Problem 31 - Thinking Mathematically

Question

Multiply in the indicated base. $$ \begin{array}{r} 623_{\text {eight }} \\ \times \quad 4 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The result of the multiplication of 623 (in base 8) by 4 is 3104 (also in base 8).

Step 1: Convert numbers to decimal form

First, convert the two multiplicants from their octal (base 8) representation into decimal (base 10). So, the octal number 623 is converted into decimal as follows: 6*8^2 + 2*8^1 + 3*8^0 = 401_(10) and the multiplication number 4 remains the same as it's the same in both octal and decimal.

Step 2: Perform multiplication

Now that we have the two numbers in decimal form, perform the multiplication. In this case, 401 x 4 which equals 1604 in decimal form.

Step 3: Convert back to octal

After the multiplication, convert the result back to base 8. Here, 1604_(10) equals 3104_(eight). Therefore, the multiplication of 623_(eight) by 4 equals 3104_(eight).

Key Concepts

Base Conversion

Converting numbers from one base to another is a fundamental skill in math and computer science. In our exercise, we convert numbers from octal (base 8) to decimal (base 10). This process allows us to perform operations like multiplication more easily.
To convert an octal number to decimal, we use the place value method, where each digit in the octal number is multiplied by the power of 8 corresponding to its position. For example, in the number 623 in base 8, you calculate:

• 6 times 8 squared
• plus 2 times 8 to the first power
• plus 3 times 8 to the zero power

This gives you the decimal number 401. This basic idea of breaking down a number based on its base is crucial for moving between number systems efficiently.

Octal Numbers

Octal numbers use base 8, which means there are eight possible digits (0-7). This base system is often used in computing. It's a compact way to represent data compared to base 10, the decimal system.
In the octal system, each digit represents the power of 8. For example, the number 623 in octal means:

• 6 times 8 squared
• plus 2 times 8 to the first power
• plus 3 times 8 to the zero power

This results in the decimal number 401. Octal numbers are useful because they can simplify binary coding, as three binary digits (bits) correspond to one octal digit. Understanding octal numbers requires recognizing their unique structure and how each digit's place affects the overall value.

Decimal System

The decimal system is the most familiar number system to most people, as it is the foundation of our everyday counting and arithmetic. It is a base 10 system, meaning it uses ten digits (0-9). Each digit's value depends on its position or place value in the number.
In the decimal system, each digit is multiplied by a power of 10 based on its position. For example, the number 401 is calculated by:

• 4 times 100 (10 squared)
• 0 times 10
• plus 1 times 1 (10 to the zero power)

In base conversion exercises, knowing the decimal system is indispensable, as it is often the intermediary or target base when converting numbers from more specialized bases like binary or octal.

Number Systems

Different number systems are used in various fields, each with its unique base and representation method. These systems convert numbers to allow understanding in different contexts.
Common number systems include:

• Decimal (base 10): Used for daily counting.
• Binary (base 2): Utilized in computer programming and digital systems.
• Octal (base 8): Often employed in computing to simplify binary representation.
• Hexadecimal (base 16): Common in programming for its concise representation of binary-coded values.

Understanding these systems is crucial for fields like computer science, electronics, and mathematics. The ability to convert and perform operations using different bases enhances problem-solving skills and adaptability across technical domains.