The document is an educational material on the binary number system, explaining its definitions, conversions, and arithmetic operations including addition and subtraction. It differentiates binary (base 2) from other numeral systems like decimal (base 10) and hexadecimal (base 16), and provides examples and methods for converting between these systems. Key concepts such as least significant bit (LSB), most significant bit (MSB), and two's complement for negative numbers are also covered.

1. Mathematics for IT
Binary
Number
System
Unit 26
Lee Chadwick

2. Binary Number System
• Topic Definitions
• A Question!
• Lesson learning outcomes
• Hex Character Range
• Binary – Hex Conversion
• Hex – Binary Conversion
• Hex – Denary Conversion

3. • Hexadecimal: "Hexadecimal" means "based on 16" (From Greek hexa:
"six" and Latin decima: "a tenth part").
• Decimal: Based on 10; Example: the numbers we use in everyday life
are decimal numbers, because there are 10 of them
(0,1,2,3,4,5,6,7,8 and 9).
• Denary: Same as Decimal – Base 10
• Binary: The word binary comes from "Bi-" meaning two. We see "bi-" in
words such as "bicycle" (two wheels) or "binocular" (two eyes).
Binary only uses 2 digits; 1 & 0
• Octal: An Octal Number uses only these 8 digits: 0, 1, 2, 3, 4, 5, 6 and
7 Examples:
• 10 in Octal equals 8 in the Decimal Number System.
• 167 in Octal equals 119 in the Decimal Number System.
Also called Base 8.
Definitions:

4. • Identify the concept of Binary
• Establish the purpose of Binary
• Compare Base 2, Base 10 and Base 16
• Convert Binary to Denary and Back
• Multiply Binary values
• Divide Binary values

5. Have a look at your kettle…
• Electronically, binary numbers are
stored/processed using off or on electrical
pulses, a digital system will interpret these
off and on states as 0 and 1.
• 0 = Off State
• 1 = On State

6. Binary Numbers Overview
Binary is a number system used
by digital devices like computers,
cd players, etc.
Binary is Base 2 unlike our counting
system decimal which is Base 10 (denary).

7. LSB versus MSB:
Significance
• Binary has only 2 different numerals (0 and 1), unlike
Decimal which has 10 numerals (0,1,2,3,4,5,6,7,8 and
9).
Here is an example of a binary number: 10011100
As you can see it is simply a bunch of zeroes and
ones, there are 8 numerals in all which make this an 8
bit binary number, bit is short for Binary Digit, and
each numeral is classed as a bit.
The bit on the far right (in this case a zero) is known
as the Least significant bit (LSB), and the bit on the
far left (in this case a 1) is known as the Most
significant bit (MSB)

8. Notations used in digital
systems:
4 bits = Nibble
(1101)
8 bits = Byte
(11010010)
16 bits = Word
(1100110111001010)
32 bits = Double word
(10101111010011101010111010111010)
64 bits = Quad Word (or paragraph)
(10101111010011101010111010111010
…………10101111010011101010111010111010)

9. Base 2
• When writing binary numbers you will
need to signify that the number is binary
(base 2).
• for example take the value 101, as it is
written, it would be hard to work out
whether it is a binary or decimal (denary)
value.

10. Expressing the difference
• to get around this problem it is common to
denote the base to which the number
belongs by writing the base value with the
number, for example:
• 1012 is a binary number and 10110 is a
decimal (denary) value.

11. And from there…
• Once we know the base then it is easy to
work out the value, for example:
1012 = 1*22 + 0*21 + 1*20 = 5 (five)
10110 = 1*102 + 0*101 + 1*100 = 101 (one
hundred and one) (Think of HTU)

12. Sign Bit
• One other thing about binary numbers is
that it is common to signify a negative
binary value by placing a 1 (one) at the left
hand side (most significant bit) of the
value, this is called a sign bit, we will
discuss this a little later

13. Converting base numbers
Base 2
1010
Base 10
10

14. Converting binary to decimal
• To convert binary into decimal is very
simple and can be done as shown below:
Say we want to convert the 8 bit value
10011101 into a decimal value, we can
use a formula like that below:
128 64 32 16 8 4 2 1
1 0 0 1 1 1 0 1

15. Binary to Denary
• As you can see we have placed the numbers 1, 2, 4,
8, 16, 32, 64, 128 (powers of two) in reverse numerical
order and then written the binary value below.
• To convert you simply take a value from the top row
wherever there is a 1 below and add the values
together, for instance in our example we would have
128 + 16 + 8 + 4 + 1 = 157. For a 16 bit value you
would use the decimal values 1, 2, 4, 8, 16, 32, 64,
128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768
(powers of two) for the conversion.
128 64 32 16 8 4 2 1
1 0 0 1 1 1 0 1

16. Alternatively…
Because we know binary is base 2 then
the above could be written as:
1*27 + 0*26 + 0*25 + 1*24 + 1*23 + 1*22 +
0*21 + 1*20 = 157.
128 64 32 16 8 4 2 1
1 0 0 1 1 1 0 1

17. Converting decimal to binary
• To convert decimal to binary is also very
simple.
• You simply divide the decimal value by 2
and then write down the remainder.
• Repeat this process until you cannot
divide by 2 anymore, for example let's take
the decimal value 157:

18. Remainders…
157 ÷ 2 = 78 with a remainder of 1
78 ÷ 2 = 39 with a remainder of 0
39 ÷ 2 = 19 with a remainder of 1
19 ÷ 2 = 9 with a remainder of 1
9 ÷ 2 = 4 with a remainder of 1
4 ÷ 2 = 2 with a remainder of 0
2 ÷ 2 = 1 with a remainder of 0
1 ÷ 2 = 0 with a remainder of 1
To convert write this remainder first.

19. Next…
• Write down the value of the remainders
from bottom to top (in other words write
down the bottom remainder first and work
your way up the list)
• Which gives:
10011101 = 157

20. Alternatively…
• Divide your first binary value into your
decimal number (157 / 128 = 1 remainder
29) Place the 1 under 128;
128 64 32 16 8 4 2 1
1 0 0 0 0 0 0 0
• Then divide 29 by 64 = 0
• Place a 0 below 64
128 64 32 16 8 4 2 1
1 0 0 0 0 0 0 0
• Then divide 29 by 32 = 0
• Place a 0 below 32
128 64 32 16 8 4 2 1
1 0 0 0 0 0 0 0
• Then divide 29 by 16 = 1 remainder 13
• Place a 1 below 16
128 64 32 16 8 4 2 1
1 0 0 1 0 0 0 0
• Then divide 13 by 8 = 1 remainder 5
• Place a 1 below 8
128 64 32 16 8 4 2 1
1 0 0 1 1 0 0 0
• Then divide 5 by 4 = 1 remainder 1
• Place a 1 below 4
128 64 32 16 8 4 2 1
1 0 0 1 1 1 0 0
• Then divide 1 by 2 = 0
• Place a 0 below 2
128 64 32 16 8 4 2 1
1 0 0 1 1 1 0 0
• Then divide 1 by 1 = 1
• Place a 1 below 1
128 64 32 16 8 4 2 1
1 0 0 1 1 1 0 1
Answer =
10011101

21. 128 64 32 16 8 4 2 1
1 0 0 0 0 0 0 0
128 64 32 16 8 4 2 1
1 0 0 0 0 0 0 0
128 64 32 16 8 4 2 1
1 0 0 0 0 0 0 0
128 64 32 16 8 4 2 1
1 0 0 1 0 0 0 0
128 64 32 16 8 4 2 1
1 0 0 1 1 0 0 0
128 64 32 16 8 4 2 1
1 0 0 1 1 1 0 0
128 64 32 16 8 4 2 1
1 0 0 1 1 1 0 0
128 64 32 16 8 4 2 1
1 0 0 1 1 1 0 1

23. Convert these to binary:
• 227
• 3
• 9
• 19
• 28
• 76
• 129

24. Adding binary numbers
• Adding binary numbers is very similar to
adding decimal numbers, first an example:
0 1 0 1 1 0 1 1
+ 0 0 1 0 0 1 1 1
1 0 0 0 0 0 1 0
1 1 1 1 1 1 1
Binary Addition Rules:
0 + 1 = 1
1 + 0 = 1
0 + 0 = 0
1 + 1 = 0 (Carry One)
1 + 1 + Carry = 1 (Carry One)

25. Break Down…
• Let's break down the previous example step by
step:
1 + 1 = 0 (carry one)
1 + 1 (+ the carry) = 1 (carry one)
0 + 1 (+ the carry) = 0 (carry one)
1 + 0 (+ the carry) = 0 (carry one)
1 + 0 (+ the carry) = 0 (carry one)
0 + 1 (+ the carry) = 0 (carry one)
1 + 0 (+ the carry) = 0 (carry one)
The last carry is placed at the left hand side of the
result giving: 10000010

26. Add these:
• 0101 + 0010
• 01101010 + 01000011
• 01010101 + 01111110
• Convert to denary and check
• Add 01010111 and 10101001
Then Comment upon your result

27. Summary of Last Session:
• Discussed binary and it’s importance
• Identified binary bit patterns
• Converted binary to denary and back
• Established rules of binary addition
• Started binary addition tasks

28. Today’s Aims
• Summarise binary progress
• Explore binary subtraction
• Discuss and apply “Twos Complement”
• Progress onto binary multiplication
• Review binary division
• Apply binary subtraction and multiplication
rules to set tasks
• What was that number?

30. Subtracting binary numbers
• The most common way of subtracting binary
numbers is done by first taking the second
value (the number to be subtracted) and
apply what is known as two's complement,
this is done in two steps:
1. complement each digit in turn (change 1 for
0 and 0 for 1).
2. add 1 (one) to the result.
• note: the first step by itself is known as one's
complement.

31. Positive to Negative
• By applying the following steps you are effectively
turning the value into a negative number, and as
when dealing with decimal numbers if you add a
negative number to a positive number then you
are effectively subtracting to the same value.
• In other words 25 + (-8) = 17, which is the same
as writing 25 - 8 = 17.
An example, let's do the following subtraction
11101011 - 01100110 (23510 - 10210)

32. Important note:
• When subtracting binary values it is
important to maintain the same amount of
digits for each number, even if it means
placing zeroes to the left of the value to
make up the digits.
• For instance in our example we have
added a zero to the left of the value
1100110 to make the amount of numerals
up to 8 (one byte) 01100110.

33. Ok, here goes…
• First we apply two's complement to 01100110
Which gives us: 10011010
0 1 1 0 0 1 1 0
(Reverse zeroes and ones)
Step One 1 0 0 1 1 0 0 1
1 0 0 1 1 0 0 1
(Take result and add 1)
Step Two + 1
1 0 0 1 1 0 1 0
1

34. Stage two
• Now we need to add 11101011 +
10011010, NOTE: When you do the
addition you always disregard the last
carry, so our example would be:
1 1 1 0 1 0 1 1
+ 1 0 0 1 1 0 1 0
1 0 0 0 0 1 0 1
1 1 1 1 1 1
Ignore the last carry

35. Our outcome?
• which gives us 10000101, now we can
convert this value into decimal, which
gives 13310
So the full calculation in decimal is:
23510 - 10210 = 13310 (correct !!)
128 64 32 16 8 4 2 1
1 0 0 0 0 1 0 1

36. Negative numbers
• The previous example is subtracting a
smaller number from a larger number, if you
want to subtract a larger number from a
smaller number (giving a negative result)
then the process is slightly different.
Usually to indicate a negative number the
most significant bit (left hand bit) is set to 1
and the remaining 7 digits are used to
express the value. In this format the MSB is
referred to as the sign bit.

37. The steps involved:
1. Apply two's complement to the larger
number.
2. Add this value to the smaller number.
3. Change the sign bit (MSB) to zero.
4. Apply two's complement to value to get
final result.
• The most significant bit (sign bit) now
indicates the value is negative.

38. For example:
10010101 - 10110100
(14910 - 18010)
1 0 1 1 0 1 0 0
(Reverse zeroes and ones)
Step One 0 1 0 0 1 0 1 1
0 1 0 0 1 0 1 1
(Take result and add 1)
+ 1
0 1 0 0 1 1 0 0
1 1
1 0 0 1 0 1 0 1 (Add this result to the
smaller value)Step Two + 0 1 0 0 1 1 0 0
1 1 1 0 0 0 0 1
1 1 1

39. Moving on…
1 1 1 0 0 0 0 1 (Change Sign Bit (MSB) to
zero)Step Three 0 1 1 0 0 0 0 1
0 1 1 0 0 0 0 1
(Reverse zeroes and ones)
Step Four 1 0 0 1 1 1 1 0
1 0 0 1 1 1 1 0
(Take result and add 1)
+ 1
1 0 0 1 1 1 1 1
Final result = 10011111
(MSB indicates a negative value)

40. Final outcome:
Now we can convert this value
into a negative decimal, which
gives -3110
So the full calculation in decimal
is 14910 - 18010 = -3110 (correct !!)

41. Try these:
• 11111111 - 00000001
• 01010101 - 00101010
• 1001 - 1111
• 10111000 – 1101 *
• 1111 – 01010101 *
*Remember!

43. Multiplying binary numbers
• Binary multiplication can be achieved in a
similar fashion to multiplying decimal values.
Using the long multiplication method, ie, by
multiplying each digit in turn, and then adding
the values together.
For example, lets do the following
multiplication: 1011 x 111 (decimal 1110 x 710)

44. Regular Multiplication? Easy!
• which gives us 1001101, now we can convert
this value into decimal, which gives 7710
So the full calculation in decimal is 1110 x 710
= 7710 (correct !!)

45. Note!
• Notice the pattern in the partial products,
as you can see multiplying a binary value
by two can be achieved by shifting the bits
to the left and adding zeroes to the right.

47. Dividing binary numbers
• Like multiplication, dividing binary values
is the same as long division in decimal.
For example, lets do the following division:
1001 ÷ 11 (decimal 910 ÷ 310)

48. Division - Further
• which gives us 0011, now we can convert
this value into decimal, which gives 310
So the full calculation in decimal is 910 ÷
310 = 310 (correct !!)

49. FYI:
• note: Dividing a binary value by two can
also be achieved by shifting the bits to the
right and adding zeroes to the left.

50. CHALLENGE:
Due:
• What is the largest binary number that can be held in:
– 8 bits
– 16 bits
– 32 bits
– 64 bits
• How many bits make one byte?
• What are the possible values that one bit can take?
• Convert the decimal numbers 20 and 10 into binary –
Add the binary numbers together - Show your
working. Then convert the result back to denary to
check your answer.

52. 0h 1h 2h 3h 4h 5h 6h 7h 8h 9h Ah Bh Ch Dh Eh Fh
0h 0 16 32 48 64 80 96 112 128 144 160 176 192 208 224 240
1h 1 17 33 49 65 81 97 113 129 145 161 177 193 209 225 241
2h 2 18 34 50 66 82 98 114 130 146 162 178 194 210 226 242
3h 3 19 35 51 67 83 99 115 131 147 163 179 195 211 227 243
4h 4 20 36 52 68 84 100 116 132 148 164 180 196 212 228 244
5h 5 21 37 53 69 85 101 117 133 149 165 181 197 213 229 245
6h 6 22 38 54 70 86 102 118 134 150 166 182 198 214 230 246
7h 7 23 39 55 71 87 103 119 135 151 167 183 199 215 231 247
8h 8 24 40 56 72 88 104 120 136 152 168 184 200 216 232 248
9h 9 25 41 57 73 89 105 121 137 153 169 185 201 217 233 249
Ah 10 26 42 58 74 90 106 122 138 154 170 186 202 218 234 250
Bh 11 27 43 59 75 91 107 123 139 155 171 187 203 219 235 251
Ch 12 28 44 60 76 92 108 124 140 156 172 188 204 220 236 252
Dh 13 29 45 61 77 93 109 125 141 157 173 189 205 221 237 253
Eh 14 30 46 62 78 94 110 126 142 158 174 190 206 222 238 254
Fh 15 31 47 63 79 95 111 127 143 159 175 191 207 223 239 255
Toconvertbetweendecimaland
hexisquitesimplewiththischart:

53. Number Systems – Task:
Number Systems Working Together
Using a structured approach, complete a range of conversions between
the number systems discussed previously. The conversions needed
are:
• Decimal to Binary, Octal, Hexadecimal
• Binary to Decimal, Octal, Hexadecimal
• Octal to Decimal, Binary, Hexadecimal
• Hexadecimal to Decimal, Binary, Octal