South 고등학교 2학년 Mathematics for Artificial Intelligence

Chapters: 5

1. AI and big data

What AI is and how it learns · History of maths in AI · Big data and AI

Artificial Intelligence: How Machines Learn to Think – Artificial intelligence (AI) is the skill of a computer system to do tasks that normally need human thinking: seeing, understanding speech, deciding and learning. An AI system is an agent that senses, thinks and acts. Old AI followed rules written by people. Modern AI mostly uses machine learning: it finds its own rule from many labelled examples (data). Neural networks are layers of simple units whose link strengths (weights) change during training. AI is used in maps, translation, health, farming and games. It can be wrong or unfair when its data is one-sided (bias), so people must check it, protect privacy and stay responsible.
Big Data – Big data is data that is too big, too fast or too mixed to store and process on one ordinary computer with normal tools. We describe it with Volume (how much), Velocity (how fast it arrives) and Variety (how many kinds). To handle it, the work is split across many machines that run at the same time (distributed processing, for example MapReduce). Big data is often stored as simple facts or as a graph of nodes and links, and it is used for weather forecasts, maps, health, shopping and training AI. It also raises questions about privacy and fairness.

Representing text with sets and vectors · Frequency vectors and similarity · How AI analyses text

Scalars and Vectors for Motion in a Plane (Class 11) – A scalar has only size; a vector has size and direction. Vectors are equal if their size and direction match. Multiplying by a number changes the length (a negative number flips it). Vectors add tail-to-head (triangle or parallelogram law); A − B = A + (−B). Any vector in a plane is A = Ax î + Ay ĵ with Ax = A cos θ, Ay = A sin θ. A·B = AB cos θ is a scalar; A×B has size AB sin θ and is perpendicular to both.
Machine Learning – Machine learning (ML) is a way for computers to learn a rule from examples instead of being told the rule. We give data made of features (inputs) and, in supervised learning, labels (answers). The machine fits a model: regression predicts a number, a decision tree asks yes/no questions to pick a class, and k-means clustering groups unlabelled data. We train on most of the data, test on data it never saw, and measure accuracy or error. A model that only memorises (overfits) fails on new data.

Images as matrices · Transforming images with matrix operations · How AI classifies images

Matrices: Order, Types, Transpose, Operations and Inverse – A matrix is a box of numbers set in rows and columns. Its order is rows × columns. Special matrices include zero, identity, diagonal, scalar, row, column and square matrices. The transpose swaps rows and columns. Symmetric means Aᵀ = A and skew-symmetric means Aᵀ = −A. We add matrices place by place, and multiply them row × column. Matrix multiplication is not commutative (AB is usually not BA). A square matrix A is invertible if some B gives AB = BA = I, and that B is unique.
Machine Learning – Machine learning (ML) is a way for computers to learn a rule from examples instead of being told the rule. We give data made of features (inputs) and, in supervised learning, labels (answers). The machine fits a model: regression predicts a number, a decision tree asks yes/no questions to pick a class, and k-means clustering groups unlabelled data. We train on most of the data, test on data it never saw, and measure accuracy or error. A model that only memorises (overfits) fails on new data.

Probability-based prediction · Trend lines with technology · Loss functions and best-fit lines · Gradient descent

Probability: Chances of a Single Event – Probability tells how likely something is, as a number from 0 to 1. When all outcomes are equally likely, P(event) = number of favourable outcomes ÷ total number of outcomes. An impossible event has probability 0 and a sure event has probability 1.
Linear Regression and the Least Squares Line – Linear regression finds the straight line ŷ = a + bx that best follows paired data (x, y). A residual is the gap between a real point and the line: e = y − ŷ. The least squares line makes the sum of squared residuals as small as possible. Its slope is b = Sxy ÷ Sxx and it always passes through the mean point (x̄, ȳ). We use it to predict y from x, but only inside the data range, and a strong link does not prove that x causes y.
Gradient Descent: Finding the Lowest Point – Gradient descent finds the minimum of a function by taking small steps downhill. At each step, new x = old x − learning rate × slope. A learning rate that is too big overshoots; too small is very slow. It is how machine learning models reduce their error.

Maths behind real AI services · AI maths project

Artificial Intelligence: How Machines Learn to Think – Artificial intelligence (AI) is the skill of a computer system to do tasks that normally need human thinking: seeing, understanding speech, deciding and learning. An AI system is an agent that senses, thinks and acts. Old AI followed rules written by people. Modern AI mostly uses machine learning: it finds its own rule from many labelled examples (data). Neural networks are layers of simple units whose link strengths (weights) change during training. AI is used in maps, translation, health, farming and games. It can be wrong or unfair when its data is one-sided (bias), so people must check it, protect privacy and stay responsible.
Mathematical Inquiry: Asking and Answering Your Own Maths Question – A mathematical inquiry is a small research project where you ask your own maths question and answer it with evidence and reasoning. It follows a cycle: choose a clear question, read what others found (literature research), plan a method (experiment, case study or development research), carry it out and record data, analyse it with maths, then reflect and report. AI tools can help with data and checking, but the thinking, honesty and sources must be yours.