When you are writing firmware for an ESP32, configuring I2C addresses, or wiring up a 74HC595 shift register, you do not have time to do base-2 math in your head. A binary chart for numbers is the foundational lookup tool that maps base-10 (decimal) to base-2 (binary) and base-16 (hexadecimal). It bridges the gap between human-readable values and the raw bitwise registers that microcontrollers use to control hardware.
Below is the definitive reference table for 8-bit and 16-bit boundaries, complete with C/C++ bitwise masks and signed/unsigned limits based on the ISO/IEC 9899 C Standard (stdint.h). Following the table, we cover the hardware realities of binary logic that pure math charts leave out.
How to Read This Binary Chart for Numbers
Before jumping to the rows you need, understand how the columns map to your specific installation or codebase. The chart is divided into mathematical representation and hardware data types.
- Decimal & Hex: The human-readable inputs. Hexadecimal (base-16) is used because one hex digit perfectly maps to a 4-bit nibble, making it easier to read than long strings of 1s and 0s.
- 8-Bit Binary: The raw physical state of an 8-bit register or port. Read from left (Most Significant Bit, MSB) to right (Least Significant Bit, LSB).
- Bitwise Mask (C/C++): The exact syntax to use in Arduino or ESP-IDF code to isolate or toggle that specific bit using the AND (
&) or OR (|) operators. - Which Column Applies to Your Installation? If you are mapping physical pin states, PWM duty cycles, or I2C addresses, use the Unsigned (
uint8_t) column. If you are calculating sensor deltas, motor encoder positions, or temperature offsets, use the Signed (int8_t) column, which relies on Two's Complement to represent negative values.
The Complete Binary, Decimal, and Hexadecimal Reference Table
This table covers the essential nibbles (0-15), critical power-of-two boundaries, and 16-bit overflow limits. Use the id anchors to bookmark your most queried rows.
| Decimal | 8-Bit Binary | Hex | Bitwise Mask | Unsigned uint8_t |
Signed int8_t |
|---|---|---|---|---|---|
| 0 | 0000 0000 | 0x00 | ~(1<<0) | 0 | 0 |
| 1 | 0000 0001 | 0x01 | (1<<0) | 1 | 1 |
| 2 | 0000 0010 | 0x02 | (1<<1) | 2 | 2 |
| 4 | 0000 0100 | 0x04 | (1<<2) | 4 | 4 |
| 7 | 0000 0111 | 0x07 | 0x07 | 7 | 7 |
| 8 | 0000 1000 | 0x08 | (1<<3) | 8 | 8 |
| 15 | 0000 1111 | 0x0F | 0x0F | 15 | 15 |
| 16 | 0001 0000 | 0x10 | (1<<4) | 16 | 16 |
| 32 | 0010 0000 | 0x20 | (1<<5) | 32 | 32 |
| 64 | 0100 0000 | 0x40 | (1<<6) | 64 | 64 |
| 127 | 0111 1111 | 0x7F | 0x7F | 127 | 127 (Max Signed) |
| 128 | 1000 0000 | 0x80 | (1<<7) | 128 | -128 (Min Signed) |
| 255 | 1111 1111 | 0xFF | 0xFF | 255 (Max Unsigned) | -1 |
16-Bit Boundaries (for uint16_t / int16_t registers):
32,767 (0x7FFF) is the maximum positive signed 16-bit integer.
65,535 (0xFFFF) is the maximum unsigned 16-bit integer, representing all 16 bits HIGH.
Hardware Realities: Logic Thresholds and Derating Ideal Binary
A binary chart for numbers assumes an ideal mathematical universe where a '1' is absolute. On the workbench, a binary '1' is just a voltage level, and it is subject to noise, trace resistance, and logic family mismatches. In power systems, derating reduces wire ampacity based on ambient heat. In digital logic, derating an ideal binary value refers to the voltage degradation and noise margins that modify the base 1 or 0 before the receiving gate interprets it.
How Derating Modifies the Base Value
Consider a common bench scenario: driving a 5V CMOS shift register (like the 74HC595) directly from a 3.3V microcontroller (like the ESP32).
- The Ideal Base Value: Your code outputs a binary '1'. The ESP32 pin drives to 3.3V.
- The Derating Factor: Standard CMOS logic defines a HIGH input voltage (VIH) as 0.7 × VCC. For a 5V chip, the VIH threshold is 3.5V.
- The Result: Your 3.3V binary '1' is derated below the 3.5V threshold. The 74HC595 reads it as a '0' or floats into an undefined state, causing erratic shifting.
To prevent your binary '1' from being derated into a '0', you have two choices. First, use a dedicated logic level shifter (like the TXB0108). Second, swap the 74HC595 for a 74HCT595. The 'T' stands for TTL-compatible, meaning the VIH threshold is hardcoded to 2.0V, allowing a 3.3V MCU to reliably drive a 5V chip without derating the binary state.
Always check the ESP32 Datasheet or your specific logic IC datasheet for the exact VIL (Voltage Input Low) and VIH (Voltage Input High) parameters. A binary chart tells you what to write; the datasheet tells you if the hardware will actually hear it.
What This Table Cannot Tell You
While this binary chart for numbers is exhaustive for mathematical conversion and standard data typing, it lacks the physical and temporal context required for advanced embedded design. Specifically, the table cannot tell you:
- Endianness (Byte Order): When you send a 16-bit value like
0x1234over SPI or I2C, does the0x12(MSB) or0x34(LSB) transmit first? The ESP32 is little-endian by default, but many external sensors expect big-endian. You must check the sensor's communication protocol. - Setup and Hold Times: When toggling a clock pin to shift in a binary byte, the data pin must be stable for a specific number of nanoseconds before and after the clock edge. The binary chart does not account for propagation delays.
- Pin Drive Strength: A binary '1' on an ESP32 GPIO pin can typically source only 40mA (and practically, you should limit it to 12mA-20mA to avoid brownouts). The chart shows the logical state, not the current capacity.
Binary Chart for Numbers FAQ
How do I use a binary chart for numbers to set microcontroller registers?
To set a specific bit in a hardware register without altering the other bits, use the Bitwise Mask column combined with the OR operator (|). For example, to set bit 3 (Decimal 8, Hex 0x08) in a port register, write: REG = REG | (1 << 3);. To clear that bit back to a binary '0', use the AND operator with the inverted mask: REG = REG & ~(1 << 3);. This prevents you from accidentally overwriting adjacent configuration bits.
Why does my binary chart for numbers show negative values for 8-bit integers?
This is due to Two's Complement, the standard method for representing signed integers in C/C++ (int8_t). In an 8-bit signed system, the Most Significant Bit (MSB) acts as the sign bit. If the MSB is '0' (values 0 to 127), the number is positive. If the MSB is '1' (values 128 to 255 in unsigned terms), the hardware interprets it as a negative number ranging from -128 to -1. If you are reading raw sensor data that can drop below zero, you must cast the variable as signed; otherwise, a temperature of -1°C will be misread as 255°C.
What is the fastest way to convert large decimal numbers using a binary chart?
Do not try to convert large decimal numbers directly to binary in your head. Convert the decimal number to Hexadecimal first, then map each hex digit to its 4-bit binary nibble using the top 16 rows of the chart. For example, to convert Decimal 255: recognize it as Hex 0xFF. Look up F (15) in the chart, which is 1111. Therefore, 0xFF is 1111 1111. This two-step method eliminates the mental math errors associated with subtracting powers of 2.
