binary conversions number systems binary to decimal decimal to binary
TRANSCRIPT
Binary Conversions
Number systems Binary to decimal Decimal to binary
Binary Humor
There are 10 kinds of people in the world - those who understand binary and those who don't.
Numbering Systems
Base 10 or decimal numbering systemBase-10 numbering systems dictate that the
numbering scheme begins to repeat after the tenth digit (in our case, the number 9).
Zero is always the first number. When we count, we usually count "00, 01, 02,
03, 04, 05 , 06, 07, 08, 09, 10, 11, 12, ...“
Numbering Systems
Base 10 or decimal numbering systemEach digit to the left and right of the decimal
point is given a name which identifies that digit's placeholder.
Each placeholder is a multiple of ten.For now lets just consider positive numbers.
Numbering Systems - Base TenEach placeholder is a base of
ten. 10º = ones
Any number to the zero power is always equal to 1.
nº=1 10º=1
10¹ = tens Any number to the first
power is always equal itself. n¹=n 10¹=10
10² = hundreds 10³ = thousands
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7 4 0 8
Numbering Systems – Base Ten
Arithmetic expression of 8 in 7408. Work right to left of decimal point. The ones position in expanded notation
calculating the exponent.10º*8=8 is the same as 1*8=8
Numbering Systems – Base Ten
Number 7 4 0 8
Position
Name
Thousands Hundreds Tens Ones
Exponential
Expression
10³*7 10²*4 10¹*0 10º*8
Calculated
Exponent
1000*7 100*4 10*0 1*8
Sum of the powers of ten.
1000*7 + 100*4 + 10*0 + 1*8 = 7408
Numbering Systems – Base two
Binary system is based on multiples of two. In binary numbering the numbering scheme
repeats after the second digit. Let's count to five in binary: “0000, 0001, 0010,
0011, 0100, 0101“ Binary numbering includes names for digit
placeholders.
Numbering Systems – Base two
Picture a odometer that is only capable of counting to two.
Numbering Systems – Base two
Binary placeholders Ones Twos Fours Eights Sixteen's Thirty-twos Sixty-fours
Decimal placeholders Ones Tens Hundreds Thousands Ten-thousands Hundred-thousands Millions
Numbering Systems – Base two
If the binary system is based on powers of 2, why is there still a "ones" position?
Remember: Anything to the zero power is always equal to 1.
In binary, the "ones" position is represented by the exponential expression 2º.
Convert Binary to Decimal
Sum of the powers of two. 8*1 + 4*1 + 2*0 + 1*1 = 13
Number 1 1 0 1
Position
Name
Eights Fours Twos Ones
Exponential
Expression
2³*1 2²*1 2¹*0 2º*1
Calculated
Exponent
8*1 4*1 2*0 1*1
Convert Binary to Decimal
Step 1 - Write the binary number in a row, separating the digits into columns.
Number 1 1 0 1
Convert Binary to Decimal
Step 2 - I want to decide whether each digit placeholder is "ON" or "OFF.“
"1" is "ON" and a "0" is "OFF.“ We don't have to calculate any digit placeholders that
are turned off.
Number 1 1 0 1
ON/OFF On On Off ON
Convert Binary to Decimal
Step 3 - Write the exponential expressions ("powers of two") that represent each placeholder and multiply each expression by 1.
We do this only for the placeholders that are turned ON. For the placeholders which are turned OFF, we simply bring down
the zero from the number itself
Number 1 1 0 1
ON/OFF On On Off ON
Exponential
Expression
2³*1 2²*1 0 2º*1
Convert Binary to Decimal
Step 4 - Calculate the exponents to get a simple multiplication expression for each placeholder.
Number 1 1 0 1
ON/OFF On On Off ON
Exponential
Expression
2³*1 2²*1 0 2º*1
Calculated
Exponent
8*1 4*1 0 1*1
Convert Binary to Decimal
Step 5 - Solve the multiplication expressions from step #4.
Number 1 1 0 1
ON/OFF On On Off ON
Exponential
Expression
2³*1 2²*1 0 2º*1
Calculated
Exponent
8*1 4*1 0 1*1
Solved Multiplication
8 4 0 1
Convert Binary to Decimal Step 6 - Add all the multiplication answers from
step #5 together to get our decimal number
Number 1 1 0 1
ON/OFF On On Off ON
Exponential
Expression
2³*1 2²*1 0 2º*1
Calculated
Exponent
8*1 4*1 0 1*1
Solved Multiplication
8 4 0 1
Add to calculate Value
8+4+0+1=13
Convert Binary to DecimalExample
Number 1 0 1 1 0 1
ON/OFF On Off On On Off On
Exponential
Expression
25 0 2³ 2² 0 2º*1
Calculated
Exponent
32*1 0 8*1 4*1 0 1*1
Solved Multiplication
32 0 8 4 0 1
Add to calculate
Value
32+0+8+4+0+1=45
Covert Decimal to Binary
Step 1 - Take the decimal number and divide it by 2.
Important: NEVER carry your divisions past the decimal point!
Decimal Number=97
Division Expression
Quotient Remainder
97/2 48 1
Covert Decimal to Binary
Step 2 - For each subsequent row, take the quotient from the previous row and divide it by two
Decimal Number=97
Division Expression Quotient Remainder
97/2 48 1
48/2 24 0
24/2 12 0
12/2 6 0
6/2 3 0
3/2 1 1
1/2 0 1
Covert Decimal to Binary
Step 3 – The remainder column only has ones or zeros.
The last cell in the remainder column of the last row must be a "1".
Read the 1s and 0s in the remainder column from the bottom to the top, we'll have our binary number!
Covert Decimal to Binary
Decimal Number=97
Division Expression
Quotient Remainder Direction
97/2 48 1
48/2 24 0
24/2 12 0
12/2 6 0
6/2 3 0
3/2 1 1
1/2 0 1
Binary Number=1100001
Read
Whiteboard Examples In Class Correction
37
DE Q R
37/2 18 1
18/2 9 0
9/2 4 1
4/2 2 0
2/2 1 0
1/2 0 1
1 0 0 1 0 1
25 24 23 22 21 20
32*1 16*0 8*0 4*1 2*0 1*1
32 0 0 4 0 1
32+0+0+4+0+1= 37
Read
The last cell in the remainder column of the last row must be a "1“ because we need to use whole numbers (nonnegative integers).1 ÷ 2 = 0 because 1 can not be divided into, 1 is the remainder.
37 (Odd Number)
DE Q R
37/2 18 1
18/2 9 0
9/2 4 1
4/2 2 0
2/2 1 0
1/2 0 1
36 (Even Number
DE Q R
36/2 18 0
18/2 9 0
9/2 4 1
4/2 2 0
2/2 1 0
1/2 0 1
Read
Read
Hexadecimal Conversation and ASCII
Hexa + Decimal
Base-16 number system It’s all Greek to me
“Sexa” = Latin = Six “Decimal” = Latin = Ten In 1963 IBM thought “Sexadecimal” was not politically
correct “Hexa” = Greek = Six Since the western alphabet contains only ten digits,
hexadecimal uses the letters A-F to represent the digits ten through fifteen.
Hexadecimal and Computing
It is much easier to work with large numbers using hexadecimal values than decimal or binary.One Hexadecimal digit = 4bitsTwo hexadecimal digits = 8 bitsEight bits=1 byteThis makes conversions between hexadecimal
and binary very easy
Counting Hexadecimal
Starting from zero, we count 00, 01, 02,03, 04, 05, 06, 07, 08, 09, 0A, 0B, 0C, 0D, 0E, 0F,10, 11, 12, 13, 14, 15, 16, 17 18, 19, 1A, 1B, 1C, 1D, 1E, 1F, 20, 21, 22, 23, 24, 25,....
Decimal Binary Hexadecimal
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
10 1010 a
11 1011 b
12 1100 c
13 1101 d
14 1110 e
15 1111 f
Convert Hexadecimal to Decimal
1 1 A =10 8
163 162 161 160
4096*1 256*1 16*10 8*1
4096 256 160 8
4096+256+160+8= 4520
Convert Decimal to Hexadecimal4520
DE Q R
4520/16 282 (.5*16)=8
282/16 17 (.625*16)=1010=A
17/16 1 (.0625*16)=1
1/16 0 (.0625*16)=1
11A8
Quotient must be a whole number. If decimal, multiply decimal portion by 16 for remainder. Remainder must be a whole number.
Read
Convert Hexadecimal to Binary
Convert each hexadecimal digit into its 4-bit binary equivalent.
1AB
Hex 1 A B
Bin 0001 1010 1011
000110101011
Convert Binary to Hexadecimal
Converteach 4bit binary digit into its hexadecimal equivalent starting from the right.
If there is an odd number of bits, add zeros to the left to make a complete 4bit digit.
110101011
Bin 0001 1010 1011
Hex 1 A B
1AB
Uses
Web pages http://www.psyclops.com/tools/rgb/
Networking MAC address
Programming C, C++, C#, Java, Assembly
Geeky T-shirts DEADB4C0FFEE
ASCII
American Standard Code for Information Interchange
Each character is 7bits + 1bit for parity = 1byte Represents English characters as numbers, with
each letter assigned a number from 0 to 127 This makes it possible to transfer data from one
computer to another. Used to store text files http://www.pcguide.com/res/tablesASCII-c.html http://nickciske.com/tools/binary.php
Conversion Lab Section I: Converting from Decimal to Binary
1) 11 2) 27 3) 54 4) 113 5) 273
Section II: Converting from Binary to Decimal 6) 101 7) 1011 8) 10100 9) 111010 10) 1010001
Conversion Lab
Section III: Convert Hexadecimal to Binary 11) 43B 12) DAB 13) 954 14) C0FFEE 15) B0A
Section IV: Convert Binary to Hexadecimal 16) 11000001111 17) 10100011110 18) 100110 19) 11011110 20) 101110110001
Conversion Lab Section V: Convert Hexadecimal to Decimal
21) FF2 22) 45 23) 19D 24) 345 25) AA
Section VI: Convert Decimal to Hexadecimal 26) 27 27) 85 28) 562 29) 4522 30) 5627