1. Numeral Systems
Rubel Biswas

2. Sexagesimal (base 60)
It is possible to count to 12 with your thumb acting as a pointer, touching each finger bone in turn.
In this system, the right hand counts repeatedly to 12 (which is called a DOZEN)
The left hand counts the number of iterations until five dozens
This gives 60 as full range.

3. Sexagesimal (base 60)

4. Sexagesimal (base 60)
It was originated with the ancient Sumerians in the years BC.
It is still used (in a modified form) for measuring time, angles, and the geographic coordinates that are angles.
The number 60 has twelve factors, namely
{ 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 }
Fractions involving sexagesimal numbers are simplified.
For example, one hour can be divided evenly into sections of 30 minutes, 20 minutes, 15 minutes, 12 minutes, 10 minutes, six minutes, five minutes, etc.

5. Interesting facts

6. Babylonian numerals
Babylonian numerals were written in cuneiform, using a wedge-tipped reed stylus to make a mark on a soft clay tablet which would be exposed in the sun to harden to create a permanent record.

7. Babylonian numerals

8. The decimal (base 10)
The decimal numeral system (also called base ten or occasionally denary) has ten as its base. It is the numerical base most widely used by modern civilizations.
Decimal notation often refers to a base-10 positional notation such as the Hindu-Arabic numeral system.
When we write numbers, the position (or "place") of each number is important.
In the number 327:
"7" is in the Units position, meaning just 7
"2" is in the Tens position meaning 2 tens (or twenty),
"3" is in the Hundreds position, meaning 3 hundreds
327 = 3x100+ 2x10 + 7x1
= 3x x101+ 7x100

9. Binary numeral system The system is base-2 number system,
It represents numeric values using two symbols 0 and 1.
More specifically, the usual base-2 system is a positional notation with a radix of 2.
Because of its straightforward implementation in digital electronic circuitry using logic gates, the binary system is used internally by almost all modern computers.
In this system
ON = 1
OFF = 0

10. Counting in Binary (1)
Counting in binary is similar to counting in any other number system.
Beginning with a single digit, counting proceeds through each symbol, in increasing order.
Decimal counting uses the symbols 0 through 9, while binary only uses the symbols 0 and 1.
000, 001, 002, , 008, 009,
010, 011, 012, ...    ...
090, 091, 092, , 098, 099
100, 101, 102, ...

11. Counting in Binary (2)
0000 1
0001 2
0010 3
0011 4
0100 5
0101 6
0110 7
0111 8
1000 9
1001 10
1010 11
1011 12
1100 13
1101 14
1110 15
1111
000 1
001 2
010 3
011 4
100 5
101 6
110 7
111 00 1 01 2 10 3 11

12. Converting Binary into Decimal2 1 21 20 22 21 20 4 2 1 23 22 21 20 8 4 2 1 8 4 2 1 1
1101 = 1x23 + 1x22+ 0x21 +1x20 =
= 1x8 + 1x4 + 0x2 + 1x1= = 13

13. Converting Decimal into Binary
Position 9 8 7 6 5 4 3 2 1
Value 29 28 27 26 25 24 23 22 21 20
Value
512
256
128 64 32 16 8 4 2 1
Example
Suppose that we want to convert 500 into binary
Start from the left, find the first number in figure above which is less that 500
Put 1 in the corresponding field 1
3. Do the following subtraction 500 – 256 = 244
4. If the result is greater than the value of the field to the right then put 1 in this field otherwise put 0 and move one step to the right.
5. Repeat this operation to the end of all fields 1

14. Converting Decimal into Binary
Another simple method is as follows
Divide the number by 2
If the remainder = 0 put 0 at the right most else put 1
Go to step 1.
Example:
500 / 2 = 250 remainder = 0
250 / 2 = 125 remainder = 0
125 / 2 = 62 remainder = 1 1
62 / 2 = 31 remainder = 0 1

15. Binary Addition Since there two number the following is valid
0 + 0 = 0
0 + 1 = 1
1 + 0 = 1
1 + 1 = 0 with a carry of 1
3 + 7 = 10
In Binary
carry 1 1 1 1 1 1

16. Binary Subtraction Following the same criteria of the addition
0 - 0 = 0
0 - 1 = 1, borrow 1
1 - 0 = 1
1 - 1 = 0
= 3
In Binary 1 1 1
borrow 1 1 1

17. Binary Multiplication
Multiplication in the binary system works the same way as in the decimal system:
0 x 0 = 0
1 x 0 = 0
0 x 1 = 0
1 x 1 = 1
101
* 11
____
_____
1111

18. Binary Division
Binary division is again similar to its decimal counterpart
Example 27/5 = (11011)/(101) 1
___________
1 0 1 )
− 1 0 1
-----
0 1 1
1 0 1
___________
1 0 1 )
− 1 0 1
1 0
Not enough, put 0 in the result and get the next number.
remainder

19. Octal (base 8)
The octal is the base-8 number system, and uses the digits 0 to 7.
Numerals can be made from binary numerals by grouping consecutive binary digits into groups of three (starting from the right)
Binary
Decimal
Octal
000
001 1
010 2
011 3
100 4
101 5
110 6
111 7
1000 8 10

20. Binary to Octal and vice versa1 74
Octal 1 1 2
Binary 1 1 1

21. Converting Decimal to Octal
Converting any decimal into octal can be done as follows
Divide the number by 8
Take the remainder
Go to step 1.
Read the number from the bottom to the top.
Example convert to octal Divide 601 by 8 = 75 and the remainder is 1 Divide 75 by 8 = 9 and the remainder is 3 Divide 9 by 8 =1 and the remainder is 1 Divide 1 by 8 = 0 and the remainder is 1 Reading from bottom to top is 11318

22. Converting Octal to Decimal
1. Keep the right number
2. Multiply all other digits by the base number times itself
the same number of times that it is to the left of your
right number.
Converting Octal into Decimal
112 = 1 x x x 80 = = 74
Which means
7410 = 1128

23. Hexadecimal (base 16)
Binary
Decimal
Hex
0 000
0 001 1
0 010 2
0 011 3
0 100 4
0101 5
0110 6
0111 7
1000 8
1001 9
1010 10 A
1011 11 B
1100 12 C
1101 13 D
1110 14 E
1111 15 F
The hexadecimal is the base-16 number system, and uses the digits 0 to 15.
Numerals can be made from binary numerals by grouping consecutive binary digits into groups of four (starting from the right)

24. Binary to Hex and vice versa1
330 4 A
Hexadecimal 1 1 1 1
Binary

25. Converting Decimal to Hex
Converting any decimal into octal can be done as follows
Divide the number by 16
Take the remainder
Go to step 1.
Read the number from the bottom to the top.
Example convert to Hex Divide 601 by 16 = 37 and the remainder is 9 Divide 37 by 16 = 2 and the remainder is 5 Divide 2 by 16 = 0 and the remainder is 2 Reading from bottom to top is 25916

26. Converting Hex to Decimal
1. Keep the right number
2. Multiply all other digits by the base number times itself
the same number of times that it is to the left of your
right number.
Converting Hex into Decimal
25916 = 2 x x x 160 = = 60110
Which means
60110 = 25916