Unit 3 – Data representation

65 Denary to Binary

Start by writing out the place values.124816326412

8

Then write 1s underneath the place values that add up to the denary number.

0 1 0 0 0 0 0 1

64 16 1 + + = 81

Binary to Denary

0 1 0 1 0 0 0 1

Start by writing the place values above each bit.124816326412

8

Then write out the place values of the 1s.

Finally add the numbers together.

Binary AdditionThe process of performing addition in binary is very similar to addition in denary.1 1 1 10 1 1 0 10 0 1 1 1 +1 0 1 0 0

Rules:0 + 0 = 00 + 1 = 11 + 1 = 0 carry 11 + 1 + 1 = 1 carry 1

0 1 1 0 01 0 1 1 1 +

1 1 1 0 01 0 1 0 1 +

OverflowWhen there isn’t enough room for a result, this is called an overflow and it produces an overflow error.

1 1 1 11 1 1 0 1

1 1 0 1 1 1 +1 1 0 1 0 0

No room for a carry, so it is lost and we get the wrong answer.Humans can easily work around this, but it isn’t so easy for a computer.

Binary SubtractionThe process of performing subtraction in binary is very similar to subtraction in denary.

1 1 0 00 1 0 0 -1 0 0 0

Rule:0 - 1 won’t go so borrow 10 (2) from the column to the left. 2 - 1 = 1.

10 10

1 1 0 10 1 1 1 -0 1 1 0

1 1 1 00 1 0 1 -

Sign and MagnitudeThis is the simplest method of representing negative numbers in binary. The most significant bit is the ‘sign bit’.

Sign bit 128 64 32 16 8 4 2 10 0 0 0 1 1 0 1 1 +2

71 0 0 0 1 1 0 1 1 -27

1 = minus 0 = plus

The Problem with Sign and MagnitudeThe problem with sign and magnitude is that addition doesn’t always work.

0 0 0 0 0 1 1 1 (+7)

1 0 0 0 0 1 0 1 (-5) +1 0 0 0 1 1 0 0 (-

12)

Two’s Complement

Two’s complement is an alternative method of representing negative numbers. This method works with binary addition.

The most significant bit is a minus number.-128 64 32 16 8 4 2 1

1 0 0 0 1 1 0 1

-128 + 8 + 4 + 1 = -115

Floating Point NumbersIn binary floating point numbers are split into two parts; the Mantissa and the Exponent.

Mantissa Exponent0111 1000 0000 0011

0 . 1 1 1 1 0 0 0

The mantissa contains the actual number and the exponent defines the position of the binary point.

Floating Point NumbersThe numbers to the right of the binary point define the fraction.

0 1 1 1 . 1 0 0 0In denary the number above is:

7 ½

1/2 1/4 1/8 1/168/16 4/16 2/16 1/16

HexadecimalProgrammers often use hexadecimal numbers as a shorthand for binary to save time. Hexadecimal (hex) numbers are based on the number 16. Each column has a value 16 times that of the one on its right.

place value 256 16 1

There are 16 different values available in hex:

Letters are used for values 10-15 to ensure that only single digits are used.

0 1 2 3 4 5 6 7 8 9 A B C D E F

Hexadecimal to Binary

Converting between hexadecimal and binary is particularly easy. You just take each character and convert it into the equivalent binary number.

Hex D B

Denary 13 11

Binary 1 1 0 1 1 0 1 1

In this example we will convert DB hex to binary.

Binary to Hexadecimal

You convert between binary and hexadecimal by doing the reverse. You just take each group of four binary digits, starting from the right and convert it into the equivalent hex number.

Binary 1 1 1 1 0 0 1 1

Hex F 3

In this example we will convert 11110011 to hexadecimal: