Chapter 2 – Number Systems and Boolean Algebra

What is a number system?

A writing system for expressing numbers using digits or symbols in a consistent manner.

What is the base (radix) of a number system?

The number of unique digits (including zero) used to represent numbers in that system.

What are the common number systems and their bases?

• Decimal (Base 10): Digits 0–9

• Binary (Base 2): Digits 0, 1

• Octal (Base 8): Digits 0–7

• Hexadecimal (Base 16): Digits 0–9 and A–F

 How do you convert from any base to decimal?

Multiply each digit by its base raised to the power of its position and sum the results.

Which number system is used internally by computers?

Binary

How do you convert binary to octal?

Group binary digits in groups of 3 from the right and convert each group to an octal digit.

How do you convert binary to hexadecimal?

Group binary digits in groups of 4 from the right and convert each group to a hex digit.

What is Boolean algebra?

A branch of algebra where variables take values of TRUE (1) or FALSE (0).

Boolean algebra uses which values?

True and False

What are the Commutative Laws of Boolean algebra?

A + B = B + A → A · B = B · A

What are the associative laws?

A + (B + C) = (A + B) + C → A · (B · C) = (A · B) · C

What are the Distributive Laws?

A + (B · C) = (A + B)(A + C → A · (B + C) = A·B + A·C

What does De Morgan’s first law state?

(A + B)’ = A’ · B’

What is the Absorption Law?

A + A·B = A → A·(A + B) = A

How do you simplify Boolean expressions?

Apply Boolean laws (commutative, associative, distributive, etc.) to reduce expressions to simpler forms.

Which Boolean form uses a sum of minterms?

Sum of Products (SOP)

What are minterms and maxterms?

• Minterm: Product (AND) of all variables in true/false form.

• Maxterm: Sum (OR) of all variables in true/false form.

How are minterms defined?

Product of all variables once (true or complemented)

What is the Sum of Products (SOP) form?

Expression represented as a sum (OR) of minterms.

What is the Product of Sums (POS) form?

Expression represented as a product (AND) of maxterms.

What are the two canonical forms?

• SOP (Σ): Sum of minterms.

• POS (π): Product of maxterms.

How do you convert between canonical forms?

Replace Σ with π and list the missing minterm/maxterm indices.

What is a standard form Boolean expression?

A Boolean expression where all terms include all variables, expanded for simplification.

What is a Karnaugh Map (K-Map)?

A graphical tool used to simplify Boolean expressions by grouping adjacent 1s.

What is a Karnaugh Map (K-Map) used for?

Boolean simplification

Which of the following is NOT a K-Map rule?

Groups can be diagonal

What are “don’t care” conditions in K-Maps?

Inputs that are irrelevant for simplification

What is the Logisim tool used for?

To design and simulate digital circuits implementing Boolean expressions.

What tool can be used to draw and test logic gate circuits?

Logisim