Английский для математики: ключевые термины

"solve for x"

To find the value of an unknown variable by performing inverse operations on both sides of an equation until the variable is isolated

"The professor asked the class to solve for x in the equation 3x + 7 = 22, reminding them to perform the same operation on both sides."

"factor out"

To rewrite an expression by extracting a common factor from each term, simplifying the expression into a product

"Before solving the quadratic, he decided to factor out 2 from all three terms, which made the remaining arithmetic much more manageable."

"expand the brackets"

To multiply each term inside parentheses by the term outside, removing the brackets and producing an equivalent expression

"The student forgot to expand the brackets correctly, so the sign on the final term was wrong and the answer was marked incorrect."

"substitute into"

To replace a variable in an expression or equation with a given numerical value or another expression in order to evaluate or simplify it

"Once she found the value of y, she was able to substitute into the second equation and quickly determine the value of x as well."

"simultaneous equations"

A set of two or more equations that share the same unknowns and must all be satisfied at the same time, solved together to find a unique solution

"The examination paper included a pair of simultaneous equations that students were expected to solve using either substitution or elimination."

"quadratic formula"

The formula x = (−b ± √(b²−4ac)) / 2a used to find the roots of any quadratic equation of the form ax² + bx + c = 0

"When the quadratic could not be factored easily, she applied the quadratic formula to obtain the two exact roots."

"coefficient"

The numerical factor multiplying a variable or power of a variable in an algebraic term

"In the expression 5x³ − 3x + 2, the coefficient of the x³ term is 5 and the coefficient of the x term is −3."

"inequality"

A mathematical statement expressing that one quantity is greater than, less than, or not equal to another, represented by symbols such as >, <, ≥, or ≤

"The constraint in the optimisation problem was expressed as an inequality, requiring that the total cost must remain below a specified budget."

"take the derivative"

To apply differentiation rules to a function in order to find its rate of change with respect to a given variable

"To find the slope of the tangent line at any point on the curve, we need to take the derivative of the function with respect to x."

"evaluate the integral"

To compute the definite or indefinite integral of a function, finding either the accumulated value over an interval or the general antiderivative

"The student used integration by parts to evaluate the integral of x·eˣ, obtaining the antiderivative after two applications of the technique."

"approaches the limit"

Describes a quantity or function value getting arbitrarily close to a specific number as the input tends toward a particular value

"As n approaches infinity, the sequence 1/n approaches the limit of zero, never actually reaching it but becoming as small as desired."

"converges to"

Describes a sequence, series, or function that tends toward a finite, well-defined value as some parameter increases without bound or approaches a given point

"The geometric series with ratio 1/2 converges to 2, which the students verified by summing the first twenty terms and observing the partial sums stabilise."

"chain rule"

A differentiation rule stating that the derivative of a composite function equals the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function

"Applying the chain rule, the lecturer differentiated sin(x²) to get 2x·cos(x²), carefully distinguishing the outer and inner functions."

"inflection point"

A point on a curve where the second derivative changes sign, indicating a transition between concave-up and concave-down behaviour

"The engineer located the inflection point of the stress-strain curve to identify the transition from elastic to plastic deformation in the material."

"partial derivative"

The derivative of a multivariable function with respect to one variable, treating all other variables as constants

"To find the direction of steepest ascent on the surface, she computed the partial derivative with respect to both x and y and formed the gradient vector."

"Taylor series"

An infinite sum of terms expressed in powers of (x − a), used to represent a smooth function as a polynomial expansion around a centre point a

"By truncating the Taylor series of cos(x) after the fourth-degree term, the physicist obtained a polynomial approximation accurate enough for small angles."

"prove that"

To demonstrate the truth of a mathematical statement by constructing a rigorous logical argument from axioms, definitions, and previously established results

"The assignment asked students to prove that the sum of the interior angles of any triangle equals exactly 180 degrees using only Euclidean postulates."

"congruent to"

Describes two geometric figures that have exactly the same shape and size and can be mapped onto each other by a rigid transformation such as rotation, reflection, or translation

"The architect confirmed that the two triangular panels were congruent to each other, meaning they could be used interchangeably in the structure."

"similar triangles"

Triangles that have the same shape but not necessarily the same size, with equal corresponding angles and proportional corresponding side lengths

"By identifying similar triangles within the diagram, the student set up a proportion that allowed her to calculate the unknown height of the tower indirectly."

"perpendicular bisector"

A line that passes through the midpoint of a line segment and meets it at a right angle, equidistant from both endpoints of the segment

"To find the circumcentre of the triangle, the surveyor constructed the perpendicular bisector of each side and identified where all three lines intersected."

"surface area"

The total area of all the outer faces or surfaces of a three-dimensional solid, measured in square units

"The packaging designer calculated the surface area of the cylindrical tin to determine how much sheet metal would be required for each unit."

"continuous function"

A function whose graph has no breaks, holes, or jumps, formally defined by the property that small changes in the input produce arbitrarily small changes in the output

"The professor noted that while the function was defined at every point, it failed to be a continuous function at x = 0 because the limit from the left did not equal the limit from the right."

"vector space"

An algebraic structure consisting of a set of elements called vectors, equipped with operations of addition and scalar multiplication satisfying a specific set of axioms

"In the abstract algebra course, the students showed that the set of all polynomials of degree at most three forms a vector space over the real numbers."

"transformation"

A function that maps every point of a geometric figure to a new position, including operations such as translation, rotation, reflection, and dilation

"The computer graphics lecture explained how a single matrix multiplication encodes a transformation of the entire coordinate plane in two dimensions."

"standard deviation"

A measure of the spread of a dataset, equal to the square root of the variance, expressing how much individual values typically differ from the mean

"With a mean score of 72 and a standard deviation of only 4, the test results were tightly clustered around the average, suggesting the questions were well-calibrated."

"null hypothesis"

In statistical testing, the default assumption that there is no effect, no difference, or no relationship between variables, which the researcher attempts to reject with evidence

"The clinical trial set up the null hypothesis that the new drug had the same efficacy as the placebo, then tested whether the data provided sufficient evidence to reject it."

"p-value"

The probability of obtaining a test statistic as extreme as the observed result, or more extreme, assuming the null hypothesis is true; a small p-value is taken as evidence against the null hypothesis

"The p-value of 0.03 was below the conventional threshold of 0.05, so the researchers concluded the result was statistically significant."

"normal distribution"

A symmetric, bell-shaped probability distribution fully characterised by its mean and standard deviation, in which approximately 68% of values fall within one standard deviation of the mean

"Height measurements in a large population tend to follow a normal distribution, with most individuals clustered near the average and fewer at the extremes."

"confidence interval"

A range of values, calculated from sample data, that is likely to contain the true population parameter with a stated probability such as 95%

"The pollsters reported that 52% of voters preferred the candidate, with a 95% confidence interval of plus or minus three percentage points."

"correlation coefficient"

A dimensionless number between −1 and 1 that measures the strength and direction of the linear relationship between two variables

"A correlation coefficient of 0.91 between the two variables indicated a strong positive linear relationship, though the statistician cautioned against inferring causation."

"random variable"

A variable whose value is determined by the outcome of a random experiment, classified as discrete if it takes countable values or continuous if it can take any value in an interval

"The number of defective items in a batch of 100 is modelled as a discrete random variable following a binomial distribution with parameters n = 100 and p = 0.02."

"expected value"

The long-run average outcome of a random variable, calculated as the sum of each possible value multiplied by its probability

"The expected value of a single roll of a fair six-sided die is 3.5, even though 3.5 is not itself a possible outcome of any single roll."

"divisible by"

Describes an integer that can be divided by another integer with no remainder, meaning the quotient is also an integer

"Every integer ending in 0 or 5 is divisible by 5, which is why divisibility rules are so useful for quick mental arithmetic."

"prime number"

A natural number greater than 1 that has exactly two distinct positive divisors: 1 and itself

"Euclid's proof that there are infinitely many prime numbers remains one of the most elegant arguments in all of mathematics, relying only on basic properties of divisibility."

"proof by contradiction"

A method of mathematical proof in which one assumes the opposite of the statement to be proved and then derives a logical contradiction, thereby establishing the original statement must be true

"Using proof by contradiction, the mathematician assumed that the square root of 2 was rational, then showed this assumption led inevitably to a contradiction."

"mathematical induction"

A proof technique for statements indexed by natural numbers, consisting of a base case verification followed by an inductive step showing that truth at n implies truth at n+1

"The lecturer demonstrated mathematical induction by first verifying the formula for n = 1, then proving that its validity for an arbitrary n guaranteed its validity for n + 1."

"greatest common divisor"

The largest positive integer that divides each of two or more integers without remainder, found efficiently using the Euclidean algorithm

"To simplify the fraction 48/36, the student found the greatest common divisor of 48 and 36, which is 12, then divided both numerator and denominator by it."

"modular arithmetic"

A system of integer arithmetic in which numbers wrap around after reaching a fixed value called the modulus, denoted by the congruence symbol ≡

"Cryptographic systems rely heavily on modular arithmetic because operations like exponentiation modulo a large prime are easy to compute but hard to reverse."

"counterexample"

A specific instance that disproves a universal statement by showing a case in which the statement fails to hold

"The conjecture that all odd numbers are prime was immediately refuted by the counterexample of 9, which is odd but equals 3 × 3."

"if and only if"

A logical connective indicating that two statements are mutually equivalent: each implies the other, so both are true together or both are false together; abbreviated as "iff"

"A quadrilateral is a parallelogram if and only if both pairs of opposite sides are parallel, which the geometry textbook proved in both directions separately."

"matrix multiplication"

An operation combining two matrices A and B to produce a third matrix C, where each entry c_ij is the dot product of the i-th row of A and the j-th column of B

"The student was careful to note that matrix multiplication is not commutative: the product AB is generally not equal to the product BA."

"determinant"

A scalar value computed from a square matrix that encodes properties such as invertibility and the scaling factor of the linear transformation represented by the matrix

"The lecturer showed that a system of linear equations has a unique solution if and only if the determinant of the coefficient matrix is non-zero."

"eigenvalue"

A scalar λ such that for a given square matrix A there exists a non-zero vector v satisfying Av = λv, meaning the transformation stretches or shrinks the vector without rotating it

"The principal components of the dataset corresponded to the eigenvectors of the covariance matrix, ordered by their associated eigenvalues from largest to smallest."

"dot product"

An operation on two vectors that returns a scalar equal to the sum of the products of corresponding components, measuring alignment and used to compute angles between vectors

"Since the dot product of the two direction vectors was zero, the engineer confirmed that the two structural beams were perfectly perpendicular."

"row reduction"

The process of applying elementary row operations to a matrix to transform it into row echelon or reduced row echelon form, used to solve linear systems and find rank

"By performing row reduction on the augmented matrix, the student identified that the system was inconsistent because a row of zeros equalled a non-zero constant."

"span"

The set of all possible linear combinations of a given set of vectors, representing the subspace that those vectors collectively generate

"Two non-parallel vectors in three-dimensional space span a plane through the origin, but a third vector is needed to span the entire three-dimensional space."

"orthogonal"

Describes two vectors whose dot product is zero, meaning they are perpendicular to each other, or a matrix whose columns are mutually perpendicular unit vectors

"The Gram-Schmidt process transforms any basis into an orthogonal basis, which simplifies many calculations in numerical linear algebra."

"rank"

The dimension of the column space of a matrix, equal to the number of linearly independent rows or columns, which determines the number of solutions to an associated linear system

"After row reduction, the rank of the matrix turned out to be 2, indicating that only two of the four equations provided independent constraints on the unknowns."