Problem 4 - A Gentle Introduction to the Art of Mathematics

Question

One method of converting from decimal to some other base is called repeated division. One divides the number by the base and records the remainder \(-\) one then divides the quotient obtained by the base and records the remainder. Continue dividing the successive quotients by the base until the quotient is smaller than the base. Convert 3267 to base- 7 using repeated division. Check your answer by using the meaning of base- 7 place notation. (For example \(54321_{7}\) means \(5 \cdot 7^{4}+\) \(\left.4 \cdot 7^{3}+3 \cdot 7^{2}+2 \cdot 7^{1}+1 \cdot 7^{0} .\right)\)

Step-by-Step Solution

Answer

3267 in base-7 is 12345_{7}. The conversion is correct when verified by the place notation formula.

Step 1 - Initial Division

Divide the given number, 3267, by the desired base, which is 7. Calculate the quotient and the remainder. \[\frac{3267}{7} = 466 \text{ with a remainder of } 5\] Record the remainder, which is 5.

Step 2 - Second Division

Now take the quotient obtained from the previous division (466) and divide it by 7. Calculate the new quotient and remainder. \[\frac{466}{7} = 66 \text{ with a remainder of } 4\] Record the remainder, which is 4.

Step 3 - Third Division

Take the quotient obtained from the previous division (66) and divide it by 7 again. \[\frac{66}{7} = 9 \text{ with a remainder of } 3\] Record the remainder, which is 3.

Step 4 - Fourth Division

Take the quotient obtained from the previous division (9) and divide it by 7 once more. \[\frac{9}{7} = 1 \text{ with a remainder of } 2\] Record the remainder, which is 2.

Step 5 - Final Calculation

Since the quotient (1) is now smaller than the base (7), record the final quotient as the leftmost digit. So the number 3267 in base-7 is the remainders read from bottom to top (along with the final quotient). Therefore, 3267 in base-7 is \(12345_{7}\).

Step 6 - Verification

Verify the conversion by calculating the decimal value from the base-7 number \(12345_{7}\): \[1 \times 7^4 + 2 \times 7^3 + 3 \times 7^2 + 4 \times 7^1 + 5 \times 7^0 \] Calculate the powers of 7: \[1 \times 2401 + 2 \times 343 + 3 \times 49 + 4 \times 7 + 5 \times 1\] Adding these results: \[2401 + 686 + 147 + 28 + 5 = 3267\] This confirms the correctness of our conversion.

Key Concepts

repeated division method

The repeated division method is a systematic technique for converting numbers from decimal (base-10) to another base. This process involves repeatedly dividing the number by the new base and keeping track of the remainders.

To better understand, let’s break down the steps:

• Start with the original number and divide it by the base. Record the quotient and remainder.
• Use the quotient from the previous step and divide it by the base again. Record the new quotient and remainder.
• Repeat this process until the quotient is smaller than the base.
• The remainders recorded, read from bottom to top, along with the final quotient, form the new base number.
  

This method ensures accuracy and makes the conversion simple to follow.

base-7

Base-7 is a numeral system with seven as the base. It means this system uses seven digits: 0, 1, 2, 3, 4, 5, and 6.

Converting a number to base-7 requires using the repeated division method. For example, when converting 3267 to base-7, the steps are:

• First division: 3267 ÷ 7 = 466 remainder 5
• Second division: 466 ÷ 7 = 66 remainder 4
• Third division: 66 ÷ 7 = 9 remainder 3
• Fourth division: 9 ÷ 7 = 1 remainder 2
• Final quotient: 1

Reading the remainders from bottom to top gives us 3267 in base-7 as: \(12345_{7}\). Verifying this result by converting it back to decimal confirms the accuracy.

place notation

Place notation is a method used to express the value of digits in various bases by assigning each position a power of the base.

In base-7, each digit's value is determined by its position and can be represented as follows:

• \text {First digit (rightmost): } d_{0} \times 7^{0}
• \text {Second digit: } d_{1} \times 7^{1}
• \text {Third digit: } d_{2} \times 7^{2}
• \text {Fourth digit: } d_{3} \times 7^{3}
• \text {Fifth digit (leftmost): } d_{4} \times 7^{4}

For example, for the base-7 number \(12345_{7}\), we calculate the decimal value as:
\[ 1 \times 7^{4} + 2 \times 7^{3} + 3 \times 7^{2} + 4 \times 7^{1} + 5 \times 7^{0} \]
This equals to:
\[ (1 \times 2401) + (2 \times 343) + (3 \times 49) + (4 \times 7) + (5 \times 1) = 3267 \]
This verification step proves the accuracy of the conversion.