Binary, Hex & Number Systems

Every digital circuit, from a single logic gate to the processor on your Arduino, works with just two voltage levels: HIGH (near the supply voltage) and LOW (near 0V). We write these as 1 and 0. A single 1-or-0 value is called a bit.

Two levels are used because they are easy to tell apart reliably, even with electrical noise. A circuit only has to decide whether a voltage is 'high enough' or 'low enough' rather than measure it precisely.

• Bit: a single binary digit, 0 or 1
• Nibble: 4 bits (one hexadecimal digit)
• Byte: 8 bits (the size of a byte variable on Arduino)

Decimal is base 10: each position is worth 10 times the one to its right (1, 10, 100…). Binary is base 2: each position is worth 2 times the one to its right (1, 2, 4, 8, 16…). To convert binary to decimal, add up the place values wherever there is a 1.

Place value:   8   4   2   1
Binary 1011:   1   0   1   1
               8 + 0 + 2 + 1 = 11 (decimal)

To go the other way, repeatedly divide by 2 and write down the remainders. Reading the remainders from bottom to top gives the binary number.

13 / 2 = 6  remainder 1   <- least significant bit
 6 / 2 = 3  remainder 0
 3 / 2 = 1  remainder 1
 1 / 2 = 0  remainder 1   <- most significant bit

13 (decimal) = 1101 (binary)

Long binary numbers are hard to read. Hexadecimal (base 16) fixes this: each hex digit stands for exactly 4 bits, so you can convert by splitting the binary number into groups of four. Hex uses the digits 0–9 and then A–F for the values 10–15.

Binary   Hex   Decimal
0000     0     0
0101     5     5
1001     9     9
1010     A     10
1100     C     12
1111     F     15

11010110 -> 1101 0110 -> D 6 -> 0xD6 (= 214 decimal)

In Arduino code you can write numbers in any base. 0b prefixes a binary literal and 0x prefixes a hex literal, so 0b11111111, 0xFF, and 255 are all the same value.

• Connect pins 2, 3, 4, and 5 each through a 220Ω resistor to an LED's long leg
• Connect all LED short legs to GND
• Line the LEDs up with pin 5 on the left, so they read like a binary number (bit 3 … bit 0)

const int ledPins[4] = {2, 3, 4, 5};  // bit 0 .. bit 3

void setup() {
  for (int i = 0; i < 4; i++) pinMode(ledPins[i], OUTPUT);
  Serial.begin(9600);
}

void loop() {
  for (int n = 0; n < 16; n++) {
    for (int bit = 0; bit < 4; bit++) {
      digitalWrite(ledPins[bit], bitRead(n, bit));  // 1 = LED on
    }
    Serial.print(n);
    Serial.print(" = 0b");
    Serial.print(n, BIN);
    Serial.print(" = 0x");
    Serial.println(n, HEX);
    delay(1000);
  }
}

bitRead(n, bit) returns the value (0 or 1) of one bit of n, where bit 0 is the rightmost. Open the Serial Monitor to see each number printed in all three bases while the LEDs count.

• Extend the counter to 8 LEDs and count 0–255.
• Make the counter count down from 15 to 0 instead of up.
• Read a number typed into the Serial Monitor and display it on the LEDs in binary.