Mathematics

Rational numbers can be expressed as a fraction where both numerator and denominator are integers, and the denominator is not zero, like 1/2. Irrational numbers cannot be expressed as simple fractions, and their decimal expansions are non-repeating and non-terminating, like √2. The set of rational numbers is countable, while irrational numbers are uncountable. Rational numbers include integers, fractions, and terminating or repeating decimals, whereas irrational numbers include square roots of non-perfect squares and certain mathematical constants like π. Understanding this distinction is crucial for solving problems in number systems and algebra.

The quadratic formula is x = [-b ± √(b² - 4ac)] / (2a), where a, b, and c are coefficients from the quadratic equation ax² + bx + c = 0. First, calculate the discriminant (D = b² - 4ac) to determine the nature of the roots. If D > 0, there are two distinct real roots; if D = 0, one real root; and if D < 0, no real roots. Substitute the values of a, b, and c into the formula to find the roots. This method is universally applicable to any quadratic equation, making it a powerful tool for solving such problems.

The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides (a² + b² = c²). It is used in construction to ensure buildings are square and to calculate distances. Architects use it to design ramps and staircases, while navigators apply it to plot the shortest path between two points. The theorem also finds applications in computer graphics for calculating pixel distances. Its real-world relevance makes it a fundamental concept in geometry and beyond.

The area of a sector can be found using the formula (θ/360) × πr², where θ is the central angle in degrees and r is the radius of the circle. First, determine the angle subtended by the sector at the center of the circle. Then, plug the angle and radius into the formula to compute the area. This formula is derived from the proportion of the sector's angle to the full circle's 360 degrees. Understanding this concept is essential for solving problems related to circles and their parts.

Mean is the average of all numbers, calculated by summing all values and dividing by the count. Median is the middle value when data is arranged in order, and mode is the most frequently occurring value in the dataset. Mean is sensitive to outliers, while median and mode are more robust in such cases. These measures of central tendency are used to summarize data sets in statistics. Choosing the appropriate measure depends on the data distribution and the presence of outliers.

First, solve one equation for one variable in terms of the other. Then, substitute this expression into the second equation to find the value of one variable. Once one variable is known, substitute it back into the first equation to find the second variable. This method is particularly useful when one equation is easily solvable for one variable. It's a systematic approach to finding the solution set of the system of equations.

Similar triangles have corresponding angles equal and sides proportional, making them useful for solving problems involving indirect measurement. They are used in constructing scale models, determining heights of inaccessible objects, and in trigonometry. The properties of similar triangles allow for the application of the Basic Proportionality Theorem (Thales' theorem). This concept is foundational in understanding geometric relationships and solving real-world problems. Mastery of similar triangles enhances problem-solving skills in geometry and beyond.

Probability is calculated as the number of favorable outcomes divided by the total number of possible outcomes. For example, the probability of rolling a 3 on a fair six-sided die is 1/6. Probability values range from 0 (impossible event) to 1 (certain event). Understanding probability is essential for predicting outcomes in various fields like statistics, finance, and science. It forms the basis for decision-making under uncertainty.

The surface area of a cylinder is 2πr(h + r), where r is the radius and h is the height. It consists of the lateral surface area (2πrh) and the areas of the two circular bases (2πr²). The formula is derived by unrolling the cylinder into a rectangle and two circles. This concept is vital for solving problems related to three-dimensional geometry. Practical applications include determining the amount of material needed to manufacture cylindrical objects.

Two lines are parallel if their slopes are equal (m1 = m2), meaning they never meet. They are perpendicular if the product of their slopes is -1 (m1 × m2 = -1), meaning they intersect at a right angle. The slope-intercept form (y = mx + c) is useful for identifying slopes. Understanding these relationships is crucial for analyzing geometric figures and solving coordinate geometry problems. This knowledge is applied in designing structures and in computer graphics.