Binary

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Binary
Written By Pat Ellison-Soper
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The Decimal System

• The Decimal Number System uses base 10.
• It includes the digits from 0 through 9.
• All numbers can be represented by any number
  times 10 to some power
• The weighted value for each position is as
  follows
• In the decimal system the number 123 is
  represented

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Decimal Floating Points

• In the decimal system digits appearing to the
  right of the floating point represent a value
  between zero and nine, times an increasing
  negative power of ten.
• For example, the value 725.194 is represented as
  follows
• 700 20 5
  0.1 0.09
  0.004 725.194
• 700.00
• 20.00
• 5.00
• 0.10
• 0.09
• 0.004
• 725.194

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Why the Binary System?

• Computer systems do not represent numeric values
  using the decimal system. Instead, they typically
  use a binary or two's complement numbering
  system.
• We have learned that a computer circuit is made
  out of transistors. Transistors have two states,
  on and off.
• With two states there is less chance for error

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The Binary System

• Modern computer systems operate using binary
  logic.
• We learned previously that the computer
  represents values using two voltage levels (0V
  for logic 0 and either 3.3 V or 5V for logic
  1).
• The binary number system works like the decimal
  number system except the Binary Number System
  uses base 2 and includes only the digits 0 and 1
• The weighted values for each position is
  determined by place as follows

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The Chart Method Binary to Decimal

• The binary system uses powers of two. The
  rightmost digit in binary represents 2 0, or
  one. Any number raised to the zero power equals
  one.
• The next digit to the left represents 2 1 or 2.
  The next digit to the left represents 2 2, or
  4, and so on.
• Each successive digit doubles the number of
  possible values.
• How numbers are stored in binary notation 1 is
  ON, and 0 is OFF. We only add the decimal values
  of the ON switches to determine the decimal
  number.
• Add 16 0 4 2 0 to equal the decimal
  number 22.
• Add 32 16 0 4 0 1 to equal the decimal
  number 53. 

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The Chart Method Decimal to Binary

• Start at the left and turn on any switch that is
  less than the value of the decimal number.
  Subtract that value from the decimal number.
• Continue to the right, turning on the switch for
  any value that does not exceed the difference.
• Example Find the binary value of the base 10
  number 29.
• We can not turn on the switch that equals 32
  because it is greater than the decimal number 29.
• The first switch we can turn on is equal to 16.
  We subtract 16 from 29. Our new value is 13.
• The first switch we can turn on is equal to 8.
  We subtract 8 from 13. Our new value is 5.
• Continue to turn on switches and subtract that
  value until you have the value of the decimal
  number.

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Decimal to Binary The Subtraction Method

• Start with a weighted place value greater than
  the number. If the number is greater than the
  weighted value, write down a 1 in the Binary
  column and subtract the weighted value.
• If the number is less than the weighted value,
  write down a 0 in the Binary column and subtract
  0.
• Continue until the result is 0.
• Convert 2,671 in Base 10 to a Binary Number
• Weighted Value Subtraction Result Binary
• 212 4096 2671 - 0 2671 0
• 211 2048 2671 - 2048 623 0 1
• 210 1024 623 - 0 623 0 10
• 29 512 623 - 512 111 0 101
• 28 256 111 - 0 111 0 1010
• 27 128 111 - 0 111 0 1010 0
• 26 64 111 - 64 47 0 1010 01
• 25 32 47 - 32 15 0 1010 011
• 24 16 15 - 0 15 0 1010 0110
• 23 8 15 8 7 0 1010 0110 1
• 22 4 7 - 4 3 0 1010 0110 11
• 21 2 3 - 2 1 0 1010 0110 111
• 20 1 1 - 1 0 0 1010 0110 1111

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Decimal to Binary Successive Division

• Successive division to change a decimal system
  number to a binary system number.
• Divide by the base number, in this case 2.
• Divide until the quotient equals 0.
• Read the binary number by reading the remainder
  numbers from bottom to top.
• Let's find the binary representation of the
  decimal number 345.
•  Division Quotient  Remainder 
• 345 / 2  172 1
• 172 / 2  86 0
• 86 / 2 43  0
• 43 / 2 21 1
• 21 / 2 10  1
• 10 / 2 5  0 
• 5 /2  2  1
• 2 /2  1  0
• 1 / 2 0  1
•  The binary representation of 345 is 0001 0101
  1001

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Binary to Decimal Multiply by Place Value

• Binary to Decimal uses Multiplication
• Example Find the base 10 value of 1 0101 1001
• Now add 256 64 16 8 1 345

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Bit Number Formats

• Binary numbers are written as a sequence of bits.
  
• We have defined boundaries for these bits.
  Computer scientists add leading zeroes to adjust
  the binary number to a desired size boundary.
  These boundaries are
• Name Size in bits Example
• Bit 1 1
• Nibble 4 0101
• Byte 8 0000 0101
• Word 16 0000 0000 0000 0101
• The rightmost bit in a binary number is bit
  position zero. Bit zero is usually referred to as
  the LSB, least significant bit. Each bit to the
  left is given the next successive bit number.
• The left-most bit is typically called the MSB,
  most significant bit. Intermediate bits are
  referred to by their respective bit numbers.

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Binary Data Organization

• The Bit
• The smallest "unit" of data on a binary computer
  is a single bit. A single bit is capable of
  representing only two different values, zero or
  one.
• The Nibble
• A nibble is a collection of bits on a 4-bit
  boundary. When we study hexadecimal numbers you
  will learn that it takes four bits to represent a
  hexadecimal digit.
• The Byte
• The most important data structure used by
  microprocessors is the byte. A byte consists of
  eight bits and is the smallest addressable data
  item in the microprocessor. This means that the
  smallest item that can be individually accessed
  by a program is an 8-bit value.
• Since a byte contains eight bits, it can
  represent 2 8, or 256, different values.
  Computers use a byte to represent
• unsigned numeric values in the range 0 gt 255
• signed numbers in the range -128 gt 127
• ASCII character codes

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More about Binary Representation

• To represent integers, which can be positive or
  negative, computers typically use a sign notation
  on the binary.
• 0 is positive and 1 is negative and this number
  precedes the the rest of the number.
• Example If 10111 is 23, then 1 10111 would be
  - 23.
• Every three decimal digits is separated with a
  comma to make larger numbers easier to read.
  Example 1, 446, 9782
• The convention for binary numbers is to add a
  space every four digits starting from the least
  significant digit on the left of the decimal
  point.
• Example The binary value 1010111110110010 would
  be written 1010 1111 1011 0010.

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Supplement Binary Measuring Units

• Units of measure with verbal prefixes for large
  quantities
• kilobytes(K) 2 10 1,024
• megabytes(M) 2 20 1,048,576
• gigabytes(G) 2 30 1,073,741,824
• terabytes(T) 2 40 1,099,511,627,776
• petabytes( P) 2 50 1,125,899,906,842,624
• exabytes(E) 2 60 1,152,921,504,606,846,976
• zettabytes(Z) 2 70 1,000,000,000,000,000,000,000
  
• yottabytes(Y) 2 80 1,000,000,000,000,000,000,00
  0,000

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Supplement Logic Explanation of 0 Power

• Why is any number raised to the zero power equal
  to one?
• Example Let's look at the list of numbers, the
  integer powers of 3 and their decimal values.
• A pattern forms here.
• Move to the left in the list, multiply by 3
• Move to the right in the list, divide by 3.
• Keep the sequence going to the right and dividing
  by 3 and the sequence looks like this

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Supplement Law of Exponents for 0 Power

• For all n, x, and y.
• n x
• n (x - y)
• n y
• 3 4
• 1 3 (4 - 4) 3 0
  Therefore 3 0 1

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Supplement Exponent Rules

• Rules of One
• Any number raised to the power of 1 equals
  itself.
• One raised to any power is one.
• Product Rule
• When multiplying two powers that have the same
  base, you can add the exponents.
• Power Rule
• To raise a power to a power, just multiply the
  exponents.
• Quotient Rule
• Divide two powers with the same base by
  subtracting the exponents.
• Zero Rule
• Any nonzero number raised to the power of zero
  equals 1.
• Negative Exponents
• Any nonzero number raised to a negative power
  equals its reciprocal raised to the opposite
  positive power.