Bit to Pebibyte Converter

What is a Bit?

A bit (short for binary digit) is the basic unit of information in information theory, computing, and digital communications. It represents a logical state with one of two possible values.

Mathematical Definition: A bit is the amount of information required to distinguish between two equally probable alternatives. In information theory (Shannon entropy), the entropy $H$ of a random variable $X$ with two equally likely outcomes is 1 bit:

$$H(X) = - \sum p(x) \log_2 p(x) = - (0.5 \log_2 0.5 + 0.5 \log_2 0.5) = 1 \text{ bit}$$

If an event has a probability $p$, the information content $I$ (in bits) of observing that event is: $$I(p) = -\log_2(p)$$

• Coin Flip: Probability 0.5. Information = $-\log_2(0.5) = 1$ bit.
• Rolling a 4 on a die: Probability 1/6. Information = $-\log_2(1/6) \approx 2.58$ bits.
• Guessing a number 1-100: Probability 0.01. Information = $-\log_2(0.01) \approx 6.64$ bits.

Physical Representation: How Computers "Store" Bits

In the abstract world of math, a bit is just a number. But in the physical world of your computer, a bit must be a tangible physical state. Engineers have developed many ways to store this "0" or "1":

1. Voltage (CPUs and RAM)

• Mechanism: Transistors act as switches that either block or allow current.
• State 1 (High): Voltage is near the supply level (e.g., 3.3V or 5V).
• State 0 (Low): Voltage is near ground level (0V).
• Speed: Extremely fast (switching billions of times per second).
• Volatility: Requires constant power. If you unplug the computer, the electrons stop flowing, and the bits vanish (Volatile Memory).

2. Electric Charge (Flash Memory / SSDs)

• Mechanism: Floating-gate transistors trap electrons in an insulated "cage."
• State 0: Electrons are trapped in the floating gate (changing the threshold voltage).
• State 1: No electrons in the floating gate.
• Speed: Fast, but slower than RAM.
• Volatility: Non-volatile. The electrons stay trapped for years even without power, which is why your USB drive remembers your files.

3. Magnetism (Hard Disk Drives - HDDs)

• Mechanism: Tiny regions (domains) on a spinning platter are magnetized.
• State 1: Magnetic north pole points in one direction.
• State 0: Magnetic north pole points in the opposite direction.
• Read/Write: A head flies over the surface detecting or flipping the magnetic field.
• Volatility: Non-volatile. Magnets stay magnetized.

4. Light / Optics (CDs, DVDs, Blu-ray)

• Mechanism: Physical pits and lands (flat areas) are stamped into a plastic disc.
• State: A laser beam scans the track.
  • Land: Reflects the laser back to the sensor.
  • Pit: Scatters the light (no reflection).
• Volatility: Non-volatile and Read-Only (for pressed discs).

5. Quantum States (Quantum Computing)

• Mechanism: Spin of an electron or polarization of a photon.
• State: Can be Up (1), Down (0), or a superposition of both.

Bit vs. Byte: The Crucial Difference

The most common source of confusion in digital metrics is the difference between the bit and the byte.

• The Bit (b) is the atom of data. It is small, fast, and granular.
  • Used for: Transmission speeds (Internet, USB, Wi-Fi).
  • Why: Serial transmission sends data one bit at a time down a wire.
• The Byte (B) is a molecule of data (8 bits). It is the smallest addressable unit of memory.
  • Used for: Storage capacity (RAM, SSDs, File sizes).
  • Why: Computers process data in chunks (bytes/words), not individual bits.

The Rule of 8: To convert Bytes to bits, multiply by 8. To convert bits to Bytes, divide by 8.

• 100 Mbps Internet (Megabits) = 12.5 MB/s download speed (Megabytes).

Ancient Origins: The Binary Concept

Long before computers, the concept of binary (two-state) systems existed:

• I Ching (9th Century BC): Ancient Chinese divination text used broken and unbroken lines (yin and yang) to form hexagrams, essentially 6-bit binary codes. The sequence of hexagrams (0 to 63) perfectly matches the modern binary count from 000000 to 111111.
• Pingala (2nd Century BC): Indian scholar who used binary numbers (short and long syllables) to classify poetic meters.
• Morse Code (1830s): Used dots and dashes to encode text. While not strictly binary (it relies on timing/pauses), it demonstrated that complex messages could be built from two simple signals.
• Braille (1824): A 6-bit binary code used for touch reading. Each character is a 2x3 grid where dots are either raised (1) or flat (0).

17th-19th Century: Mathematical Foundation

• Gottfried Wilhelm Leibniz (1679): The German polymath formalized the modern binary number system. He saw a spiritual significance in it: 1 represented God and 0 represented the void. He showed that any number could be represented using only 0s and 1s. He was amazed to discover that his binary system matched the I Ching hexagrams.
• George Boole (1847): The English mathematician published "The Mathematical Analysis of Logic," creating Boolean Algebra. This system of logic (True/False, AND, OR, NOT) became the operating manual for modern computer processors a century later. Boole proved that logic could be reduced to simple algebra.

20th Century: The Birth of the Bit

• 1937: Claude Shannon, a master's student at MIT, wrote "A Symbolic Analysis of Relay and Switching Circuits." He proved that electrical switches (relays) could implement Boolean algebra to perform any logical or numerical operation. This is arguably the most important master's thesis of the 20th century—it bridged the gap between abstract logic and physical machines.
• 1947: John W. Tukey, a statistician at Bell Labs, was working with early computers. Tired of writing "binary digit," he shortened it to "bit." (He also coined the term "software"!).
• 1948: Claude Shannon published "A Mathematical Theory of Communication." This paper founded Information Theory. He adopted Tukey's term "bit" as the fundamental unit of measure for information entropy. Shannon defined the bit not just as a digit, but as a measure of uncertainty resolution.

The 8-Bit Standard

In the early days of computing, machines used various "word" sizes (groups of bits) ranging from 4 to 60 bits.

• 4-bit (Nibble): Intel 4004 (first microprocessor).
• 6-bit: Common for early character sets (64 characters is enough for uppercase + numbers).
• 36-bit: Common in scientific mainframes (DEC PDP-10).
• 60-bit: CDC 6600 Supercomputer.

The 8-bit byte became the industry standard with the IBM System/360 in 1964. IBM chose 8 bits because it allowed for 256 characters (EBCDIC), enough to store uppercase, lowercase, numbers, and symbols. The success of the System/360 forced the rest of the industry to standardize on 8-bit bytes, cementing the relationship that 1 Byte = 8 bits.

1. Internet Speed (Bandwidth)

Internet Service Providers (ISPs) universally sell speed in bits per second.

• Mbps (Megabits per second): The standard unit for home internet.
  • Basic: 25 Mbps
  • Fast: 100-500 Mbps
• Gbps (Gigabits per second): "Gigabit internet" or Fiber.
  • Ultra-fast: 1 Gbps (1,000 Mbps)

Why not Bytes? Historically, data transmission happens serially (one bit after another). Measuring the raw stream count (bits) is technically more accurate for the engineer managing the wire. For the consumer, it also produces larger, more impressive marketing numbers (100 Mbps sounds faster than 12.5 MB/s).

2. Audio Quality (Bit Depth & Bitrate)

• Bit Depth: Determines the dynamic range (loudness resolution) of audio.
  • 16-bit audio (CD quality): 65,536 volume levels ($2^{16}$).
  • 24-bit audio (Studio quality): 16.7 million volume levels ($2^{24}$).
• Bitrate: The amount of data consumed per second of audio.
  • 128 kbps: Standard streaming quality.
  • 320 kbps: High-quality MP3.
  • 1,411 kbps: Uncompressed CD audio (WAV).

3. Color Depth (Images)

The number of bits used to represent the color of a single pixel.

• 1-bit: Black and White.
• 8-bit: 256 colors (old GIF / VGA graphics).
• 24-bit: 16.7 million colors (Standard "True Color" JPG/PNG).
• 30-bit / 10-bit color: 1 billion colors (HDR video, professional photography).

4. Cryptography

Security strength is measured in bits (key length).

• 128-bit encryption: Considered strong for most commercial uses.
• 256-bit encryption: Military-grade standard (AES-256).
• 2048-bit RSA: Asymmetric encryption keys need to be much longer to offer equivalent security to symmetric keys.