MATH -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=1×10^1x+6×10^0   (1 is the first digit, 10 is the basis, 6 is the second digit

32=3×10^1+2×10^0

120=1×10^2+2x\10^1+0^0

3024= 3×10^3+0x10^2+2×10^1+4×10^0

 

12560= 1×10^4+2×10^3+5×10^2+6×10^1+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:   00001                         11:    01011
2:   00010                         12:    01100
3:   00011                         13:    01101
4:   00100                         14:    01110
5:   00101                         15:    01111
6:   00110                         16:    10000
7:   00111                         17:    10001
8:   01000                         18:    10010
9:   01001                         19:    10011
10: 01010                         20:   10100

 

Lets see some binary numbers and convert them to decimal

1000 = 1×2^3+0x2^2+0x2^1+0x0^0=8+0+0+0=8

 

 

1111=1×2^3+1×2^2+1×2^1+1×2^0=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.