Numbering System
Why binary?
• The original computers were designed to be high-speed calculators.
• The designers needed to use the electronic components available at the time.
• The designers realized they could use a simple coding system--the binary system--
to represent their numbersRepresenting Information
in Computers
• All the different types of information in computers can be represented using binary
code.
- Numbers
- Letters of the alphabet and punctuation marks
- Microprocessor instruction
- Graphics/Video
- SoundBits and Bytes
• A binary digit is a single numeral in a binary number.
• Each 1 and 0 in the number below is a binary digit:
– 1 0 0 1 0 1 0 1
• The term “binary digit” is commonly called a “bit.”
• Eight bits grouped together is called a “byte.”Computer Number Systems
• Decimal Numbers
• Binary Numbers
• Hexadecimal NumbersNumbering Systems
Decimal Number System
• The prefix “deci-” stands for 10
• The decimal number system is a Base 10
number system:
– There are 10 symbols that represent quantities:
• 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
– Each place value in a decimal number is a
power of 10.Background Information
• Any number to the 0 (zero) power is 1.
– 4
0 = 1 16
0 = 1 1,482
0 = 1.
• Any number to the 1st power is the
number itself.
– 10
1 = 10 49
1 = 49 827
1 = 827Numbering Systems
Decimal Number System
1 4 9 2
10
3 10
2 10
1 10
0
1000 100 10 1Numbering Systems
Decimal Number System
1 x 1000 = 1000
4 x 100 = 400
9 x 10 = 90
2 x 1 = + 2
1492
1492Numbering Systems
Binary Numbers
• The prefix “bi-” stands for 2
• The binary number system is a Base 2 number system:
– There are 2 symbols that represent quantities:
• 0, 1
– Each place value in a binary number is a power
of 2.Numbering Systems
Binary Number System
1 0 1 1
2
3 2
2 2
1 2
0
8 4 2 1Numbering Systems
Binary Number System
1 x 8 = 8
0 x 4 = 0
1 x 2 = 2
1 x 1 = + 1
11
1011Numbering Systems
Binary Number System
1 0 1 1 0 1 0 1
2
7 2
6
25
24
23
22 2
1 2
0
128 64 32 16 8 4 2 1Numbering Systems
Converting Binary Numbers to Decimal
• Step 1
– Starting with the 1s place, write the binary place
value over each digit in the binary
number being converted.
16 8 4 2 1
1 0 1 0 1Numbering Systems
Converting Binary Numbers to
Decimal
• Step 2
– Add up all of the place values that have a “1” in
them.
16 8 4 2 1
1 0 1 0 1
16 + 4 + 1 = 21You Try It!
• Convert the binary number 1 1
0 0 1 0 1 to decimal.
1 1 0 0 1 0 1
64 32 16 8 4 2 1
64 + 32 + 4 + 1=101Numbering Systems
Converting Decimal Numbers to
Binary
• There are two methods that can be used to convert decimal numbers to binary:
– Repeated subtraction method
– Repeated division method
• Both methods produce the same result and you should use whichever one you are
most comfortable with.Numbering Systems
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