How do you write binary

The binary system

What is the binary system?

We usually use numbers made up of the digits 0-9. After the 9, when "counting up" this digit comes back to 0 and the next higher digit is incremented by one. For example, after the number 3019 comes the number 3020. On the ones digit, the 9 became a 0, and on the tens digit, the 1 became a 2.
After the 1999 comes the 2000: Here the change continues, so to speak, from the ones digit to the thousands digit.

Since a new digit is started from 10 and every tenfold (100, 1000 ...), our number system is called the system of ten or Decimal system.

in the Binary system there are only the digits 0 and 1. After the 1 comes the 0 again, and at the same time the next digit is incremented by one. After the 10000 (read: "one-zero-zero-zero-zero", not "ten thousand") comes the 10001 and then the 10010. The 1 in the last digit has become the zero, and the 0 has become the second digit from the right the 1.

Here, too, they may propagate this change over many places, which even happens very often: For example, 1101010000 comes after 1101001111.

The result is the following series of binary numbers: 0, 1, 10, 11, 100, 101, 110, 111, 1000, 1001, 1010, 1011, 1100, 1101, 1110, 1111, 10000, 10001 ...

When "juxtaposing" the numbers of the decimal system and the binary system that belong together, one discovers different rules:

Decimal-Binary system

Observations:

  • In the last position, the digits change from 0 to 1 and back each time.
  • In the 2nd position from the right, the digits change every second time.
  • In the third position from the right, the digits change every fourth time.
  • On the 4th position from the right, the digits change every eighth time.
  • etc.

  • New positions will be filled for the first time at 1, 2, 4, 8, 16, 32 ...
    each of these numbers is double that of its predecessor. This is why this number system is also called the two-part system or binary system.
    The first digit from the right is the ones digit, the second from the right is the two digit, the third from the right is the four digit, the fourth from the right is the eight digit, etc.

    It is best to imagine binary numbers entered in such a place value table:

    Binary number10245122561286432168421
        1011011
     1011101001
      110010010
    10000100000

At the Converting numbers from the binary system to the decimal system it is checked in places whether there is a 1 on the spot (e.g. on the four digit). If so, the value of the digit (in the example: 4) is added.

Example:
(The small 2 at the back right of the number means: "This number is a binary number")

10110112

The ones sit on the ones, the two, the eight, the sixteen and the 64 digits. (See also the first example in the table)

64 + 16 + 8 + 2 + 1 = 91

Therefore: 10110112 = 9110


At the Convert from decimal to binary if you proceed in reverse:
The decimal number is put together by adding the appropriate digits in the binary system and the associated binary number is found.

Examples:

  • 2010 = 16 + 4 = 101002  (With this binary number, the 16's and 4's are occupied!)
  • 1310 = 8 + 4 + 1 = 11012
  • 30010 = 256 + 32 + 8 + 4 = 1001011002
  • 200010 = 1024 + 512 + 256 + 128 + 64 + 16 = 111110100002

The best way to do this is:
The decimal number 300 is to be converted into the binary system.

  1. Find the largest digit in the binary system that fits into the number.
    It's 256. There, write a 1 on the position board.
  2. Calculate what's left of 300 after subtracting 256. It's 44.
  3. Continue with 44, i.e. look again for the largest binary digit that fits in, write a 1 in the table and subtract the digit from 44. Repeat until nothing is left:
    • The 32 fits into the 44. Remainder: 12
    • The 8th fits into the 12th Remainder: 4
    • The 4 is itself a binary digit. Remainder: 0

    So are occupied with a one: the 256, the 32, the 8 and the 4 digit.
    The remaining places are filled with zeros:

    decimal number Binary number5122561286432168421
    300 = 100101100 100101100