How to Convert Base 4 to Base 6?

Converting numbers between different bases can seem tricky, but with a systematic approach, it becomes manageable. Let’s break down the process of converting a number from base 4 to base 6 step-by-step.

1. Convert Base 4 to Base 10
   The first step is to convert the base 4 number to a base 10 (decimal) number. Here’s how you can do it:

Example

Let’s convert the base 4 number 123_4 to base 10.

1. Write down the number and its positional values:
   • $1 times 4^2$
   • $2 times 4^1$
   • $3 times 4^0$
2. Calculate each term:
   • $1 times 4^2 = 1 times 16 = 16$
   • $2 times 4^1 = 2 times 4 = 8$
   • $3 times 4^0 = 3 times 1 = 3$
3. Sum these values to get the decimal number:
   • $16 + 8 + 3 = 27$

So, 123_4 is 27_{10} in base 10.

2. Convert Base 10 to Base 6
   Next, we convert the base 10 number to a base 6 number. This involves repeatedly dividing the number by 6 and keeping track of the remainders.

Example

Let’s convert 27_{10} to base 6.

1. Divide 27 by 6 and record the quotient and remainder:
   • $27 , div , 6 = 4$ with a remainder of $3$
2. Divide the quotient (4) by 6:
   • $4 , div , 6 = 0$ with a remainder of $4$
3. Read the remainders from bottom to top to get the base 6 number:
   • Remainders: $4, 3$
   • So, 27_{10} is 43_6 in base 6.

Conclusion

To summarize, converting a base 4 number to base 6 involves two main steps: first, convert the base 4 number to a base 10 number, and second, convert that base 10 number to a base 6 number. By following these steps, you can accurately convert between any number bases.

Practice Problem

Try converting the base 4 number 302_4 to base 6.

1. Convert 302_4 to base 10:
   • $3 times 4^2 + 0 times 4^1 + 2 times 4^0 = 3 times 16 + 0 times 4 + 2 times 1 = 48 + 0 + 2 = 50$
2. Convert 50_{10} to base 6:
   • $50 , div , 6 = 8$ with a remainder of $2$
   • $8 , div , 6 = 1$ with a remainder of $2$
   • $1 , div , 6 = 0$ with a remainder of $1$
   • Read remainders from bottom to top: 122_6

So, 302_4 is 122_6 in base 6.

Citations

1. 1. Khan Academy – Number Systems
2. 2. Math Is Fun – Number Bases
3. 3. Purplemath – Converting Between Bases