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.
- 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.
- Write down the number and its positional values:
- $1 times 4^2$
- $2 times 4^1$
- $3 times 4^0$
- Calculate each term:
- $1 times 4^2 = 1 times 16 = 16$
- $2 times 4^1 = 2 times 4 = 8$
- $3 times 4^0 = 3 times 1 = 3$
- Sum these values to get the decimal number:
- $16 + 8 + 3 = 27$
So, 123_4 is 27_{10} in base 10.
- 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.
- Divide 27 by 6 and record the quotient and remainder:
- $27 , div , 6 = 4$ with a remainder of $3$
- Divide the quotient (4) by 6:
- $4 , div , 6 = 0$ with a remainder of $4$
- 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.
- Convert 302_4 to base 10:
- $3 times 4^2 + 0 times 4^1 + 2 times 4^0 = 3 times 16 + 0 times 4 + 2 times 1 = 48 + 0 + 2 = 50$
- 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.