# What is Base 7 and How to Convert It to Decimal?

Base 7 is a positional numeral system with seven as its base. It uses 7 different digits for representing numbers. The digits for base 7 are 0, 1, 2, 3, 4, 5, and 6.

Base 7 is also known as septenary or heptal. It is one of the rare bases that has a name other than “base-n”. Some examples of numbers in base 7 are:

- 10
_{7}= 7_{10} - 21
_{7}= 15_{10} - 100
_{7}= 49_{10} - 123
_{7}= 66_{10} - 654
_{7}= 306_{10}

To convert a number from base 7 to decimal (base 10), we can use the following formula:

**decimal = â (digitÃbase)**

This means that we multiply each digit of the base 7 number by 7 raised to the power of its position (starting from right digit number 0), and then add up the results. For example, to convert 123_{7} to decimal, we do:

**(3Ã7) + (2Ã7) + (1Ã7) = (3Ã1) + (2Ã7) + (1Ã49) = 3 + 14 + 49 = 66 _{10}**

To convert a number from decimal to base 7, we can use the following method:

- Divide the decimal number by 7 and write down the remainder.
- If the quotient is not zero, repeat the previous step with the quotient.
- The base 7 number is the sequence of remainders read from bottom to top.

For example, to convert 66_{10} to base 7, we do:

Decimal Number |
/7 |
= Quotient |
+ Remainder |

66 | /7 | =9 | +3 |

9 | /7 | =1 | +2 |

1 | /7 | =0 | +1 |

The base-7 number is: (Remainders read from bottom to top) |
|||

=123_{7} |

We can also use online tools like Base Converter, Base-7 Conversion Tool, or Base-7 to Base-10 Converter, to easily convert between different bases.

In conclusion, base 7 is a numeral system that uses seven digits to represent numbers. It can be converted to decimal by multiplying each digit by its corresponding power of seven and adding them up. It can be converted from decimal by dividing the number by seven repeatedly and writing down the remainders in reverse order.

## Why Use Base 7?

Base 7 is not a very common base, but it has some interesting properties and applications. For example:

- Base 7 is the smallest base that can represent all the days of the week with a single digit. This makes it useful for calendars and scheduling systems. For instance, the International Fixed Calendar used base 7 to divide each month into four weeks of seven days.
- Base 7 is also the smallest base that can represent all the notes of the musical scale with a single digit. This makes it useful for music theory and notation. For instance, the solfÃ¨ge system used base 7 to name the seven notes of the diatonic scale: do, re, mi, fa, sol, la, and si.
- Base 7 is also related to some mathematical patterns and sequences. For example, the Thue-Morse sequence, which alternates between 0 and 1 based on the parity of the number of 1s in the binary representation of a number, can be generated by adding 3 in base 7 and ignoring the carry. For example:

Decimal Number |
Binary Number |
Number of 1s |
Parity |
Thue-Morse Digit |
Base-7 Number |

0 | 0 | 0 | Even | 0 | 0 |

1 | 1 | 1 | Odd | 1 | 3 |

2 | 10 | 1 | Odd | 1 | 6 |

3 | 11 | 2 | Even | 0 | 12_{7} |

4 | 100 | ||||

The Thue-Morse sequence is: (Thue-Morse digits read from left to right) |
|||||
---|---|---|---|---|---|

=0110100110010110… |

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