Algebra va Geometriya

Definition: Branch of mathematics dealing with symbols and the rules for manipulating those symbols.
Key Concepts:
- Variables: Letters representing numbers (e.g., x, y).
- Expressions: Combinations of variables and constants (e.g., 3x + 2).
- Equations: Mathematical statements asserting equality (e.g., 2x + 3 = 7).
- Functions: Relationships between sets that assign each input one output (e.g., f(x) = x^2).
Types:
- Linear Algebra: Study of vectors, vector spaces, and linear transformations.
- Polynomial Algebra: Focus on polynomials and their properties.

Definition: Study of shapes, sizes, and properties of figures and spaces.
Key Concepts:
- Points, Lines, and Angles: Basic building blocks of geometry.
- Shapes:
  - 2D (e.g., triangles, circles, rectangles).
  - 3D (e.g., spheres, cubes, cylinders).
- Theorems:
  - Pythagorean theorem (a² + b² = c² for right triangles).
  - Congruence and similarity.
Measurement:
- Area and perimeter for 2D shapes.
- Volume and surface area for 3D shapes.

Definition: Study of change and motion, using limits and derivatives.
Key Concepts:
- Limits: Approach of a function as the input approaches a point.
- Derivatives: Rate of change of a function; slope of the tangent line.
- Integrals: Accumulated area under a curve; opposite of differentiation.
Applications: Used in physics, engineering, economics for modeling and problem-solving.

Definition: Study of data collection, analysis, interpretation, presentation, and organization.
Key Concepts:
- Descriptive Statistics: Summarize and describe data (mean, median, mode).
- Inferential Statistics: Draw conclusions from data samples (hypothesis testing, confidence intervals).
- Probability: Measure of the likelihood of events occurring.
Data Types:
- Qualitative (categorical) vs. Quantitative (numerical).
- Discrete vs. Continuous data.

Definition: Study of the relationships between angles and sides of triangles.
Key Concepts:
- Trigonometric Ratios: Sine, cosine, and tangent definitions associated with right triangles.
- Unit Circle: Tool for understanding the behavior of trigonometric functions.
- Identities: Fundamental equations involving trigonometric functions (e.g., Pythagorean identity: sin²θ + cos²θ = 1).
Applications: Used in physics, engineering, architecture, and various fields to solve problems involving angles and distances.

Algebra - bu matematikning ramzlar va ushbu ramzlarni manipulyatsiya qilish qoidalari bilan shug'ullanadigan bo'limi.
Asosiy tushunchalar:
- O'zgaruvchilar: Sonlarni ifodalovchi harflar (masalan, x, y).
- Iboralar: O'zgaruvchilar va doimiylar kombinatsiyasi (masalan, 3x + 2).
- Tenglamalar: Tenglikni ta'kidlovchi matematik bayonotlar (masalan, 2x + 3 = 7).
- Funktsiyalar: Har bir kirishga bir chiqish tayinlaydigan to'plamlar o'rtasidagi munosabatlar (masalan, f(x) = x^2).
Turlari:
- Chiziqli Algebra: Vektorlar, vektor fazolari va chiziqli o'zgarishlarni o'rganish.
- Ko'phadli Algebra: Ko'phadlar va ularning xossalariga e'tibor qaratilgan.

Geometriya shakllarning, o'lchamlarning va raqamlar va bo'shliqlarning xossalarini o'rganadi.
Asosiy tushunchalar:
- Nuqtalar, chiziqlar va burchaklar: Geometriyaning asosiy elementlari.
- Shakllar:
  - 2D (masalan, uchburchaklar, doiralar, to'rtburchaklar).
  - 3D (masalan, sharlar, kublar, silindrlar).
- Teoremalar:
  - Pifagor teoremasi (to'g'ri uchburchak uchun a² + b² = c²).
  - Moslik va o'xshashlik.
O'lchov:
- 2D shakllar uchun maydon va perimetr.
- 3D shakllar uchun hajm va sirt maydoni.

Hisoblash o'zgarish va harakatni, limitlar va hosilalarni qo'llagan holda o'rganadi.
Asosiy tushunchalar:
- Limitlar: Funktsiyaning kiritish ma'lum bir nuqtaga yaqinlashganda qiymatiga yaqinlashishi.
- Hosilalar: Funktsiyaning o'zgarish tezligi; teginuvchi chiziqning qiyaligi.
- Integrallar: Egri chiziq ostidagi to'plangan maydon; differentsiallashning teskarisi.
Ilovalar: Fizika, muhandislik, iqtisodda modellashtirish va muammolarni hal qilish uchun ishlatiladi.

Statistika ma'lumotlarni to'plash, tahlil qilish, talqin qilish, taqdim etish va uyushtirishni o'rganadi.
Asosiy tushunchalar:
- Ta'rifiy statistika: Ma'lumotlarni umumlashtirish va tavsiflash (o'rtacha, mediana, moda).
- Inferensial statistika: Ma'lumotlar namunalaridan xulosalar chiqarish (gipotezani sinash, ishonchli oralig'i).
- Ehtimollik: Hodisalar ro'y berish ehtimoli o'lchovi.
Ma'lumotlar turlari:
- Sifatli (kategorik) va miqdoriy (raqamli).
- Diskret va doimiy ma'lumotlar.

Trigonometriya uchburchaklarning burchaklari va tomonlari o'rtasidagi bog'liqliklarni o'rganadi.
Asosiy tushunchalar:
- Trigonometrik nisbatlar: To'g'ri uchburchak bilan bog'liq sinus, kosinus va tangens ta'riflari.
- Birlik doirasi: Trigonometrik funktsiyalarning xatti-harakatini tushunish vositasi.
- Tenglamalar: Trigonometrik funktsiyalarni o'z ichiga olgan asosiy tenglamalar (masalan, Pifagor tengligi: sin²θ + cos²θ = 1).
Ilovalar: Fizika, muhandislik, me'morchilik va burchaklar va masofalarni o'z ichiga olgan turli sohalarda muammolarni hal qilish uchun ishlatiladi.