This document provides an introduction to binary numbers and their use in computers. It discusses how binary numbers are represented by bits and bytes and used to represent data. The document outlines how to convert between decimal and binary numbers, perform basic binary arithmetic like addition and multiplication, and how binary numbers are used to transmit data as streams of bits.

1. An Introduction to Computer
Hardware - BKHS
Binary Numbers
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2. Introduction
 Importance
of the Binary Number
 Use of binary numbers in computers
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3. Agenda
 Binary
Theory
 Binary to Decimal Conversion
 Decimal to Binary Conversion
 Data Flow using Binary Numbers
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4. Overview
 Binary
numbers are used extensively
in digital electronics
 Binary numbers are the foundation of
other numbering systems such as
Hexadecimal and Octal when used in
digital electronics.
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5. Binary Numbers Defined
 A single
“bit” is the foundation
 Only 2 states possible
 Hi – Lo, On – Off, True – False, Open
– Closed
 8 “bits” makeup a single “byte”
 Data typically stored in “bytes”
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6. Converting Decimal to Binary
 LSB
– least significant bit
 MSB – Most significant bit
 Divide the number by 2
 If no remainder, record a zero (0) for LSB
 If there is a remainder, record a one (1)
for LSB
 Divide the previous answer by 2
 If no remainder, record a zero in the next
bit position (to the left of the LSB)
 If there is a remainder, record a one.
 Repeat previous 3 steps until the answer
is no longer divisible by 2.
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7. Decimal to Binary Example
Convert 5267 to binary
5267/2 = 2633
2633/2 = 1316
1316/2 = 658
658/2 = 329
329/2 = 164
164/2 = 82
82/2 = 41
41/2 = 20
20/2 = 10
10/2 = 5
5/2
=2
2/2
=1
1/2 =0
r-1
r-1
r–0
r–0
r–1
r–0
r–0
r–1
r–0
r–0
r–1
r–0
r–1
LSB = 1
next = 1
next = 0
next = 0
next = 1
next = 0
next = 0
next = 1
next = 0
next = 0
next = 1
next = 0
MSB = 1
Binary Number – 1010010010011
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8. Converting Binary to Decimal
Convert 1101001 to decimal
Each bit position is calculated using the formula:
(value in position) x 2^(position #)
so, if bit position 2 = 1 then, applying the formula
1 x 2^2 = 4
Any bit position containing a zero is skipped
Bit position 0 is the LSB. LSB = 1, so 2^0 = 1, add it.
Bit position 1 is 0, so skip it
Bit position 2 is 0, so skip it also
Bit position 3 is 1, so 2^3 = 8, add it.
Bit position 4 is 0, so skip it
Bit position 5 is 1, so 2^5 = 32, add it.
Bit position 6 is the MSB, MSB = 1 so 2^6 = 64, add it.
1 + 8 + 32 + 64 = 105.
Decimal value = 105
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9. Adding Binary Numbers
1010 +1111 ______
 Step one:
Column 2^0: 0+1=1.
Record the 1.
Temporary Result: 1; Carry: 0
 Step two:
Column 2^1: 1+1=10.
Record the 0, carry the 1.
Temporary Result: 01; Carry: 1
 Step three:
Column 2^2: 1+0=1 Add 1 from carry: 1+1=10.
Record the 0, carry the 1.
Temporary Result: 001; Carry: 1
 Step four:
Column 2^3: 1+1=10. Add 1 from carry: 10+1=11.
Record the 11.
Final result: 11001
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10. Binary Multiplication
Multiplication in the binary system works the same
way as in the decimal system:
 1*1=1
 1*0=0
 0*1=0
101
* 11
-----101
1010
-----1111
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11. Data Streams using Binary numbers
In the diagram, a start bit is sent, followed by eight data
bits, no parity bit and one stop bit, for a 10-bit character
frame. The number of data and formatting bits, and the
transmission speed, must be pre-agreed by the
communicating parties.
After the stop bit, the line may remain idle indefinitely, or
another character may immediately be started:
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