Understanding Base Conversion and Multiplication: A Guide to 12 Base 7 Multiplied by 4

When working with different number bases, understanding how to convert numbers and perform operations (like multiplication) can be challenging. In this guide, we will explore the process of converting 12 in base 7 to base 10, multiplying it by 4, and then converting the result back to base 7. By the end of this article, you will have a solid grasp on the concept and can apply it to other similar problems.

Step 1: Understanding Base 7 Conversion to Base 10

To convert the number 12 in base 7 to base 10, we need to understand the positional value of each digit. In base 7, the number 12 can be expressed as follows:

1 x 71 2 x 70 1 x 7 2 x 1 7 2 9 in base 10

Step 1.1: Breakdown of the Conversion

1 x 71: This represents the coefficient of 7 to the first power (7 x 1 7). 2 x 70: This represents the coefficient of 7 to the zero power (7 x 1 1).

When added together, these values sum to 9 in base 10.

Step 2: Multiplying by 4 in Base 10

Now, we take the base 10 equivalent, 9, and multiply it by 4:

9 x 4 36 in base 10

Step 2.1: Verification of the Multiplication

This multiplication can be verified as follows:

4 x 9 36 (which is the same as 4 x 10 4 x 0 36)

Step 3: Converting 36 from Base 10 to Base 7

To convert 36 from base 10 to base 7, we use the process of repeated division by 7, keeping track of the remainders:

36 ÷ 7 5 remainder 1 5 ÷ 7 0 remainder 5

Reading the remainders from bottom to top, we get 51 in base 7.

Conclusion

Thus, 12 in base 7 multiplied by 4 in base 10 is 51 in base 7. Here is a summary of the process:

Convert 12 base 7 to base 10: 1 x 7 2 x 1 9. Multiply 9 by 4: 9 x 4 36. Convert 36 base 10 to base 7: 51.

Additional Examples and Practice

For further understanding, let's look at one more example:

Example: Convert 34 base 5 to base 10, multiply by 3, and convert back to base 5

Convert 34 base 5 to base 10: 3 x 5 4 x 1 15 4 19 in base 10. Multiply 19 by 3: 19 x 3 57 in base 10. Convert 57 base 10 to base 5: 57 ÷ 5 11 remainder 2, 11 ÷ 5 2 remainder 1, 2 ÷ 5 0 remainder 2. Reading from bottom to top gives 212 in base 5.

This example further reinforces the steps and processes involved in base conversion and multiplication.

Key Takeaways:

To convert from one base to another, use the positional value of each digit. To perform arithmetic operations, convert to base 10, perform the operation, and then convert back to the desired base.

By practicing these steps and understanding the underlying principles, you can confidently handle base conversion and multiplication problems in different number systems.