Download how to see numerical systems

HOW TO SEE NUMERICAL SYSTEMS
We can build infinite numerical systems. What change is the basis. The most known are
decimal , binary and hexadecimal
In decimal system(the usual one )the basis is the number 10 (we have ten fingers)
It is assumed in math that any number powered to zero equals 1
So you have in your mind the term “powered to”
2^4=2x2x2x2=16
5^3=5x5X5=125
….
Now we go how to build numerical systems starting with decimal system:
In decimal system we have ten symbols 1,2 ,3 ,4 , 5 , 6 , 7 , 8 , 9 ,0
In binrary we have only 2 symbols 1, 0
In hexadecimal 16 simbols 1, 2 , 3 , 4 , 5 , 6 , 7 , , 8 , 9 , 0 , A, B, C, D, E, F
Returning to decimal system
Number examples expressed in its basis (in this case 10)
16=1x10^1x+6x10^0 (1 is the first digit, 10 is the basis, 6 is the second digit
32=3x10^1+2x10^0
120=1x10^2+2x\10^1+0^0
3024= 3x10^3+0x10^2+2x10^1+4x10^0
12560= 1x10^4+2x10^3+5x10^2+6x10^1+0^0
Numbers in bold are the power counted from left to right of the digit position
beging in zero
And what about binary system ?
We only have two digits 1 , 0
How can we write any number in a binary system basis?
It is the same logical structure!
Binary System
Unlike the decimal system, only two digits - 0, 1 - suffice to represent
a number in the binary system. The binary system plays a crucial role
in computer science and technology. The first 20 numbers in the
binary notation are 1, 10, 11, 100, 101, 110, 111, 1000, 1001, 1010,
1011, 1100, 1101, 1110, 1111, 10000, 10001, 10010, 10011, 10100,
the origin of which may be better understood if they are re-written in
the following way:
1:
2:
3:
4:
5:
6:
7:
8:
9:
10:
00001
00010
00011
00100
00101
00110
00111
01000
01001
01010
11: 01011
12: 01100
13: 01101
14: 01110
15: 01111
16: 10000
17: 10001
18: 10010
19: 10011
20: 10100
Lets see some binary numbers and convert them to decimal
1000 = 1x2^3+0x2^2+0x2^1+0x0^0=8+0+0+0=8
1111=1x2^3+1x2^2+1x2^1+1x2^0=8+4+2+1=8+4+1=15
Or using the digits written like above
Any binary number can be converted into thedecimal system by
summing the appropriate multiples of the different powers of two. For
example, starting from the right, 10101101 represents (1 x 20) + (0
x 21) + (1 x 22) + (1 x 23) + (0 x 24) + (1 x 25) + (0 x 26) + (1 x 27)
= 173. This example can be used for the conversion of binary
numbers into decimal numbers.
For the conversion of decimal numbers to binary numbers, the same
principle can be used, but the other way around. Thus, to convert,
first find the highest power of two that does not exceed the given
number, and place a 1 in the corresponding position in the binary
number. For example, the highest power of two in the decimal
number 519 is 29 = 512. Thus, a 1 can be inserted as the 10th digit,
counted from the right: 1000000000.
In the remainder, 519 - 512 = 7, the highest power of 2 is 22 = 4, so
the third zero from the right can be replaced by a 1: 1000000100.
The next remainder, 3, consists of the sum of two powers of 2: 21 +
20, so the first and second zeros from the right are replaced by 1:
519 = 10000001112.