Boolean Logic & Binary Cheat Sheet

A printable reference covering Boolean operators, truth tables, binary place value, base conversion, bitwise logic, and binary addition for grades 9-12.

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Boolean logic and binary are the foundation of how computers store data, make decisions, and process instructions. This cheat sheet helps students connect true or false statements with the 1s and 0s used inside digital systems. It is useful for programming, circuits, algorithms, and understanding how low-level computing works. Students need these ideas to read conditions, simplify logic, and work confidently with different number bases. The core ideas include Boolean operators such as AND, OR, and NOT, along with truth tables that show every possible input and output. Binary numbers use powers of 2, so each bit position has a place value such as 1, 2, 4, 8, and 16. Base conversion uses repeated division, place-value expansion, or grouping bits into hexadecimal digits. Binary addition follows normal carrying rules, but each column can only contain 0 or 1.

Key Facts

In Boolean logic, true is usually represented by 1 and false is usually represented by 0.
The AND operator is true only when both inputs are true, so A AND B = 1 only if A = 1 and B = 1.
The OR operator is true when at least one input is true, so A OR B = 1 if A = 1 or B = 1 or both are 1.
The NOT operator flips a value, so NOT 1 = 0 and NOT 0 = 1.
Binary place values are powers of 2, so 1011 in base 2 = 1x8 + 0x4 + 1x2 + 1x1 = 11 in base 10.
To convert decimal to binary, divide by 2 repeatedly and read the remainders from bottom to top.
Binary addition rules are 0 + 0 = 0, 0 + 1 = 1, 1 + 1 = 10, and 1 + 1 + 1 = 11.
Four binary bits match one hexadecimal digit, so 1111 in base 2 = F in base 16.

Vocabulary

Bit: A bit is the smallest unit of digital data and can have the value 0 or 1.
Boolean expression: A Boolean expression is a statement that evaluates to either true or false.
Truth table: A truth table lists every possible input combination and the output for a Boolean expression.
Logic gate: A logic gate is a circuit symbol or operation that produces an output from Boolean inputs.
Byte: A byte is a group of 8 bits often used to store one character or small integer value.
Hexadecimal: Hexadecimal is a base-16 number system that uses digits 0 through 9 and letters A through F.

Common Mistakes to Avoid

Treating 10 in binary as ten: 10 in base 2 equals 2 in base 10 because the left digit is the 2s place.
Forgetting that AND requires both inputs to be true: A AND B is 0 if either input is 0, even when the other input is 1.
Confusing OR with exclusive OR: normal OR is true when one or both inputs are true, while XOR is true only when the inputs are different.
Reading decimal-to-binary remainders in the wrong order: the remainders must be read from bottom to top after repeated division by 2.
Ignoring carries in binary addition: 1 + 1 is not 2 as a single digit, it becomes 10 with a carry to the next column.

Practice Questions

1 Convert 110101 in base 2 to base 10.
2 Convert 45 in base 10 to binary.
3 Add the binary numbers 1011 and 0110.
4 Explain why computers can represent both logic decisions and numerical data using only 0s and 1s.