3.3.2 Converting Between Number Bases

• To convert a decimal number to a binary number, use the column headings. You need to find the highest column heading that you can take away from the number and start there.
• To convert the decimal number 57 into binary.
• The highest column heading we can take out of 57 is 32 (The next one is 64, too high):
  1. We can start by placing a 1 in the column headed 32 (57 - 32 = 25).
  2. We can then place a 1 in the column headed 16 (25 - 16 = 9).
  3. Finally place 1s in the columns 8 and 1 (9 - 8 = 1).

• This gives us the value: 57 = 00111001

Here are some more examples:

• You will notice in the examples above that we always write the binary numbers using eight bits.
• This is common practise, it is not incorrect to write the first binary value as 10111 rather than 00010111 (Without the leading zeros), but the second example is more commonly used as values are often stored in bytes.

• As mentioned previously, humans are not very good at remembering long strings of numbers. For example which of these are easier to remember: 0110111 or 6F
• So to make it easier, we can represent every group of 4 bits (Nibble) with a single digit.
• The smallest value you can hold in 4 bits is 0000. The largest value is 1111. This means that we need to represent the decimal values 0 to 15 with a single digit.
• As we only have 10 single digits in decimal (0 to 9), so to get around this proble we use uppercase letters A to F for the remaining six digits.
• The following table shows the hexadecimal digits and their decimal equivalents:

Useful Conversion Table

Note for above table

• Decimal Number 15 - A single digit replaces 4 bits, 15 is the biggest number you can have with 4 bits.
• Decimal Number 16 - So 16 is one group of 16 and no units.
• Decimal Number 255 - 255 is 15 groups of 16 + 15 units i.e. ((15 x 16) + 15 = 240 + 15).

Hexadecimal Explained

• The following video explains how hexadecimal number systems work.

• In GCSE computing you will only need to work with 8-bit binary numbers, which can be represented as two hex digits.
• The left-hand hex digit represents groups of 16, the right-hand hex digits represents the units.
• For example, to convert
• There are several methods for converting Decimal to Hexadecimal:
• Method 1: Convert to Binary/Hex
• Using the tables and video above try and convert the following decimal number: 112 first into Binary and then into Hexadecimal?
• Step 1: First we need to use the binary table to work out the Binary number.
• Step 2: Does 128 go into 112? No, so we place a 0 under the 128 value, then does 64 go into 112? Yes, 112 - 64 = 48 - so we place a 1 under the 64 value.
• Step 3: We can then check does 32 goes into 48? Yes, 48 - 32 = 16 - so we place a 1 under the 32 value.
• Step 4: Repeat this until there is no remainder.

• This gives us 01110000 - We then need to split the byte (8 bits) into two nibbles (4 bits).
• Then starting at the least significant bit (The right side).
• Read the 4 bits (0000) - Look at the conversion table above gives us a 0.
• Read the next 4 bits (0111) - Again look at the coversion table gives us a 7.
• So 112 in Decimal converts to 01110000 in Binary - Converts to 70 in Hexadecimal.
• Method 2: Divide by 16
• To convert the decimal number 182 into hex the first step is to work out how many groups of 16 there are in 182.
• Secondly work out how many units are left over.
• 182 / 16 = 11 remainder 6
• 11 is B in hex, 6 is just 6, so 182 in decimal = B6 hex.
• Method 3: Denary as a Middle Step
• With experience, you'll not need a table, as you'll come to recognise the hex equivalent for each nibble.
• However, if you don't have a conversion table or can't remember the values, you can convert between binary and hex, by using denary as a middle step.
• Lets start with the binary number 01011101
• Split it into nibbles - 0101 1101
• Convert to denary - 1*1 + 0*2 + 1*4 + 0*8 and = 1*1 + 0*2 + 1*4 + 1*8
• So we get - 5 and 13
• Now convert to Hex - 5 and D
• So the Hex is 5D

Converting Hexadecimal to Binary

• Converting a hexadecimal number into binary is a simple matter of converting each hex digit into a group of 4 binary digits. To convert the hex number A7 to binary:
• Use the Binary Conversion table above to look at the four left-hand binary digits that represent the hex number A —> 1010 and then look at the next four binary digits that represent the hex number 7 ---> 0111.
• So the hex number A7 can be represented by the binary value 10100111

More Examples:

• B5 ---> 1011 0101
• FA ---> 1111 1010

Converting Hexadecimal to Decimal

• To convert a hexadeicmal number into decimal, multiply the heading or place value by the hex digit value, to convert the hex number A7 to decimal. (Remember column place values are: 16 and 1, and A = 10)
• A = (10 x 16) + 7 = (7 x 1) = 167

More Examples:

• 7F —> 112 + 15 = 127
• CD —> 192 + 13 = 205