Agenda

• Shortcuts converting among numbering systems– Binary to Hex / Hex to Binary– Binary to Octal / Octal to Binary

• Signed and unsigned binary numbers

• Addition / Subtraction of Binary Numbers

• Two’s Complement

Numbering System Shortcuts

It is very simple to convert binary numbers to octal or hexadecimal numbers since 8 is 2^3, and 16 is 2^4

In other words:

• 1 Octal digit = 3 binary digits

• 1 Hex digit - 4 binary digits

Binary to Octal

Notice the Pattern:• Largest 3 digit binary is 111

• 1 octal digit will represent a 3 digit binary number

• Highest Octal digit is 7

• Therefore: 1112 = 78

Binary to Octal

Relationship:Octal Binary

0 000

1 001

2 010

3 011

4 100

5 101

6 110

7 111

Does this table look familiar?

Practical Example

• applying octal values of rwx the chmod command (e.g., chmod 751).

• chmod 777

• chmod 755

• chmod 711

• chmod 644

Binary to Hexadecimal

Notice the Pattern:• Largest 4 digit binary is 1111

• 1 hex digit will represent a 4 digit binary number

• Highest hex digit is F

• Therefore: 11112 = F16

Binary to Hexadecimal

Relationship:Hexadecimal Binary Hexadecimal Binary

0 0000 8 1000

1 0001 9 1001

2 0010 A 1010

3 0011 B 1011

4 0100 C 1100

5 0101 D 1101

6 0110 E 1110

7 0111 F 1111

Convert Hex to Binary

Steps:• Convert Hex number to groups of

powers of 2.

• Convert to Binary number (Remember to drop leading zeros for first set of binary numbers - i.e. first left set)

Convert Hex to Binary

11F616

= 1 1 F 6

= 000(1) 000(1) (8)(4)(2)(1) 0(4)(2)0

= 1 0001 1111 0110= 10001111101102

000

Drop Leading zeros

Convert Binary to Hex

Steps:• Separate into 4 bit groups starting from

the right.

• Calculate decimal equivalent (in placeholders in powers of 2)

• Convert to Hexadecimal number

Convert Binary to Hex

10001111101102

=1 0001 1111 0110

=0001 0001 1111 0110= 1 1 (8)(4)(2)(1) 0(4)(2)0

= 1 1 15 6= 11F616

Converting Octal to Hexadecimal

• The easiest method to convert between Octal and Hexadecimal is to convert to binary as an intermediate step

• Regroup binary in groups of 4 for hexadecimal and 3 for octal

Storing Numbers

• Numeric information (stored as a non negative number) is often store in a computer in binary.

Eg.

• 1 byte (0 - 255 numbers)

• 2 bytes (0 - 65535)

• 4 bytes (0-4294967295)

Data Formats

Unsigned Binary• Data stored as a binary number, with

no way to express a negative quantity

Data Formats

Signed Binary• Data stored as a binary number, but

using a leading zero to represent a positive number, and the two’s complement of a binary number for a negative number

Adding / Subtracting Binary Numbers

• Addition:0 + 0 = 0

0 + 1 = 1, 1 + 0 = 1

1 + 1+ = 10

• Subtraction:0 - 0 = 0

1 - 1 = 0, 1 - 0 = 1

0 - 1 (Must borrow from next placeholder)

Therefore 10 - 1 = 1

Two’s Complement

• Simple method of subtracting two binary numbers by adding.

• Two Complement– Flip binary numbers (0 becomes 1, visa versa)– then add 1– Result becomes negative

• Therefore, short-hand method of representing negative integers