NUMBER SYSTEMS

The BASE of a number system

Determines the number of digits available

In our number system we use 10 digits: 0-9

The base in our system is 10

It is called decimal or denary

The BASE of a number system

Computers are bi-stable devices A bi-stable device has only two possible

states: ON or OFF Hence a base 2 number system is

enough for such devices A base 2 number system is called

binary The 2 possible digits are 0 and 1

The BASE of a number system

An other number system used with

computers is Hexadecimal

The base for this system is 16

The 16 possible digits are 0 – 9 and A - F

The BASE of a number system

System Base Digits used

Decimal base 10e.g. 510

0, 1, 2, 3, 4, 5, 6, 7, 8, 9

Binary base 2e.g. 102

0, 1

Hexadecimal

base 16e.g.

3C16

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

Place values The position of each number within a

series of numbers is very important

The position of the number determines

how large it is

All number bases have their place values

Place values

In base 10, the place values are as follows:

106 105 104 103 102 101 100

1 000 000 100 000 10 000 1000 100 10 1

Values go up in powers of 10 as you move from left to right

Place values

In base 2, the place values are as follows:

Values go up in powers of 10 as you move from left to right.

29 28 27 26 25 24 23 22 21 20

512 256 128 64 32 16 8 4 2 1

Binary to Decimal Conversion

To convert binary numbers to decimal we need to follow these four steps:

1.Write down the place values, starting from the

right hand side

2.Write each binary digit under its place value

3.Multiply each binary digit by its place value

4.Add up the total number

Binary to Decimal ConversionExample Convert the binary number 10101 to decimal.

24 23 22 21 20

Step 1 Place values

16 8 4 2 1

Step 2 Binary digits

1 0 1 0 1

Step 3 1 × 16 = 16

0 × 8 = 0

1 × 4 = 4

0 × 2 = 0

1 × 1 = 1

Step 4 Decimal 16 + 0 + 4 + 0 + 1 = 21

Answer: 101012 = 2110

Decimal to Binary Conversion

To convert a decimal number to binary

1.Successively divide the decimal number by 2

and record the remainder

2.Use the numerator for the next division until

the result of the division is 0

3.The remainder (1 or 0) of each division

makes up the binary number

Decimal to Binary Conversion

To convert a decimal number to binary

1. The remainder of the first division gives the LEAST

significant bit

2. The remainder of the next division gives the NEXT

bit etc.

3. This method produces the bits in reverse order -

reading down the way gives the bits in order of

right to left

Decimal to Binary ConversionExampleConvert 3710 in binary

2 37

2 18 with a remainder of 1

2 9 with a remainder of 0

2 4 with a remainder of 1

2 2 with a remainder of 0

2 1 with a remainder of 0

0 with a remainder of 1Answer : 3710 = 1001012

Decimal to Binary Conversion

An alternative method is to use the place values. To convert a decimal number to binary follow these steps:

1. Write down the place values up to the one which is just greater than the decimal number we need to convert

Decimal to Binary Conversion

Let us consider the decimal number 37, we would need to write down the place values up to 64

26 25 24 23 22 21 20

Place values 64 32 16 8 4 2 1

Decimal to Binary Conversion

2. Work through the place values deciding on whether to place a 1 or a 0 under the place value

Decimal to Binary Conversion

Since 37 is less than 64 we write a 0 under the 64. We would need a 32 so we write a 1 under the 32

26 25 24 23 22 21 20

Place values 64 32 16 8 4 2 1

Binary number

0 1

Decimal to Binary ConversionThis would leave us with 37 – 32 = 5. Now we are left with 5 to distribute under the other place values. Since 5 is less than 16 and 8 we write a 0 under both. We can now place a 1 under 4. This leaves us with 5 – 4 = 1. Hence we put a 0 under 2 and a 1 under 1.

26 25 24 23 22 21 20

Place values 64 32 16 8 4 2 1

Binary number

0 1 0 0 1 0 1

So 3710 = 1001012

The Hexadecimal (Hex) number system

Base 16 16 symbols: 0 – 9 & A – F Place values increase in powers of 16

The advantage of the hexadecimal system is its usefulness in converting directly from a 4-bit binary number

Equivalent numbers in different bases

Decimal

Binary

Hex

0 0000 0

1 0001 1

2 0010 2

3 0011 3

4 0100 4

5 0101 5

6 0110 6

7 0111 7

8 1000 8

9 1001 9

Decimal

Binary

Hex

10 1010 A

11 1011 B

12 1100 C

13 1101 D

14 1110 E

15 1111 F

Hex to Decimal Conversion

To convert Hex numbers to decimal we need to follow these four steps:

1.Write down the place values, starting from the

right hand side

2.Write each hex digit under its place value

3.Multiply each hex digit by its place value

4.Add up the total number

Hex to Decimal Conversion

ExampleConvert 3CD16 to its decimal equivalent.

162 161 160

Step 1 Place value 256 16 1

Step 2 Hex 3 C D

Step 3 256 × 3= 768

16 × 12= 192

1 × 13= 13

Step 4 Decimal 768 + 192 + 13 = 973

Answer: 3CD16 = 97310

Decimal to Hex Conversion

To convert a decimal number to hex Successively divide the decimal number

by 16 and record the remainder Use the numerator for the next division

until the result of the division is 0 The remainder of each division makes

up the hex number

Decimal to Hex Conversion

Example 1Convert 4110 to hex

Answer : 4110 = 2916

16 41

16 2 with a remainder of 9

0 with a remainder of 2

Decimal to Hex Conversion

Example 3Convert 10910 to hex

Answer : 10910 = 6 1316

= 6D16

16 109

16 6 with a remainder of 13

0 with a remainder of 6

Decimal to Hex Conversion

An alternative method is to use the place values. To convert a decimal number to hex follow these steps:

1. Write down the place values up to the one which is just greater than the decimal number we need to convert

Decimal to Hex Conversion

Let us consider the decimal number 356, we would need to write down the place values up to 4096

163 162 161 160

Place values 4096 256 16 1

Decimal to Hex Conversion

2. Work through the place values deciding on weather to place a 0 or a value from 1 - 15 under the place value.

Decimal to Hex Conversion

Since 356 is less than 4096 we write a 0 under the 4096. We would need a 256 so we write a 1 under the 256.

163 162 161 160

Place values 4096 256 16 1

Hex number 0 1

Decimal to Hex Conversion

This would leave us with 356 – 256 = 100. Now we are left with 100 to distribute under the other place values. Since 100 ÷ 16 =6 r 4 we write a 6 under 16. This leaves us with 4 ones. Hence we put a 4 under 1.

163 162 161 160

Place values 4096 256 16 1

Hex number 0 1 6 4So 35610 = 16416

Binary to Hex Conversion

To convert numbers from binary to hex and vice versa, we need to use the conversion table shown below

Binary

Hex

0000 0

0001 1

0010 2

0011 3

0100 4

0101 5

Binary

Hex

1011 B

1100 C

1101 D

1110 E

1111 F

Binary

Hex

0110 6

0111 7

1000 8

1001 9

1010 A

Binary to Hex Conversion

For each digit in the hex number, write down the equivalent 4-bit binary digit.

ExampleConvert C316 to its binary equivalent

Answer : C316 = 110000112

Hexadecimal C 3

Binary 1100 0011

Hex to Binary Conversion

Divide the binary number into 4-bit groups starting from the right (LSB). If we end up with a group of less than 4 bits on the left, add 0s to fill up the required places. Then translate each group into its equivalent hexadecimal number below.

Hex to Binary Conversion

ExampleConvert 10111010102 to hexadecimal

Answer : 10111010102 = 2EA16

Binary 0010 1110 1010

Hexadecimal 2 E A

Binary Arithmetic

Addition of Binary Numbers

13  +

17

30

Note:

3 + 7 = 10 which we write as 0 carry 1.

0 1 1 0 1  +

1 0 0 0 1

1 1 1 1 0

Similarly in binary:

1 + 1 =  0 carry 1.

Addition of Binary Numbers

13  +

17

  1

31 Note:

3 + 7 + 1= 11 which we write as 1 carry 1.

0 1 1 0 1  +

1 0 0 0 1

0 0 0 0 1

1 1 1 1 1

Similarly in binary:

1 + 1 + 1 =  1 carry 1.

Numeric Overflow

13  +

17

11

41

Note:

The number 41 cannot be represented using 5 bits since with 5 bits we can represent the range from 0 to 31.

0 1 1 0 1  +

1 0 0 0 1

0 1 0 1 1

0 1 0 0 11

Numeric overflow occurs if a number too large to be represented is encountered.  For example a 5 bit register is used to store the decimal number 41. 

Negative Numbers

Negative numbers can be represented in binary using one of the following ways:

• Sign and Magnitude Codes (SM)

• Two's Complement (2C)

• The Most Significant Bit is used to represent the sign of the number 

Sign and Magnitude Codes

• The other bits represent the magnitude of the number.  

Example: Sign and Magnitude

Convert 24 to Sign and Magnitude representation using an 8-bit register format.

24In binary

1 1 0 0 0

24Using 7 bits

0 0 1 1 0 0 0

24SM

0 0 0 1 1 0 0 0

O since number is positive

Example: Sign and Magnitude

Convert - 24 to Sign and Magnitude representation using an 8-bit register format.

24In binary

1 1 0 0 0

24Using 7 bits

0 0 1 1 0 0 0

24SM

1 0 0 1 1 0 0 0

1 since number is negative

Two's Complement (2C)

The bits have the same place values as binary numbers. 

However the Most Significant Bit is also used to represent the sign of the number.

Example: Two's Complement

Complement 67 using a 9-bit register

67In binary

0 1 0 0 0 0 0 1 1

Reverse Bits 1 0 1 1 1 1 1 0 0

Add 1 1 0 1 1 1 1 1 0 1

This gives -67

Range for 2C Representation

1-bit register 0 represents 0

1 represents -1

2-bit register

0 0 represents 00 1 represents 11 0 represents -21 1 represents -1

1-bit register -1 , 0

2-bit register -2, -1, 0, 1

3-bit register

0 0 0 represents 00 0 1 represents 10 1 0 represents 20 1 1 represents 3

3-bit register -4, -3, -2, -1, 0, 1, 2, 3

1 0 0 represents -41 0 1 represents -31 1 0 represents -21 1 1 represents -1

Range for 2C Representation

In general an N-bit register can represent binary numbers in the range

- 2N-1 ... + 2N-1- 1.

Binary SubtractionExample: Subtract 54 from 103

103 - 54 Can be written as 103 + (- 54)

Step 1: Check how many bits you need

Therefore we are going to use 8 bits in both cases.

103 8 bits (range for +ve numbers is 0 …2n-1)

-54 7 bits (range for 2C numbers is 2n-1 to + 2n-1-1)

Binary Subtraction

Step 2: Convert - 54 to a 2C binary number

-54 1 1 0 0 1 0 1 02

Step 3: Convert 103 to a binary number

103 0 1 1 0 0 1 1 12

Step 4: Add binary 103 to - 54

0 1 1 0 0 1 1 12 +

   1 1 0 0 1 0 1 02

1 0 0 1 1 0 0 0 12

Overflow bit

The 9th bit or overflow bit will be ignored thus the 8-bit answer is correct. 0 0 1 1 0 0 0 12

Binary Multiplication

• Shifting the number left multiplies that number by 10  

Decimal System Binary System

• Shifting the number left multiplies that number by 2 

420 = 42 x 10. 111000 =  11100 x 2