The document discusses different number systems including decimal, binary, octal, and hexadecimal. It provides details on each system such as the base and valid digits used. The decimal system uses base 10 with digits 0-9. The binary system uses base 2 with digits 0-1. Octal uses base 8 with digits 0-7. Hexadecimal uses base 16 with digits 0-9 and A-F. Conversion between these systems involves repeatedly dividing the number by the new base and recording the remainders as the new number's digits. Understanding these number systems is important for fields like computer science that commonly use binary.

1. Applied physics III
Number systems
1
by Benson Mbuya

2. Decimal System
• Most people today use decimal representation
to count. In the decimal system there are 10
digits:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9
• These digits can represent any value, for
example:
754.
The value is formed by the sum of each digit,
multiplied by the base (in this case it
is 10 because there are 10 digits in decimal
system) in power of digit position (counting
2
from zero):

3. Decimal System
• Position of each digit is very important! for
example if you place "7" to the end:
547
it will be another value:
3

4. Decimal System
4

5. The decimal system
• In the decimal number system, each digit 0
through 9 represents either itself, or else itself
times some power of 10.
• The value of the digit depends on its position or
place in the number.
• In the case of the number 35912, the digit 2 is
in the units place, and has a value of simply 2.
• The digit 1 is in the 10s place; it takes the value
of 1 x 101 =10.
• The digit 9 is in the 100s place, having a value
of 9 × 102 = 900.
5

6. The decimal system
• The digit 5 is in the 1,000s place, and
represents 5 10 3 5000.
• The digit 3 is in the 10,000s place, and
• has a value of 3 104 30,000.
• The total value is the sum of all these:
2 + 10+ 900 + 5,000 + 30,000 = 35,912.
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7. The decimal system
• A decimal number can be represented in digital
form.
• Any decimal number has a binary equivalent
composed of digits that are all either 0 or 1
• four-digit number ‘‘1,234’’ was built out of four
digits, any of which could have the 10 values
‘‘0’’, ‘‘1’’, ‘‘2’’, ‘‘3’’, ‘‘4’’, ‘‘5’’, ‘‘6’’, ‘‘7’’, ‘‘8’’ and ‘‘9’’.
• The number of different values for each digit is
referred to as its ‘‘base’’ or‘‘radix’’.
• It is important to note that the first value is
always zero and the last value is the base minus
one.
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8. The Binary system
• Computers use binary system, binary system
uses 2 digits: 0, 1
And thus the base is 2.
• Each digit in a binary number is called a BIT,
• 4 bits form a NIBBLE,
• 8 bits form a BYTE,
• two bytes form a WORD,
• two words form a DOUBLE WORD(rarely
used):
8

9. The Binary system
• There is a convention to add "b" in the end of a
binary number, this way we can determine that
101b is a binary number with decimal value of
5.
• The binary number 10100101b equals to
decimal value of 165:
9

10. The Binary system
10

11. Octal system (Base 8)
• the system is based on the number eight (as
opposed to ten). Remember how in base-10 we
had ten digits? Now, in base-8, we are limited to
only eight digits: 0, 1, 2, 3, 4, 5, 6, and 7.
There’s no such thing as 8 or 9.
• after we get to 7, we run out of different digits
for any higher number. We need a way to
represent eight of something. So we add
another digit, change the 7 back to 0, and end
up with 10. Our answer of 10 in base-8 now
represents what we would normally think of as
8 in base-10.
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12. Octal system (Base 8)
• What’s the next number after 778? If you said
1008, you’re correct. We know from what we’ve
learned so far that the first 7 in 778 represents
groups of 8, and the second 7 represents
induvidual items. If we add these all up, we
have 7*8 + 7*1 = 63. So we have a total of
6310. So 778=6310. We all know 6410comes after
6310.
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13. Converting From Base-8 to Base-10
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14. Converting from Base-10 to Base-8
• Let's start with an example: 15010
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15. Hexadecimal System
• Hexadecimal System uses 16 digits:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F
And thus the base is 16.
• Hexadecimal numbers are compact and easy to
read.
• It is very easy to convert numbers from binary
system to hexadecimal system and vice-versa,
every nibble (4 bits) can be converted to a
hexadecimal digit using this table:
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16. Hexadecimal System
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17. Hexadecimal System
• There is a convention to add "h" in the end of a
hexadecimal number, this way we can determine that
5Fh is a hexadecimal number with decimal value of 95.
We also add "0" (zero) in the beginning of hexadecimal
numbers that begin with a letter (A..F), for
example 0E120h.
The hexadecimal number 1234h is equal to decimal
value of 4660:
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18. Converting from Decimal System to
Hexadecimal
• In order to convert from decimal system, to any
other system, it is required to divide the decimal
value by the base of the desired system,
• each time you should remember the result and
keep the remainder, the divide process
continues until the result is zero.
• The remainders are then used to represent a
value in that system.
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19. Converting from Decimal System to
Hexadecimal
• Let's convert the value of 39 (base 10)
to Hexadecimal System (base 16):
• As you see we got this hexadecimal
number: 27h.
All remainders were below 10 in the above
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example, so we do not use any letters.

20. Converting from Decimal System to
Hexadecimal
• Here is another more complex example:
let's convert decimal number 43868 to
hexadecimal form:
The result is 0AB5Ch, we
are using the above
table to convert
remainders over 9 to
corresponding letters.
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21. Converting from Hexadecimal to Binary
• Using the same principle we can convert to
binary form (using 2 as the divider), or convert
to hexadecimal number, and then convert it to
binary number using the above table:
• As you see we got this binary
number: 1010101101011100b
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22. references
• Chapter 30, Teach yourself electronics, 3rd ed, Stan
Gibilisco.
• Chapter 4, digital electronics demystified, Myke Predko.
• Chapter 18, Schaums outline: basic mathematics with
applications to science and technology, Kruglak, Moore.
• http://www.electronics.dit.ie/staff/tscarff/number_syste
ms/number_systems.html
• http://active.tutsplus.com/tutorials/codingconcepts/number-systems-an-introduction-to-binaryhexadecimal-and-more/
• http://www.math.grin.edu/~rebelsky/Courses/152/97F/
Readings/student-binary
• http://active.tutsplus.com/tutorials/codingconcepts/number-systems-an-introduction-to-binary22
hexadecimal-and-more/