Digital computers encode data and programs using which number system?

Introduction / Context:
Internally, digital computers represent and process data using electronic circuits that can be in one of two stable states, often described as on or off. These two states correspond naturally to the digits zero and one. As a result, computers use a specific number system that relies only on these two digits to encode numbers, characters and instructions. This question asks you to identify that number system.

Given Data / Assumptions:

• We are talking about digital computers, not human calculations.
• Digital circuits operate using two distinct voltage levels representing two states.
• The options include decimal, binary, hexadecimal, octal and Roman numerals.
• We seek the number system that aligns with two state representation.

Concept / Approach:
The binary number system uses only two digits, zero and one. It is the natural choice for digital electronics because each bit can easily correspond to an on or off state. Inside the computer, every piece of data, from numbers and text to images and programs, is ultimately encoded as patterns of bits. Decimal, hexadecimal and octal systems are sometimes used by programmers for convenience, but internal storage and processing are always based on binary. Roman numerals are a historical system with no direct role in digital encoding.

Step-by-Step Solution:
Step 1: Recall that digital computers use bits, which can be zero or one.Step 2: Recognize that a number system using two digits is called the binary system.Step 3: Compare this with the decimal system, which uses ten digits from zero to nine.Step 4: Understand that hexadecimal uses sixteen symbols and octal uses eight symbols, but these are derived representations.Step 5: Conclude that the encoding of data and programs is fundamentally in binary.

Verification / Alternative check:
You can verify this by looking at low level data formats or machine code listings, which show sequences of bits. Memory capacity is measured in bits and bytes, where one byte is eight bits. Logic gates and flip flops in hardware operate on binary values. Higher level representations in decimal, hexadecimal or octal exist only to make it easier for humans to read and interpret binary patterns. This confirms that the primary system used by digital computers is the binary number system.

Why Other Options Are Wrong:

• Option A, decimal, is the system most familiar to humans but not used directly in digital circuits for encoding bits.
• Option C, hexadecimal, is a shorthand representation used by programmers to group binary bits but is not the fundamental hardware encoding.
• Option D, octal, is another shorthand grouping of binary bits and is rarely used today compared to hexadecimal.
• Option E, Roman numeral system, has no role in modern digital computer encoding.

Common Pitfalls:
Some learners may choose decimal because they know computers can handle decimal numbers or because humans mostly use decimal. Others may be tempted by hexadecimal due to its frequent appearance in programming contexts such as color codes and memory addresses. However, it is crucial to distinguish between internal representation and human friendly notation. Internally, everything is binary.

Final Answer:
The correct answer is Binary number system.