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The hex system, or hexadecimal, is a base 16 number system. Because the decimal system has only 10 digits, the additional 6 digits are represented by the first 6 letters in the alphabet. For example, a hex value of B would be represented in decimal form as 11, or a binary value of 1011. It is an easy way to express binary numbers in modern computers where a byte is usually defined as containing eight binary digits.

The decimal system is one of today’s oldest and most commonly used numbers systems. It is also known as base 10 numbering because it is based on 10 single digits:0,1,2,3,4,5,6,7,8,9. For example, a decimal value of 11 would be represented as a hex value of B, or as a binary value of 1011.

Now, how do you manually convert hex to a decimal? First, you must know that all the letters in a hex have decimal equivalents as listed in the table below.

Hex base 16 | Decimal base 10 | Calculation |
---|---|---|

0 | 0 | - |

1 | 1 | - |

2 | 2 | - |

3 | 3 | - |

4 | 4 | - |

5 | 5 | - |

6 | 6 | - |

7 | 7 | - |

8 | 8 | - |

9 | 9 | - |

A | 10 | - |

B | 11 | - |

C | 12 | - |

D | 13 | - |

E | 14 | - |

F | 15 | - |

10 | 16 | 1×16^{1}+0×16^{0} = 16 |

11 | 17 | 1×16^{1}+1×16^{0} = 17 |

12 | 18 | 1×16^{1}+2×16^{0} = 18 |

13 | 19 | 1×16^{1}+3×16^{0} = 19 |

14 | 20 | 1×16^{1}+4×16^{0} = 20 |

15 | 21 | 1×16^{1}+5×16^{0} = 21 |

16 | 22 | 1×16^{1}+6×16^{0} = 22 |

17 | 23 | 1×16^{1}+7×16^{0} = 23 |

18 | 24 | 1×16^{1}+8×16^{0} = 24 |

19 | 25 | 1×16^{1}+9×16^{0} = 25 |

1A | 26 | 1×16^{1}+10×16^{0} = 26 |

1B | 27 | 1×16^{1}+11×16^{0} = 27 |

1C | 28 | 1×16^{1}+12×16^{0} = 28 |

1D | 29 | 1×16^{1}+13×16^{0} = 29 |

1E | 30 | 1×16^{1}+14×16^{0} = 30 |

1F | 31 | 1×16^{1}+15×16^{0} = 31 |

20 | 32 | 2×16^{1}+0×16^{0} = 32 |

30 | 48 | 3×16^{1}+0×16^{0} = 48 |

40 | 64 | 4×16^{1}+0×16^{0} = 64 |

50 | 80 | 5×16^{1}+0×16^{0} = 80 |

60 | 96 | 6×16^{1}+0×16^{0} = 96 |

70 | 112 | 7×16^{1}+0×16^{0} = 112 |

80 | 128 | 8×16^{1}+0×16^{0} = 128 |

90 | 144 | 9×16^{1}+0×16^{0} = 144 |

A0 | 160 | 10×16^{1}+0×16^{0} = 160 |

B0 | 176 | 11×16^{1}+0×16^{0} = 176 |

C0 | 192 | 12×16^{1}+0×16^{0} = 192 |

D0 | 208 | 13×16^{1}+0×16^{0} = 208 |

E0 | 224 | 14×16^{1}+0×16^{0} = 224 |

F0 | 240 | 15×16^{1}+0×16^{0} = 240 |

100 | 256 | 1×16^{2}+0×16^{1}+0×16^{0} = 256 |

200 | 512 | 2×16^{2}+0×16^{1}+0×16^{0} = 512 |

300 | 768 | 3×16^{2}+0×16^{1}+0×16^{0} = 768 |

400 | 1024 | 4×16^{2}+0×16^{1}+0×16^{0} = 1024 |

There are other number system tables with more values for octas, hexes, decimals, and binaries, but the table above provides everything we need to understand how to convert hexadecimal to decimal.

To manually convert a hexadecimal to a decimal, you must start by multiplying the hex number by 16. Then you raise it to a power of 0 and increase that power by 1 each time according to the equivalent hexadecimal number.

We start from the right of the hexadecimal number and go left when applying the powers. Every time you multiply a number by 16, the power of 16 increases.

When converting a hexadecimal C9 to decimal your work should look like this:

**Example**

*9 = 9 * (16 ^ 0) = 9*

*C = 12 * (16 ^ 1) = 192*

Then, we add the results

*192 + 9 = 201 _{10 }decimal*

*Let’s try to understand how:*

- First, we converted all of our hex numbers into their decimal equivalents. C is equal to the decimal 12 (see table above) and 9 is equal to the decimal 9.

- Then we multiplied the numbers 12 and 9 starting with the last number in the question by 16 and its power. Remember, the powers start at zero.

- Our first multiplication had 0 power and the second multiplication had 1. If there were a third, it would have had a power of 2.

- The (^) symbol represents "raised to the power of." Therefore, the first terms in brackets read, "16 to the power of 0." This means that sixteen were multiplied by themselves zero times. Anything raised to zero power is 1. Therefore, 9 were multiplied by one.

- In the second bracket, the term read, "16 to the power of 1." A number raised to one's power is equal to that number. Therefore, 12 were multiplied by 16. We got 192 when we multiplied these.

- We then added the results to get our number of decimal equivalents, which was 201.

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