# Sixteen Number System and Binary Conversion / Hexadecimal Numbers

The sixteen-digit system groups binary numbers into four sets that allow the conversion of 16 different binary digits.

The main disadvantage of binary numbers is that the binary string equivalent of a large de de-de de-deserial-10 number can be quite long.

When working with large digital systems such as computers, it is common to find binary numbers consisting of 8, 16, and even 32 digits, which makes it difficult to both read and write, especially without generating a large number of errors when working a large number of 16 or 32 bits.

A common way to overcome this problem is to organize binary numbers into four-bit groups or clusters.These 4-bit groups use another enumeration system commonly used on computers and digital systems called **Hexadecimal Numbers.**

The sixteen-numbering system uses **the base 16** system and is a popular choice to represent long binary values because their format is quite compact and much easier to understand compared to long binary arrays of 1s and 0s.

Since there is a base-16 system, the hexagon numbering system therefore uses 16 (sixteen) different digits with a combination of numbers 0 through 15.In other words, there are 16 possible number symbols.

However, there is a potential problem with the use of this step representation method, which arises from the fact that decimal numbers 10, 11, 12, 13, 14, and 15 are normally written using two adjacent symbols.For example, if we write 10 as hexadecimal, we must decide whether to use the decimal number or the binary number of two (1 + 0).To overcome this difficult problem, hexadecimal numbers that define the values of ten, eleven, are replaced with the uppercase letters A, B, C, D, E, and F, respectively. In addition to the numbers in the sixteen-number system, there are six more digits in addition to the numbers in the 10-based numbers. (A=10, B=11, C=12, D=13, E=14, F=15)

a ={0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F}

As we have said before, binary sequences can be quite long and difficult to read, but we can make life easier by dividing these large binary numbers into double groups, making them much easier to write and understand.For example, binary number group 1101 0101 1100 1111_{2} is much easier to read and understand than 1101010111001111_{2}when all put together.

In the daily use of the decimal number system, we use groups of three digits or 000s from the right side to make it easier for us to understand a very large number, such as a million or trillion, and the same is true in digital systems.

Sixteen Numbers is a more complex system than using only binary or dex numbers, and is often used when dealing with computers and memory address locations.By dividing a binary number into 4-bit groups, each group or 4-digit set can now have a possible value between " 0000 " (0) and " 1111 " (8+4+2+1 = 15), and we can show a total of **16** different number combinations from 0 to 15.

From our first tutorial on Binary Numbers, we remember that a group of 4-bit digits is called "nibble", and 4 bits are also required to produce a hexadecimal number; half a byte.Two hexadecimal numbers are then required to produce an integer between 00 and FF

Also, because there is a fourth force of 16 , 2 (or ^{2 4)} in the decimal system, there is a direct relationship between the numbers 2 and 16, so a hexagon has a value equal to four binary digits.

Because of this relationship, four digits in a binary number can be represented by a single hexadecimal digit.This makes converting between binary and hex numbers very easy, and can be used to write large binary numbers with hexagons and much fewer digits.

De de deity number | 4-bit Binary Number | Hexadecimal Number |

0 | 0000 | 0 |

1 | 0001 | 1 |

2 | 0010 | 2 |

3 | 0011 | 3 |

4 | 0100 | 4 |

5 | 0101 | 5 |

6 | 0110 | 6 |

7 | 0111 | 7 |

8 | 1000 | 8 |

9 | 1001 | 9 |

10 | 1010 | A |

11 | 1011 | B |

12 | 1100 | C |

13 | 1101 | NS |

14 | 1110 | E |

15 | 1111 | F |

16 | 0001 0000 | 10 (1+0) |

17 | 0001 0001 | 11 (1+1) |

Using the original binary number in 1101 0101 1111 _{2} above, this can be converted to an equivalent hexadecimal D5CF number, which is much easier to read and understand than the long line of 1s and 0s we had before.

Therefore, using hexadecimal notation, digital numbers can be written using fewer digits and with the possibility of a much lesser error occurring.Similarly, converting hexadecimal-based numbers back to binary numbers is the opposite of the process.

Then the main feature of a Hexadic number system is that there are 16 different counting digits from 0 to F, and each digit has a weight or value of 16, starting with the least significant bit (LSB).To distinguish hexadecimal numbers from decimal numbers, a prefix "#" , (Hash) or "$" (Dollar sign) is used before the actual sixteen-number value , #D5CF or $D 5CF.

Since the base of a hexadecimal system is 16, which also represents the number of individual symbols used in the system, the subscript is 16, it is used to define a number expressed as hexadecimal.For example, the previous hexadecimal number is expressed as follows: D5CF_{16}

MSB | Hexadecimal Number | LSB | ||||||

16^{Of 8} | 16^{Of 7} | 16^{Of 6} | 16^{Of 5} | 16^{Of 4} | 16^{Of 3} | 16^{OF 2} | 16^{Of 1} | 16^{of 0} |

4.3G | 2.6G | 16 million | 1 million | 65k | 4k | 256 | 16 | 1 |

Adding these additional sixteen digits to convert both decimal and binary numbers to sixteen numbers is very easy if there are 4, 8, 12, or 16 binary digits to convert.However, if the number of binary bits is not multiple of four, we can also add zeros to the left of the most meaningful bit, MSB .

For example, 11001011011001 _{2} is a fourteen-bit binary number that is too large for only three hexadecimal digits, but too small for a number of four hexadecimal digits.The answer is to add a left-hand zero until there is a complete four-bit binary number or multiples of them.

## Adding 0 to Binary Numbers

Binary number | 0011 | 0010 | 1101 | 1001 |

Hexadecimal Number | 3 | 2 | NS | 9 |

The main advantage of the sixteen number is that it is very compact, and the use of the base 16 means that the number of digits used to represent a particular number is usually less than the binary or decimal number.In addition, converting between hex numbers and binary numbers is quick and easy.

### Sixteen Number System Question Example 1

Convert binary number 1110 1010_{2}to the sixteen-point equivalent.

Number = 11101010 _{2} | |||

Divide lice into four, starting on the right side | |||

= | 1110 | 1010 | |

Find the Deity equivalent of each group | |||

= | 14 | 10 | (in de dethending) |

Convert to Sixteen using the table above | |||

= | E | A | (in hexadecimal) |

then the hexagon equivalent of binary number 1110 1010_{2} is #EA_{16} |

### Sixteen Number System Question Example 2

Convert sixteen numbers #3FA7_{16}to binary equivalent, as well as dex equivalent by using subscripts to define each enumeration system.

#3FA7_{16} |

= 0011 1111 1010 0111_{2} |

= (8192 + 4096 + 2048 + 1024 + 512 + 256 + 128 + 32 + 4 + 2 + 1) |

= 16.295_{10} |

Then, the decimal number **of 16,295** can be represented as follows:

#3FA7 in_{a 16-under} order and

0011 1111 1010 0111 _{2} binary order.

## Summarize

**The hexagon** or **Hex** enumeration system is widely used in computers and digital systems to reduce large strings of binary numbers into four-digit sets for us to easily understand.The word "hexagon" means sixteen, since this type of digital numbering system uses 16 different numbers from 0 to 9 and A to F.

Hexadecimal numbers group binary numbers into four-digit sets.To convert a binary array to an equivalent *hexadecimal number,* we must first group the binary digits into a 4-bit array.These binary sets can have any value between 0 10 ( 0000_{2} ) and_{15 10} (_{1111 2} ), representing the hexagon equivalent of_{0} to F.

In the next tutorial on number systems, we will look at converting entire binary number **sequences**to another digital numbering system called Octal Numbers, or vice versa.