Advanced Math for Computer Science Mastery

Explore how boolean algebra underpins computer hardware and digital circuits, using operators not, and, or with 0/1 logic, truth tables, operator priority, De Morgan's laws, and simplification rules.

Explore how transistors act as electronic switches and form logic gates such as and, or, not, including three-input variants, and how these gates decompose into two-input structures.

Explore how boolean expressions with constants 0/1, binary variables and operators map to circuits or truth tables, then practice simplifying and converting diagrams to algebraic expressions and gate circuits.

Apply the distributive law and boolean identities to simplify circuit expressions and demonstrate constructing gate expressions from A, B, C, and D using or blocks.

Convert a digital circuit to a boolean expression, simplify it with boolean laws, and redraw a minimal circuit that uses fewer blocks, cutting costs and power.

Examine the set-reset flip-flop, a two-stable-state latch that stores one bit and uses S and R inputs to control the Q output.

Explore the decimal numerical system, its Arabic digits 0–9, and place-value structure using powers of ten; learn how grouping and base ten relate to binary and hexadecimal representations in computing.

Explore the binary numerical system, base two, and master converting between binary and decimal using powers of two and division by two, then perform binary addition.

Learn how two's complement notation represents positive numbers by using a leading zero, and negative numbers by inverting bits and adding one. Note overflow can occur if bits are insufficient.

Explore hexadecimal notation and base 16, used in digital electronics and computer engineering, with digits 0-9 and a-f, and convert hex to binary using per-digit 4-bit patterns.

Explore how computers encode characters and numbers using binary codes, from ASCII seven-bit to eight-bit, and Unicode sixteen-bit, highlighting ASCII compatibility and memory efficiency for numeric representation.

Explore how sounds are represented in computers via analog-to-digital conversion, sampling rate, and bit depth, capturing frequency and amplitude in binary for mp3 and cd formats.

Explore how digital images are represented by sampling and scanning into a 2D pixel grid, where pixel density and bits per pixel determine quality.

Explain how parity checking and parity bits enforce odd or even parity to detect single errors, and use the Hamming distance to identify and correct errors in digital codes.

Write the circuit's boolean expressions and simplify with De Morgan's, double complement, distributive, associative, idempotent, and observation laws. Resetting the flip-flop with the reset input changes q to zero.

Explore operations on sets, including union, intersection, and complement, through disjoint examples and universal set references.

Explore the Cartesian product of sets and one-to-one correspondences across elements. Learn finite and infinite sets by cardinal number, Cantor's formula, equivalence, and proper subsets.

Explore automata theory as a core of the theory of computation, detailing symbols, alphabets, strings, and languages with examples of counting strings of length n.

explore finite automata as machines that understand a defined alphabet and move between states via a transition function, including deterministic and nondeterministic forms with start, accepting, and dead states.

Explore deterministic finite automata by mapping states to graph symbols, identifying start, final, and dead states, and constructing DFAs for languages: strings starting with zero, and length-two strings.

Explore building a deterministic finite automaton that accepts binary strings starting with 0 and ending with 1. The lecture shows state design, transitions, and final states in a six-state automaton.

Explore nondeterministic finite automata (NFA) through five tuples: q, Σ, q0, f, δ. Compare with deterministic automata and note how a state can have multiple next states, including empty input.

Explore nondeterministic finite automata by building an nfa that accepts strings containing the substring 01, and contrast with a dfa that accepts strings ending in 01.

Explore the core operations on languages — union, concentration, and Kleene star — with examples showing how combined strings form new languages and infinite sets of strings.

Explore how regular languages over an alphabet arise from the empty language and the universal language, and are closed under union, concatenation, and star, with zeros and ones examples.

Explore regular expressions and their correspondence to regular languages, covering union, concatenation, and star, with precedence rules and practical examples.

Explore divisibility rules and divisor notation, showing when one number divides another and how multiples establish divisors. Learn about quotient and remainder with the example 3 divided by 2.

Explore the Euclidean algorithm for finding the greatest common divisor of two numbers, including gcd with zero and the role of division and remainders in deriving the result.

Learn modular arithmetic by defining a mod n as the remainder of a divided by n, and use congruent modulo to compare remainders and determine divisibility.

Explore prime numbers and their properties, including Fermat's little theorem and Euler's totient function, with practical totient calculations for numbers such as 12, 23, 17, and 35.

Explore prime numbers with a simple divisibility test and the Miller-rabin algorithm. Express n-1 as 2^k q, choose random a, and test primality for primes and likely primes.

Explore modular addition and multiplication, using remainders to simplify calculations and verify key laws like commutativity, associativity, and distributivity with concrete examples in mod n.

Explore cryptography fundamentals by learning how to encrypt messages with public and private keys. Discover how asymmetric cryptography uses two keys to encrypt and decrypt plaintext into ciphertext.

Present three encryption schemes between A and B using public and private keys, including using B's public key, A's public key, and a double public-key scheme, with corresponding private-key decryption.

Explain rsa encryption: choose p and q, compute n and phi(n), pick e coprime to phi(n), determine d, and use public (n,e) and private (n,d) for encryption and decryption.

Generate RSA keys and demonstrate encryption and decryption using public and private keys, illustrating the role of n, e, d, and the Euler function phi(n) with worked examples.

Discover how the Diffie-Hellman key exchange enables two parties to securely establish a shared session key using a primitive root and modular exponentiation, avoiding RSA's slower encryption.

Practice Diffie-Hellman key exchange by computing the public keys ya and yb from xa and xb modulo q=71 with primitive root 7, then derive and verify the shared key.

Compute ya = 3^97 mod 353 and yb = 3^233 mod 353 to obtain ya = 40 and yb = 248, then derive the shared key as 160.

Explore the Dijkstra algorithm for finding the shortest path between vertices in a graph, illustrated with a graphical setup, distance updates, and predecessor tracking, plus a practical code example.

Implement the Dijkstra algorithm in C within a practical project, building the graph as an adjacency matrix, managing distance and processed sets, and printing the shortest paths.