Problem 3 - Thinking Mathematically

Question

Convert the numeral to a numeral in base ten. \(52_{\text {eight }}\)

Step-by-Step Solution

Answer

The decimal equivalent of the octal number \(52_8\) is 42.

Step 1: Identify each digit and its corresponding power of 8

The number given is \(52_8\). The digit 2 (i.e., the rightmost digit) corresponds to \(8^0\) and the digit 5 (i.e., the next digit to the left) corresponds to \(8^1\). Note that these powers start from 0 and increase by 1 as we move from right to left.

Step 2: Multiply and sum

Multiply each digit by its corresponding power of 8 and sum all the results. These calculations give us \(2*8^0 + 5*8^1\).

Step 3: Compute the sum

Now perform the calculations: \(2*1 + 5*8 = 2 + 40 = 42\)

Key Concepts

Numeral Systems

Numeral systems are the sets of symbols and rules we use to represent numbers. These systems can vary greatly, with different bases or 'radix' at their core. For instance, the decimal system, familiar to most of us, is base ten, meaning it has ten unique digits (0 to 9) and each digit's place value is a power of ten.

However, there are numerous other systems such as binary (base two), octal (base eight), and hexadecimal (base sixteen). These systems are crucial in various fields, particularly in computing. The octal system was a popular choice in computing due to its simplicity for converting to and from binary, though it has largely been supplanted by hexadecimal.

Understanding how to convert numbers between different numeral systems is important for computer science, electronics, and mathematics. It's a foundational skill that enhances one's ability to comprehend and work with different types of numerical data.

Powers of Eight

When working with the octal numeral system, it's essential to understand the 'powers of eight.' Each place value in an octal number represents a power of eight, starting with the rightmost digit associated with \(8^0\), which is 1. As we move one place to the left, the place value is \(8^1\), then \(8^2\), and so on, each multiplied by eight from the previous value. This is similar to how the decimal system uses powers of ten.

For example:

• The rightmost digit is multiplied by \(8^0 = 1\).
• The next digit to the left is multiplied by \(8^1 = 8\).
• The second digit from the right is multiplied by \(8^2 = 64\), and so on.

This pattern continues with each digit being increasingly significant due to the exponent’s increase. Mastery of this concept enables us to convert octal numbers to decimal numbers accurately.

Mathematical Computation

The process of performing mathematical computation involves various operations such as addition, subtraction, multiplication, and division. When converting numbers from one base to another, particularly from octal to decimal, we apply multiplication of each octal digit by the appropriate power of eight and then sum the results.

In our example, for the octal number \(52_8\), we calculated \(2 \times 8^0\) and \(5 \times 8^1\) and added them together to reach the decimal equivalent of 42. This process, while straightforward for small numbers, can become complex with larger numbers or different bases.

Using these steps ensures proper conversion and helps in understanding the significance of each digit’s position in contributing to the overall value of the number. A firm grasp of these calculations is fundamental in mathematics and applicable in various practical and theoretical contexts.