Binary Numbers 1 to 15: The 4-Bit Digital Logic Reference Chart

Binary numbers 1 to 15 represent the complete non-zero counting sequence of a 4-bit digital system, where each digit (bit) is either a 0 (low voltage/off) or a 1 (high voltage/on), yielding 15 active states from 0001 to 1111. In a real circuit, mastering this specific 4-bit sequence dictates how you configure microcontroller GPIO port registers, set hardware addresses on I2C expanders, or wire physical DIP switches for device selection. Beginners most commonly confuse the raw binary string (e.g., 1010) with its decimal equivalent (10) or its hexadecimal shorthand (0x0A), a mix-up that routinely leads to masked pins, off-by-one addressing errors, and bricked communication buses.

The 4-Bit Binary Numbers 1 to 15 Reference Table

When working at the bench, you rarely have time to do base-2 math in your head. The table below maps the 4-bit binary sequence to its decimal and hexadecimal equivalents, alongside the ideal 5V TTL logic voltages you should measure with a multimeter when probing physical pins. Note that 0000 is included for completeness, as it represents the "all pins low" baseline state.

Worked Numeric Example: Decoding with the SN74HC154

To see how binary numbers 1 to 15 physically manifest in a circuit, let us look at the Texas Instruments SN74HC154, a standard 4-line to 16-line decoder. This chip takes a 4-bit binary input and pulls exactly one of its 16 output pins LOW (0V), while the rest remain HIGH (5V). This is heavily used in memory addressing and multiplexed LED displays.

Suppose you need to activate output pin Y11. You must apply the decimal value 11 to the input pins. Following the reference table, decimal 11 translates to the binary sequence 1011.

Here is the exact physical wiring and voltage state required to achieve this:

• Input D (MSB): Wired to 5V (Logic 1)
• Input C: Wired to GND (Logic 0)
• Input B: Wired to 5V (Logic 1)
• Input A (LSB): Wired to 5V (Logic 1)

When you power the circuit, the internal logic gates evaluate the 1011 state. Output pin Y11 will immediately drop to roughly 0.1V (sinking current), while Y0 through Y10 and Y12 through Y15 will sit at 5V. If you accidentally wire Input C to 5V instead of GND, you are feeding the chip 1111 (decimal 15), and Y15 will activate instead. This single-bit wiring error is the most common cause of "ghost" addresses in DIY digital logic builds.

Where You Meet This in Practice

You will rarely write out "1011" in modern embedded software, but the 4-bit boundary (0 to 15) dictates the architecture of almost every microcontroller and digital peripheral you interact with.

Direct Port Manipulation in Microcontrollers

When you need to toggle multiple pins simultaneously without the overhead of digitalWrite(), you write directly to the microcontroller's hardware registers. On an ATmega328P (Arduino Uno), PORTD controls pins 0 through 7. If you only want to control the lower 4 bits (pins 0-3) without disturbing the serial TX/RX pins (pins 0 and 1, which are actually part of PORTD but often reserved), you use a bitwise AND mask. To set the lower 4 pins to binary 0111 (decimal 7) while preserving the upper 4 bits, you write:

PORTD = (PORTD & 0xF0) | 0x07;

Here, 0xF0 masks the upper nibble, and 0x07 injects the binary 0111 into the lower nibble. Understanding the binary numbers 1 to 15 is mandatory for writing these hex masks correctly. For a deeper dive into AVR register mapping, refer to the official Arduino Port Manipulation documentation.

Binary Coded Decimal (BCD) Displays

Chips like the CD4511BE BCD-to-7-segment latch decoder accept 4 binary inputs to drive a 7-segment LED display. However, standard BCD only defines the numerals 0 through 9. If you feed the binary numbers 10 to 15 (1010 through 1111) into a CD4511BE, the chip recognizes these as invalid BCD states and intentionally blanks the display (all segments turn off). If your DIY clock project occasionally shows blank digits, check your counting logic to ensure you are resetting the counter at 9 (1001) rather than letting it roll into the 10-15 range.

I2C and SPI Hardware Addressing

Many I2C peripherals, such as the MCP23017 GPIO expander, use physical address pins (A0, A1, A2) to define their bus address. While 3 pins yield 8 addresses (0-7), chips with 4 address pins utilize the full 1 to 15 binary range to allow up to 16 identical devices on a single bus. Setting the physical DIP switches to 1100 (decimal 12) shifts the base I2C address by 12 hex steps, a calculation you must perform manually when initializing the Wire library in your firmware.

Frequently Asked Questions

Why do binary numbers 1 to 15 require exactly four bits?

In base-2 mathematics, each additional bit doubles the number of available states. One bit gives 2 states (0-1), two bits give 4 states (0-3), and three bits give 8 states (0-7). To represent the decimal number 15, you need 16 total states (including zero). Since 2 to the power of 4 equals 16, exactly four bits are required. In digital logic, this 4-bit grouping is formally called a "nibble" (half of an 8-bit byte), which is why hexadecimal (base-16) maps perfectly to 4-bit binary.

How do I convert binary numbers 1 to 15 to hexadecimal for embedded C code?

Hexadecimal uses the numbers 0-9 for the first ten states, and the letters A-F for the remaining six. Binary 1010 is 0xA, 1011 is 0xB, 1100 is 0xC, 1101 is 0xD, 1110 is 0xE, and 1111 is 0xF. When writing C or C++ code for an ESP32 or Arduino, you prefix the hex value with 0x. Therefore, if you want to set a 4-bit PWM threshold to decimal 14, you can write it as 0x0E or 0b1110 (using the binary prefix 0b supported by modern GCC compilers). Hex is preferred in professional codebases because it aligns cleanly with 8-bit, 16-bit, and 32-bit memory boundaries.

What happens if I send a value greater than 15 to a 4-bit digital input?

If you attempt to write a 5-bit value like decimal 16 (10000) to a 4-bit hardware register or a 4-bit physical decoder, the most significant bit (the leftmost '1') is truncated or overflows. The hardware only reads the lower 4 bits (0000), effectively wrapping the value back to zero. In software, this is known as a bitmask overflow. If you are reading a 4-bit DIP switch into a microcontroller variable, always apply a bitwise AND mask (value & 0x0F) to strip away any stray high-bit noise that might have been introduced by floating GPIO pins or electromagnetic interference on the breadboard.