Binary and Hexadecimal Number Systems

Understanding Binary and Hexadecimal Number Systems

Binary (base‑2) and hexadecimal (base‑16) are the two most common numeral systems used in computer science and programming. Mastering these systems is essential for tasks such as low‑level debugging, memory address calculations, and designing efficient algorithms. This course breaks down the core concepts tested in the quiz, providing clear explanations, examples, and practical tips that are both educational and SEO‑friendly for learners searching for "binary number system" or "hexadecimal conversion".

1. Range of a 4‑bit Unsigned Binary Number

What does "4‑bit unsigned" mean?

An unsigned binary number uses all bits to represent magnitude only—no sign bit is reserved. With n bits, the smallest value is 0 and the largest value is 2ⁿ − 1. For n = 4:

• Minimum: 0 (binary 0000)
• Maximum: 2⁴ − 1 = 15 (binary 1111)

Therefore, a 4‑bit unsigned binary number can represent values from 0 up to 15 in decimal.

2. Converting Hexadecimal to Binary

Hexadecimal digit to 4‑bit binary mapping

Each hexadecimal digit maps directly to a 4‑bit binary group:

• 0 → 0000
• 1 → 0001
• 2 → 0010
• 3 → 0011
• 4 → 0100
• 5 → 0101
• 6 → 0110
• 7 → 0111
• 8 → 1000
• 9 → 1001
• A → 1010
• B → 1011
• C → 1100
• D → 1101
• E → 1110
• F → 1111

Applying this to the hexadecimal value BD07C:

• B → 1011
• D → 1101
• 0 → 0000
• 7 → 0111
• C → 1100

The correct binary grouping is 1011 1101 0000 0111 1100. This matches the quiz option that shows the last group as 1100, not 1110 or 0011.

3. Two's‑Complement Representation

Positive numbers

In a two's‑complement system, the binary pattern for a positive integer is identical to its unsigned representation. For a 4‑bit word, +5 is 0101 (since 5 = 4 + 1).

Negative numbers

To obtain the two's‑complement of a negative value, invert all bits (one's complement) and add 1. Starting with +5 (0101):

1. Invert → 1010
2. Add 1 → 1011

Thus, the binary pattern that represents –5 in a 4‑bit two's‑complement system is 1011.

4. Binary Addition and Carry

Adding three 1‑bits

Binary addition follows the same principle as decimal addition, but the base is 2. Adding 1 + 1 + 1 yields:

• 1 + 1 = 10 (binary 2) → write 0, carry 1
• Carry 1 + the third 1 = 10 again → write another 0, carry 1

The final result is 11 (binary 3) with a carry out of the most‑significant column. The quiz answer 11 is therefore correct.

5. Detecting Overflow in Two's‑Complement Addition

What is overflow?

Overflow occurs when adding two numbers with the same sign produces a result with a different sign. In two's‑complement arithmetic, this indicates that the true mathematical sum exceeds the representable range.

Example: Adding two positive 4‑bit numbers 0110 (+6) and 0101 (+5) yields 1011, which is interpreted as –5. The sign flipped from positive to negative, signalling an overflow condition.

Why “carry out” is not the same

In unsigned addition, a final carry out indicates the sum exceeds the maximum unsigned value. In two's‑complement, overflow is detected by examining the carry into and out of the sign bit: if they differ, overflow has occurred.

6. Hexadecimal Notation for a Binary Group

Grouping the bits

Take the binary sequence 0011 0101 0111 0011. Split it into 4‑bit nibbles from left to right:

• 0011 → 3
• 0101 → 5
• 0111 → 7
• 0011 → 3

Combining the hexadecimal digits gives 0x3573. This matches the correct quiz choice.

7. Sign Extension – Preserving Value When Expanding Bit‑Width

Why sign extension matters

When a signed integer is stored in a larger register, the original value must remain unchanged. The standard technique is to replicate the most‑significant (sign) bit into the newly added high‑order bits. This is called sign extension.

Correct method

For a 4‑bit signed number, extending to 8 bits involves copying the sign bit four times. For example, 1011 (‑5) becomes 1111 1011. The other options—adding 0xFF, shifting left, or inserting zeros—alter the numeric value.

Key takeaway: Replicate the sign bit into the new high‑order bits to preserve the original signed value.

8. Converting Decimal to Binary Using the Weight‑Sum Method

Weight‑sum (or repeated subtraction) steps

The weight‑sum method repeatedly subtracts the largest power of two that does not exceed the remaining decimal value, recording a 1 for that position and 0 for all lower powers that are not used.

Convert 156 to binary:

1. 2⁷ = 128 ≤ 156 → write 1, remainder 156‑128 = 28
2. 2⁶ = 64 > 28 → write 0
3. 2⁵ = 32 > 28 → write 0
4. 2⁴ = 16 ≤ 28 → write 1, remainder 28‑16 = 12
5. 2³ = 8 ≤ 12 → write 1, remainder 12‑8 = 4
6. 2² = 4 ≤ 4 → write 1, remainder 0
7. 2¹ = 2 > 0 → write 0
8. 2⁰ = 1 > 0 → write 0

The resulting 8‑bit binary string is 10011100, which matches the quiz answer.

9. Summary of Key Concepts

• Unsigned 4‑bit range: 0 – 15.
• Hex‑to‑binary conversion: Replace each hex digit with its 4‑bit binary equivalent.
• Two's‑complement negative: Invert bits and add 1.
• Binary addition of three 1s: Result is 11 (binary 3) with a carry.
• Overflow detection: Same‑sign operands produce a different‑sign result.
• Hex representation of a binary group: Group into nibbles and translate.
• Sign extension: Replicate the sign bit when increasing bit‑width.
• Weight‑sum decimal‑to‑binary: Subtract powers of two, recording 1s and 0s.

Understanding these fundamentals equips programmers to work confidently with low‑level data representations, debug bit‑wise errors, and design algorithms that manipulate binary and hexadecimal values efficiently.

10. Practice Quiz Review

Below is a concise recap of each quiz question with the correct answer highlighted. Use this as a self‑check after studying the sections above.

• A 4‑bit unsigned binary number can represent values up to 15.
• Hex BD07C converts to binary 1011 1101 0000 0111 1100.
• Two's‑complement of –5 (4‑bit) is 1011.
• Adding 1 + 1 + 1 yields binary 11.
• When the sign flips after adding two same‑sign numbers, an overflow has occurred.
• Binary 0011 0101 0111 0011 is hexadecimal 0x3573.
• Correct 4‑bit to 8‑bit signed extension: replicate the sign bit.
• Decimal 156 converts to binary 10011100 using the weight‑sum method.

Review each concept, practice additional conversions, and you’ll master the binary and hexadecimal number systems essential for any computer‑science curriculum.