Notion(Base) with conversion

1. About notion(base system)

Base system is a calculation method that specifies the unit in which the number of digits changes when counting.
  For example, the normally used number is a decimal system using 10 digits from 0 to 9. In a day and night, time with 12 intervals is in the decimal system, 1 hour has a period of 60 minutes, and 1 minute has a period of 60 seconds.

  Unlike humans, in the case of computers, the only things recognized as electronic products are that electricity flows (1) and does not flow (0), and all operations are performed in binary format accordingly. However, the larger the number, the longer it is difficult for humans to read, so it is common to write binary numbers in hexadecimal format, where every four digits is converted to a single number. Accordingly, it is necessary to know the number system for three base systems, binary system, and hexadecimal system, and to be able to convert between base systems.

  Decimal - 0₁₀, 1₁₀, 2₁₀, 3₁₀, 4₁₀, 5₁₀, 6₁₀, 7₁₀, 8₁₀, 9₁₀ (10 digits, then continue in the order of 10₁₀-11₁₀-12₁₀.)
  The last ₁₀ is added to distinguish it from binary and hexadecimal numbers, but it is the same as the usual number.

  Binary - 0₂, 1₂ (2 digits, then continue in the order of 10₂-11₂-100₂.)
  To distinguish binary numbers, ‘0b’ is sometimes written in front like 0b100 instead of of the last ₂.

  Hexadecimal - 0₁₆, 1₁₆, 2₁₆, 3₁₆, 4₁₆, 5₁₆, 6₁₆, 7₁₆, 8₁₆, 9₁₆, A₁₆, B₁₆, C₁₆, D₁₆, E₁₆, F₁₆ (16 digits, then continue in the order of 10₁₆-11₁₆-12₁₆)
  To distinguish hexadecimal numbers, ‘0x’ is sometimes written in front like 0x12 instead of the last ₁₆.

  Comparing the table below:

Decimal	010	110	210	310	410	510	610	710
Binary	02	12	102	112	1002	1012	1102	1112
Hex	016	116	216	316	416	516	616	716
Decimal	810	910	1010	1110	1210	1310	1410	1510
Binary	10002	10012	10102	10112	11002	11012	11102	11112
Hex	816	916	A16	B16	C16	D16	E16	F16
Decimal	1610	1710	1810	1910	2010	2110	2210	2310
Binary	100002	100012	100102	100112	101002	101012	101102	101112
Hex	1016	1116	1216	1316	1416	1516	1616	1716

 

2. Base Conversion - Decimal to Binary

Before the conversion, it should be noted that the numbers in each base system are constructed as powers of the base system as the digits are increased.
  For example, the decimal number 123₁₀ is expressed as (1₁₀×10²)＋(2₁₀×10¹)＋(3₁₀×10⁰). In the same way, the binary number 10100₂ can be expressed as (1₂×2⁴)＋(0₂×2³)＋(1₂×2²)＋(0₂×2²)＋(0₂×2⁰). It can be seen that the table above is the same.

  When converting a decimal number to a binary number, the method of dividing 2 repeatedly is usually used. To convert 123₁₀ to a binary number, do as follows.

2 )123
  2 ) 61······1
  2 ) 30······1
  2 ) 15······0
  2 ) 7······1
  2 ) 3······1
  2 ) 1······1
       0······1
  After dividing by 2, write the remainder on the right until the quotient becomes 0, and then write the recorded remainders in order from the bottom to the result of converting to binary number 1111011₂.

  Another way to convert a decimal number to a binary number is to subtract a power of 2, which is to subtract from the largest number that can be subtracted. If it is 123₁₀, subtraction is repeated starting from 64₁₀, which is the smallest and largest power, and if it can be subtracted, it is recorded as 1, and if it cannot be subtracted, it is recorded as 0 and repeated until it becomes 0.
  128₁₀ : Can't subtraction
  64₁₀ : 123₁₀－64₁₀＝59₁₀
  32₁₀ : 59₁₀-32₁₀＝27₁₀
  16₁₀ : 27₁₀-16₁₀＝11₁₀
  8₁₀ : 11₁₀-8₁₀＝3₁₀
  4₁₀ : Can't subtraction
  2₁₀ : 3₁₀-2₁₀＝1₁₀
  1₁₀ : 1₁₀-1₁₀＝0₁₀

 

12810	6410	3210	1610	810	410	210	110
0	1	1	1	1	0	1	1

 

As a result of the calculation, 1111011₂ is the same.

  Get used to either method, whichever way you are comfortable with.

 

3. Base Conversion - Binary to Decimal

If you understand the notation for powers of numbers, it's simple. If the binary number 11000110₂ is (1₂×2⁷)＋(1₂×2⁶)＋(0₂×2⁵)＋(0₂×2⁴)＋(0₂×2³)＋(1₂×2²)＋(1₂×2²)＋(0₂×2⁰) and ends by adding only non-zero digits, 2⁷＋2⁶＋2²＋2¹. That is, 128₁₀＋64₁₀＋4₁₀＋2₁₀＝198₁₀.

 

4. Base Conversion - Between Decimal and Hexadecimal

The same method can be used when converting decimal number to binary number, but it is omitted because it is more convenient to convert to binary number rather than double-digit division and then convert again to hexadecimal number.
  When converting a hexadecimal number to a decimal number, it can also be used as a notation for power, but it is omitted because it is convenient to calculate through binary numbers, while multiplication by two digits is often inconvenient.
  In conclusion, it is a conversion that can be done, but is not good.

 

5. Base Conversion - Between Binary and Hexadecimal

Between binary and hexadecimal, there is a formula: ‘4 binary digits = 1 hexadecimal digit’. In fact, 2⁴＝16, so naturally, the process of converting the number 1111011₂ is as follows.
  First, cut off 4 digits from the 1's digit. That is, 111 and 1011 become 111₂ and 1011₂.
  Each of the four-digit numbers is converted to hexadecimal according to the comparison table, and then written. It becomes 7B₁₆.

	1	1	1	1	0	1	1
7				B

 

Converting hexadecimal to binary is also reversed. When converting hexadecimal number C6₁₆, convert C₁₆ and 6₁₆ into 4-digit binary number and write it as it is. It becomes 11000110₂.

C				6
1	1	0	0	0	1	1	0

 

Because the process is so simple, if you remember only 16 corresponding numbers, you can instantly convert from binary to hexadecimal and from hexadecimal to binary.

 

6. Conclusion

Just learn how to calculate this yourself, and usually use a calculator on your computer or a converter on the Internet.