Problem 44 - Thinking Mathematically

Question

Convert each base ten numeral to a numeral in the given base. 129 to base three

Step-by-Step Solution

Answer

The base three representation of 129 is 11210.

Step 1: Initialization

Begin with the base ten number 129.

Step 2: Division by Target Base

Divide 129 by 3. The result is 43 with a remainder of 0. The remainder becomes the least significant digit (rightmost position) in the base three representation.

Step 3: Repeated Division by Target Base

Divide 43 by 3. The result is 14 with a remainder of 1. This remainder becomes the next digit in the base three representation.

Step 4: Continue Division by Target Base

Divide 14 by 3. The result is 4 with a remainder of 2. This remainder becomes the next digit.

Step 5: Continue Division by Target Base until Quotient is Zero

Continue division by 3 again. The result is 1 with a remainder of 1. These remainders add on the base three representation.

Step 6: Final Division

Since the quotient is 1, do one final division by 3. The result is 0 with a remainder of 1. This remainder becomes the most significant digit (leftmost position) in the base three representation.

Step 7: Assemble the Base Three Representation

Read the remainders in reverse order of when they were obtained. The first remainder from Step 2 is the rightmost digit, etc. The final remainder from Step 6 is the leftmost digit, thus the base three representation of 129 is 11210.

Key Concepts

Understanding Base Ten Numeral System

Base ten numerals make up our everyday number system. It is also known as the decimal system. In this system, each position represents a power of ten. For instance, the number 129 can be broken down into 1 hundred, 2 tens, and 9 units.

• 129 = \(1 \times 10^2 + 2 \times 10^1 + 9 \times 10^0\)

This means the value in each position is determined by raising ten to increasing powers starting from zero at the rightmost position. Moreover, only digits from 0 to 9 are used. This makes it intuitive for counting, arithmetic operations, and more.

Exploring Base Three Numerals

The base three numeral system, often referred to as the ternary system, uses only three digits: 0, 1, and 2. Each position in a base three number represents a power of three.

• For example, the number 11210 in base three can be broken down as:


• \(1 \times 3^4 + 1 \times 3^3 + 2 \times 3^2 + 1 \times 3^1 + 0 \times 3^0\)

The value adjusts as the positions increase from right to left. This system may sound more complex than the decimal, but it follows a similar position-driven pattern, which makes conversions systematic and logical.

Conversion Using the Division Method

The division method is a straightforward technique used for converting numbers from base ten to another base. Here's how it works:

• Continuously divide the base ten number by the new base (in this case, 3).
• Record the remainder of each division step.
• The process stops when you reach a quotient of zero.

The trick is to list the remainders in reverse order from the last calculation to the first. This reordering transforms them into digits of the new base numeral. For the example of 129, you'd see:

• 129 divided by 3 gives a quotient of 43 with a remainder of 0.
• 43 divided by 3 gives 14 with a remainder of 1.
• 14 divided by 3 gives 4 with a remainder of 2.
• 4 divided by 3 gives 1 with a remainder of 1.
• Finally, 1 divided by 3 results in 0 quotient with a remainder of 1.

Reading the remainders backwards forms the number 11210 in base three.

Mathematics Problem Solving Made Simple

Problem solving in mathematics is often about breaking down a challenge into clear, logical steps. Understanding the method and reasoning behind each step is key to mastering new concepts.

• Begin by identifying the form of the given number and the desired outcome.
• Use systematic methods like division to transform the number.
• Organize information to make sense of the calculations and results.

In base conversions, it's essential to stay methodical to prevent mistakes. Each division and remainder should be clear to make the final reassembling accurate. Remember, practice with these processes builds intuition, making math problem solving easier and even enjoyable.