had to deal with positive numbers, but how does a computer represent negative whole numbers?

We use Two’s complement (note the spelling of complement).

Two’s compliment allows us to represent both positive and negative numbers and there’s two methods to do it.

This method is carried out using three steps. 

1. Find the positive value
2. Switch the 1s and 0s
3. Add 1

Let’s convert -116 using 8-bit Two’s complement
 

Step 1– Find positive 116 in Binary

Step 2 - Switch all the 1s and 0s:

• Our next step is to switch the 1s and 0s around.

• In a Two’s complement number if the left most bit is 1 then its  a negative number, if it’s a 0 then it’s a positive number. 

Step 3 - Add 1

• We need to move to the right hand side and add a 1 where we see our first 0 and then replace the 1s with a 0

That method isn't the only method, there are actually several. 

The first method has a lot of steps involved in it.

The second method is easier and can also be used as a verification technique if you do choose to use the first method.

Step 1 - Start negative

• Start with your column headings. Make sure you’re left most bit (128) is negative!

• Put a 1 under 128

Step 3 - Make 116

• Put 1s under each of the numbers that added to -128 make up -116.

• 1000 0000 = -128
• 0100 0000 = 64

• A floating-point number is made of two parts called the mantissa and exponent
• The mantissa dictates the precision of a number, the more bits allocated to the mantissa, the more precise a number can be
• If you want to be store a large range of numbers then you need to allocate more bits to the storage of the exponent, as the exponent dictates the range of numbers that can be represented

The number 20.125 can be represented as .20125 x 10²

20.125 can be represented as .20125 x ^{base places point moved}

But a computer works in Binary so:

At Higher level you will be given a real binary number and be asked to identify the mantissa and exponent.

We are going to start by converting real numbers into binary.

Let's convert 10.75 to floating point.

Once we have our decimal number in binary we have to represent it as a floating point number.

Step 1: Move the decimal point before the first 1 and write as floating point number:

• This means it's moved three steps.

Step 3: Convert 3 (the number of steps the decimal moved) into a binary number.

Sometimes you might be given a number where you need to move the decimal point to the right instead of the left. For example:

0.0001101011

When you move the decimal point to the right the exponent becomes negative. So our number would be:

.1101011 x 2^-3

Convert the following binary number into Floating Point Representation, assigning 16 bits to the Mantissa (incl. sign bit) and 8 bits for the Exponent.

0.0000110101

Convert the following binary number into Floating Point Representation, assigning 16 bits to the Mantissa (incl. sign bit) and 8 bits for the Exponent.

0.0000110101

Step 1: Move the decimal point so it is before the first 1

0.0000110101 = .110101

Step 3: Convert -4 (the number of steps the decimal moved) into a binary number.

We’ve already seen how numbers are stored in a computer system and that computers can only store numbers as a series of 1s and 0s.

When it comes to storing text a computer works in the exact same way. They store text with its own unique binary number

Unicode is the standard that has been developed for text representation.

We should already know from National 5 about Extended ASCII.

We should already know from National 5 about Extended ASCII.

The Unicode system uses at least 16 bits to represent each character (there are now 32 bit Unicode variations as well).

That means we can store and represent 65,356 different characters codes.

The table below compares E-ASCII and Unicode

Bitmapped graphics are a type of graphic that store images as colour code for a two-dimensional grid of pixels.

Pixel is a portmanteau of picture and element which are the dots that make up the graphic

Resolution is the number of pixels going across (the width) the image multiplied by the number of pixels going up and down (the height) the image.

It's useful to know how the file size of a standard and uncompressed bitmap graphic is calculated and it's really easy to understand.

You can use the following formula to calculate it:

Vector graphics store images as a list of objects, each of which is described by its attributes

Object attributes include: starting (x, y) co-ordinates, line colour, fill colour, line thickness etc.

Vector graphics are resolution independent which means the resolution can be increased without a loss in quality.