Bits and Number Systems

INFORMATION AND COMPUTERS

• Multiple bit representation of information:
• numbers (32 bits) -> double/float; int
• Characters (8 bits) -> ASCII chars
• Numerical equivalence of multiple bits:
• The computer doesn't know/care what the bits are supposed to be used for, it just sees bits
• numbers/chars can be manipulated by same instructions.

ANALOGUE AND DIGITAL

• Analogue – Data that represented on a continuous scale (real numbers)
• e.g. voltage, height, distance, amount of magnetisation on audio tapes.
• Good for real world measurements, transients (spikes), transistors.
• Digital – Data that is represented as whole numbers (count)
• e.g. class size, coins in your pocket, sound samples on CD recordings,
• Good for noise rejection, computers.

DIGITAL ADVANTAGES

• Digital systems are easier to:
• design and modify
• store data
• maintain accuracy and precision
• program operations
• to protect from noise
• to create on an IC chip
• However the real world is mostly analogue
• Translation is inaccurate.
• The more bits the more precise.
• Data in computer is stored digitally as 0s and 1s in a binary code.
• The 1s and 0s are represented in many ways:
• by a voltage e.g. 0 or 5 volts (actually we don’t use exact voltages) eg < 0.8 volts == 0 > 2.4 volts == 1 (TTL)
• by the absence or presence of a pit (CD) Signal-to-noise ratio.
• by the absence or presence of a magnetic field (disks) http:// www.linfo.org/
• Why binary? shannon- hartley_theore
• Simple, robust, universal laws (Shannon-Hartley), flexible m.html
• c.f. sending decimal using voltages…. long cables… interference.)
• groups of bits can represent numbers, characters, computer instructions.

• Hardware has to be able to manipulate groups of bits differently depending on what they represent.
• Amazingly, we can use combinations of Gates to do this.

8-Bit Example

Bit7 Bit6 Bit5 Bit4 Bit3 Bit2 Bit1 Bit0
1 0 1 0 1 0 1 1

• Possible interpretations:
• Character ‘1/2’ assuming extended ASCII/ISO
• The number -85

Bit7 Bit6 Bit5 Bit4 Bit3 Bit2 Bit1 Bit0
1 1 0 0 0 0 0 1

NUMBER SYSTEMS

• We use a “positional number system” #big-endian
• in decimal numbers, the further left a digit is, the greater the power of 10 it is multiplied by.
• e,g, 348 = 3 x 102 + 4 x 101 + 8 x 100
• This system applied to other Radix (or Bases)
• eg. Decimal radix = 10 digits 0..9
• Binary radix = 2 digits 0..1
• Octal radix = 8 digits 0..7
• Hexadecimal radix = 16 digits 0..F

BINARY NUMBERS

• Computers work in binary numbers (2 values) e,g, binary number 010011102 is =0 * 27 + 1 * 26 + 0 * 25+ 0 * 24 + 1 * 23+ 1 * 22+ 1 * 21+ 0 * 20 =0 * 128 + 1 * 64+ 0 * 32+ 0 * 16+ 1 * 8 + 1 * 4 + 1 * 2 + 0 * 1 = 1 * 64+ 1 * 8 + 1 * 4 + 1 * 2 = 78
• Exercise:- convert to decimal
• 011102 = 0*24 + 1*23 + 1*22 + 1*21 + 0*20 = 8 + 4 + 2 = 14
• 100112 = 1*24 + 0*23 + 0*22 + 1*21 + 1*20 = 16 + 2 + 1 = 19
• notation: in maths use 011102 to indicate a binary number
• some systems use other ways such as %01110 or b01110

Binary Counting

• Hex is another abbreviation for binary numbers.
• Each byte represented by 2 hex digits; each hex digit represents 4 bits in an 8-bit byte.
• Hex to binary can be done one digit at a time: e.g. what is 4E16 in binary? 4 E 0100 1110 4*16 14*1 =4x161+ 14x160 = 64+14 = 7810

HEXADECIMAL NUMBERS

• 0=0, 1=1, 2=2, 3=3, 4=4, 5=5, 6=6, 7=7, 8=8, 9=9
• Memorise: A=10, B=11, C=12, D=13, E=14, F=15
• mathematically use 123016 or 1230H to indicate a hexadecimal number
• some systems use other ways such as 0x1110 or 01110h
• Or sometimes hex is default (not decimal!)

leading 0x means hex

HEX - BINARY (4 Bits To A Hex Digit)

BIT-WISE OPERATIONS

• Can base decisions on a single bit #button state
• e.g. if (bit represents True) then …
• Usually 1 means True, 0 False.
• Unary operation: NOT
• Complement (“invert”, “flip”) the value
• Binary* operations: AND, NAND, OR, NOR, XOR
• N-bit operations: e.g. 4-input AND, OR
• All can be implemented using binary building blocks (“2-input gates”)

SUMMARY

• Modern computers operate with digital representations of information:
• Easier to work with but has Implications
• Number systems and conversions to know:
• Binary ß à decimal
• Hex ß à Binary
• Hex ß à Decimal
• And the formula in general!
• Next Lecture: Gates