MATH

IMPORTANCE OF BINARY SYSTEM

I will try to explain in simple words the importance of binary system.

The switch has only two options, on or off.

The morse code is another example because It also works with only two digits, a dot or a dash.

Computers can only talk in binary

Electronic circuits have only 2 states off and on

Each binary sequence has a special meaning

The binary system is essential in technology.

Bits and Bytes

It is very important to know the difference between a bit and a byte because these two can get easily confused.

10101110 is a byte a sequence of 8 zeros and ones, if you take a group of 8 bits, you have a byte.

Any character you type on your keyboard is interpreted by your computer as a byte, an 8 digit binary number.

A good example is how ASCII code (American Standard Code for Information Interchange) works

. For example, the letter “A” is expressed as the ASCII code 65. But 65 is a decimal number, so if you convert it to a binary number, you get 01000001. These 8 digits, or one byte, are known to your computer as the letter “A”.

One good example is monitoring data transfer speed. When you download a file from the Internet, you probably have noticed that your browser indicates the transfer rate in KBps. The letter “B” is capitalized. We are talking about bytes.(not bits)

Sometimes we are talking in kilobits per second that is written kbps ( note the lower letter b)

If we see the transfer rate is for instance 46 kbps,and since we know that 8 bits equal one byte, we divide 46 by 8 and get a theoretical maximum of 5,75 Kilo Bytes per second.(B capital letter)

 

                                                        Another thing

A CD stores data using the binary system in the following matter: When writing data to a CD, the laser does nothing else than following a spiral-shaped “guide groove” while burning the data as a long string of pits into a reflective dye layer on the CD. When the CD is read later, a laser follows the same spiral and reads either a pit where the light does not get reflected very well, or a smooth area called land where the light reflects very well. This system is nothing else but our good old binary system, using only two possible states, pit or land, to read and record data in binary.

Storing data on a magnetic media, such as a hard drive, also uses the binary system. In a very simplified manner of speaking, each data bit gets stored on a drive as a tiny magnetic field. Each magnetic field has two poles, North and South. When the disk spins around and data is read by the read head, the magnetic field either has the North or the South pole aligned first. Again, only two possible stages, North or South, 0 or 1.

                                      Another interesting thing

I bought a 8.4GB hard drive, but when I formatted it, Windows tells me that I have only 7.82 GB available. Did they sell me the wrong drive?”

No!

They sold you the correct drive, but it was not labeled quite correctly. The marketing department of the drive manufacturer doesn’t know too much about bits and bytes and the binary system. To make it easier to calculate, they assume that 1 KB is 1000 Bytes, 1 MB is 1000 KB, etc. which is wrong. So when they have a drive that can hold 8,400,000,000 Bytes, they just call it 8.4 GB and say that’s close enough for government work. Not so.The multiplication factor is not 1000 since we’re not using the decimal system, it is 1024 instead (2 to the power of 10).

To figure out the correct size of that drive, divide 8,400,000,000 Bytes by 1024 and you’ll get 8,203,125 KB. Divide that by 1024, and you get 8,010 MB. Divide that by 1024 and you get 7.82 GB which is the actual size of your hard drive in GB as reported by your operating system.


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.


how to see numerical systems

HOW TO SEE NUMERICAL SYSTEMS