Convert the following hexadecimal numbers into their decimal...

Convert the following hexadecimal numbers into their decimal equivalents: (a) $7 \mathrm{~A}_{16}$ (b) $3 \mathrm{~F}_{16}$ (a) $7 \mathrm{~A}_{16}=7 \times 16^{1}+\mathrm{A} \times 16^{0}=7 \times 16+10 \times 1$ $=112+10=122$ Thus $\mathbf{7 A}_{16}=\mathbf{1 2 2}_{10}$ (b) $3 \mathrm{~F}_{16}=3 \times 16^{1}+\mathrm{F} \times 16^{0}=3 \times 16+15 \times 1$ $=48+15=63$ Thus $3 \mathrm{~F}_{16}=63_{10}$

Convert the following hexadecimal numbers into their decimal equivalents:
(a) $7 \mathrm{~A}_{16}$
(b) $3 \mathrm{~F}_{16}$
(a) $7 \mathrm{~A}_{16}=7 \times 16^{1}+\mathrm{A} \times 16^{0}=7 \times 16+10 \times 1$ $=112+10=122$
Thus $\mathbf{7 A}_{16}=\mathbf{1 2 2}_{10}$
(b) $3 \mathrm{~F}_{16}=3 \times 16^{1}+\mathrm{F} \times 16^{0}=3 \times 16+15 \times 1$ $=48+15=63$
Thus $3 \mathrm{~F}_{16}=63_{10}$

Key Concepts

Base Conversion

Base conversion involves translating numbers from one numeral system to another, using the principle of positional notation. To convert a number from a non-decimal base to decimal, each digit is multiplied by the power of the base corresponding to its position, and the resulting products are summed. This technique is essential when working with different numeral systems, especially in computing where binary, hexadecimal, and decimal systems often interact.

Positional Notation

Positional notation is a method of representing or encoding numbers where the position of each digit in a number determines its contribution to the overall value. Each digit is multiplied by a power of the base that corresponds to its position, and the sum of these values gives the decimal equivalent. This principle applies universally to numeral systems, whether base-16 (hexadecimal) or base-10 (decimal).

Hexadecimal Number System

The hexadecimal numeral system is a base-16 number system that uses sixteen distinct symbols: 0–9 to represent values zero to nine and A–F (or a–f) to represent values ten to fifteen. This system is widely used in computer science and digital electronics because it neatly represents binary-coded values, with each hexadecimal digit corresponding exactly to four binary digits.

Decimal Number System

The decimal numeral system is a base-10 number system that utilizes the ten digits from 0 to 9. It is the most commonly used number system for everyday arithmetic and calculations, serving as the standard way of representing numbers in many contexts outside of computing.

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