Problem 45 - Thinking Mathematically

Question

Convert each base ten numeral to a numeral in the given base. 386 to base six

Step-by-Step Solution

Answer

386 in base ten is 1402 in base six.

Step 1: Identify Powers of Base

To convert the numerical value 386 to base six, one should first understand the powers of the base six. Conversely, in base six numbering system, the position values are typically based on ascending powers of 6 starting from right to left i.e., \[6^0, 6^1, 6^2, ...\] and so on.

Step 2: Iterations to Reduce the Base 10 number

Begin by dividing 386 by the highest possible power of 6. In this case, start with \[6^3 = 216\], since \[6^4 = 1296\] would exceed 386. Divide 386 by 216, obtaining 1 full 216 with a remainder of 170, which is less than the next power of six. Write down 1 for the \[6^3\] place.

Step 3: Continuing Dividing Process

Now divide the remainder 170 by the next power of six, \(6^2 = 36\). We can get 4 full 36's with a remainder of 26, so we write down 4 for the \(6^2\) place.

Step 4: Final Division Steps to Obtain Base Six Notation

Continuing in this manner, divide the remainder 26 by the next power of six, \(6^1 = 6\). We get 4 full 6's with a remainder of 2, so we write down 4 for the \(6^1\) place. Finally, the remainder 2 can't be divided further. So 2 is used in the \(6^0\) place.

Step 5: Final Result

386 in base 10 equals 1402 in base six as the process results in the following places: \(6^3\) bound with 1, \(6^2\) bound with 4, \(6^1\) bound with 0, and \(6^0\) bound with 2.

Key Concepts

Understanding Base Ten

The base ten system, also known as the decimal system, is the most commonly used number system. It is the system we use in everyday life for counting and measuring. Each position in a base ten number represents a power of 10. For example, in the number 386, we have:

• The digit 3 in the hundreds place, which represents \(3 \times 10^2\)
• The digit 8 in the tens place, which represents \(8 \times 10^1\)
• The digit 6 in the units place, which represents \(6 \times 10^0\)

Thus, 386 can be expanded as \(3 \times 100 + 8 \times 10 + 6 \times 1\). This understanding of places and powers is crucial before attempting conversions to another base.
In the context of base conversion, we must redefine these places in terms of the powers of the new base system, which in this case is base six.

Exploring Base Six

In the base six numbering system, each digit is a position based on powers of 6. This differs significantly from what we use in base ten for our daily activities. Let's look at how base six numbers work. In base six:

• Each place value is a power of 6: \(6^0, 6^1, 6^2, 6^3\), and so forth.
• Digits can range from 0 to 5, since the next digit would overflow into the next higher position.

For example, the number 1402 in base six is expanded as follows:

• The digit 1 in the leftmost place represents \(1 \times 6^3 = 216\)
• The digit 4 represents \(4 \times 6^2 = 144\)
• The digit 0 represents \(0 \times 6^1 = 0\)
• The digit 2 represents \(2 \times 6^0 = 2\)

By understanding base six, we are now prepared to delve into how each digit in a base ten number is transformed into its corresponding base six equivalent.

Significance of Powers of Numbers in Base Conversion

Powers of numbers play a vital role in converting from base ten to another base, such as base six. These powers help us establish place values in the new base. In the conversion process:

• We begin with the highest power of 6 that is less than the number we are converting. This gives us the position of the digit in that particular power place.
• For 386, the highest power less than 386 is \(6^3 = 216\).
• Each subsequent step involves dividing the remainder by decreasing powers of 6: \(6^2 = 36\), \(6^1 = 6\), and \(6^0 = 1\).

Understanding these powers helps in breaking down the number into parts that align with base six positions, allowing for an easier and more organized conversion.

Performing the Division Process in Base Conversion

The division process is essential in base conversion. It involves sequentially dividing the base ten number by powers of the target base, starting from the highest power. Here's how it's done:

• Identify the highest power of six that does not exceed the given base ten number. For 386, this power is \(6^3 = 216\).
• Divide 386 by 216 to find out how many 216's fit into 386. The quotient provides the digit for that power place.
• Subtract the product of the quotient and the power from 386, obtaining the remainder.
• Repeat this process using the next lower power of six. Continue until all place values are determined, ending with the smallest power, \(6^0\).

This step-by-step division yields the final base six number, ensuring that each digit correctly represents its respective place value in the new base system.