Question

Given our discussion of positional numbering systems in Section 4.2.1, see whether you can determine the decimal value of the following numbers: a. 133 (base 4) b. 367 (base 8 , also called octal) c. 1BA (base 16, also called hexadecimal. B is the digit that represents 11 ; \(\mathrm{A}\) is the digit that represents \(10 .)\)

Short answer

(a) 31 (b) 247 (c) 442

Step-by-step solution

Understanding Base 4 Conversion

To convert from base 4 to decimal, each digit is multiplied by 4 raised to the power of its position, starting from zero on the right. The number 133 in base 4 means: \(1 \times 4^2 + 3 \times 4^1 + 3 \times 4^0\).

Calculate Decimal Value for 133 (Base 4)

Calculate each part separately: \(1 \times 4^2 = 16\), \(3 \times 4^1 = 12\), and \(3 \times 4^0 = 3\). Add them together to get the decimal value: \(16 + 12 + 3 = 31\).

Understanding Base 8 Conversion

To convert from base 8 to decimal, each digit is multiplied by 8 raised to the power of its position. The number 367 in base 8 means: \(3 \times 8^2 + 6 \times 8^1 + 7 \times 8^0\).

Calculate Decimal Value for 367 (Base 8)

Calculate each part: \(3 \times 8^2 = 192\), \(6 \times 8^1 = 48\), \(7 \times 8^0 = 7\). Add them to get the decimal value: \(192 + 48 + 7 = 247\).

Understanding Base 16 Conversion

To convert from base 16 to decimal, each digit is multiplied by 16 raised to the power of its position. The number 1BA in base 16 means: \(1 \times 16^2 + B \times 16^1 + A \times 16^0\) where B is 11 and A is 10.

Calculate Decimal Value for 1BA (Base 16)

Calculate each part: \(1 \times 16^2 = 256\), \(11 \times 16^1 = 176\), \(10 \times 16^0 = 10\). Add them to get the decimal value: \(256 + 176 + 10 = 442\).

Key concepts

Positional Notation

Positional notation, also known as place-value notation, is a way to represent numbers using digits. Each digit has a value based on its position, or place, in the number. The value of a digit depends on two factors: the digit itself and its position. In positional notation, a base or radix is chosen. For example, in the decimal system, the base is 10. In the octal system, it's 8, and in the hexadecimal system, it's 16. The positional value is the base raised to the power of the position number. For instance, in the decimal number 245, the digit 2 is in the hundreds place, so it represents 2 times 10 squared, or 200. Understanding positional notation is key to converting between different number bases.

Base Conversion

Base conversion is the process of changing a number from one base to another. Each number system uses a different base which affects how numbers are represented. Converting numbers between bases is essential for tasks that involve different systems. For base conversion, break down the number in the original base using positional notation, and then calculate its value by representing it in terms of the target base. The key is to understand the roles of each digit and the respective base's power. A good practice is to convert numbers to the decimal system first, as it is often easier to work with, and then to other bases. This method involves multiplying each digit by the base raised to the power of the digit's position.

Decimal System

The decimal system is the most widely used number system in the world. It is a base 10 system, meaning it uses ten digits, from 0 to 9. Each position in a decimal number represents a power of 10. For example, in the number 542, the 5 is in the hundreds place, the 4 is in the tens place, and the 2 is in the units place. As such, its value is calculated as \(5 \times 10^2 + 4 \times 10^1 + 2 \times 10^0 = 542\). The decimal system is foundational in arithmetic and everyday computations. Understanding its structure and conversion methods makes it easier to understand other numeral systems.

Octal System

The octal system is a base 8 number system. It uses digits from 0 to 7. Each position in an octal number carries a power of 8. To convert from octal to decimal, multiply each digit by 8 raised to the power of its positional index starting from 0. For instance, the octal number 367 converts to decimal as follows: - \(3 \times 8^2 = 192\) - \(6 \times 8^1 = 48\) - \(7 \times 8^0 = 7\) Adding these values gives the decimal equivalent of 247. This conversion demonstrates the positional importance of each digit in the octal system.

Hexadecimal System

The hexadecimal system, or base 16, utilizes sixteen digits: 0-9 and letters A-F. In this system, A represents 10, B represents 11, up to F which represents 15. Each position in a hexadecimal number holds a power of 16.When converting from hexadecimal to decimal, multiply each digit by 16 raised to the power of its position. For instance, the hexadecimal 1BA translates to decimal as follows:- \(1 \times 16^2 = 256\) - \(11 \times 16^1 = 176\) where B is represented as 11 - \(10 \times 16^0 = 10\) where A is represented as 10 Summing these yields a decimal value of 442. This conversion highlights how larger values can be represented compactly in the hexadecimal system.