Math Flashcards
All whole numbers both positive and negative as well as zero
Integers
Point where two curves, lines or edges meet
Vertex
Branch of mathematics that focuses on calculating and understanding the likelihood of an event
Probability
The three legs of a right triangle are associated with this theorem which states that the sum of the squares of the side lengths of the triangle equals the square of the length of the hypotenuse (the side of the triangle opposite of the right angle)
Pythagorean theorem
This type of number consists of a real part and an imaginary part.
Complex number
Operation involving repeated multiplication of a number based on its superscript.
Exponentiation
Natural numbers whose only factors are one and themselves.
Prime numbers
A transcendental number equal to approximately 3.14159. It denotes the ratio of a circle’s circumference to its diameter
Pi
Are mathematical quantities that have both magnitude and direction; examples include velocity and acceleration.
Vectors
Is a mathematical operation that sets a number x to the power of another number y, such as x raised to the y power.
Exponentiation
Calculated by dividing the sum of all numbers in the set with the number of values, also called the mean.
Average
These are values found on an infinite number line and include all rational and irrational numbers.
Real numbers
The expression of a number or polynomials into simpler terms.
Factoring
A series of numbers formed by adding the two preceding numbers.
Fibonacci Sequence
A Greek mathematician who devised namesake type of “Screw” used to lift water
Archimedes
The next highest perfect square number after one, as it is the product of 2 times 2.
Four
These shapes, also known as tetragons, have four sides and have interior angles that sum to 360 degrees.
Quadrilaterals
A symmetrical plane curve in the shape of the letter U. It can be formed by graphing a quadratic function
Parabola
A diagram constructed by its namesake French mathematician in which each number is equal
to the sum of the two numbers above.
Pascal’s Triangle
A statistical property that describes the condition in which the outcome of one event has no
relevance or effect on the outcome of another event.
Indenpendence
A round three-dimensional shape with no edges.
Sphere
Numbers are called this when the product of their squares is negative.
Imaginary
What term describes an inferential argument made to determine the truth of a mathematical
statement?
Proof
What rectangular array of numbers or symbols is used to represent a single mathematical
object?
Matrix
This quantity is proportional to temperature, but inversely proportional to pressure
from the combined gas law. This quantity has a value of 22.4 for one mole of an ideal gas. For
the point, name this quantity describing the space occupied by a substance, usually expressed
in liters. What geometric term refers to the quantity of 3D space held within an enclosed surface,
such as the amount of water in a glass?
Volume
A dodecagon has how many sides?
12
Isosceles, Scalene, and Equilateral are all types of which shape?
Triangle
All of the interior angles of a triangle sum to how many degrees?
180
Which shape is defined as the set of all points in a two-dimensional plane
that are equidistant from one point in the center?
Circle
These expressions are classified as analytic because they are a finite power series.
The Fundamental Theorem of Algebra states that every one of these expressions with degree n and complex coefficients has n complex roots. For the point, name these expressions that contain coefficients and variables, like x squared minus one.
Polynomial
Unlike addition, this operation is not commutative for matrices in general. The use of numerous instances of this operation upon one number is called exponentiation. Often thought of as a form of repeated addition, this is, for the point, what operation that when applied to 2 and 3 yields 6?
Multiplication
This function appears three times in the formula for combinations, and it is estimated by Stirling’s approximation. This function applied to 3 is 6, and when applied to 4 yields 24.
Defined as the product of all positive integers less than or equal to n, for the point, this is what function denoted by an exclamation point?
Factorial
One inequality named for this figure is reversed in Minkowski space. The intersection of altitudes in this shape is known as its orthocenter, and its area can be computed using Heron’s formula. The angles in any example of this polygon sum to 180 degrees in Euclidean space. For the point, name this polygon with three sides.
Triangle
Kepler’s conjecture concerns the packing of these figures in three-dimensional space, and the sum of angles of a triangle on this shape exceeds 180 degrees. The surface area of one of these objects is 4 pi r squared. For the point, name this 3D shape with a volume of four thirds pi r cubed, exemplified by structures such as marbles and globes.
Sphere
The rank-nullity theorem is used on these objects, and local curvature is described
by the “Hessian” type of these objects. The equation A v equals lambda v describes the eigenvector and eigenvalue of these objects. Expansion by minors is used to find the determinant of these objects, which can be used to invert them. For the point, name these mathematical objects which are arrays of numbers in rows and columns.
Matrix
In polar coordinates, the equation “r equals the constant A” produces one of these shapes. The Cartesian product of two of these shapes gives a torus. This conic section defines an eccentricity of zero, and a square cannot be constructed with the same area as this shape with only a compass and straightedge. Pi times the radius squared gives the area of, for the point, what 2D shape consisting of all points equidistant from a fixed center?
Circle
This function is generalized by the gamma function. Stirlings formula is an approximation of this function. This function applied to a positive integer n is given by n times quantity n minus one all
the way down to one. This function gives the number of ways to arrange n distinct objects in a row. For the point, name this function symbolized by an exclamation point next to a number
Factorial
One of these things named for Poisson can model events that occur with a fixed rate. The central limit theorem details how independent random variables can tend to one of these things that is
described by the 68-95-99.7 rule. That example of these statistical things is sometimes named for Gauss and has a bell curve shape. For the point, name this statistical device which can be uniform or normal and describes the spread of data.
Probability distribution
The name for this sort of process is thought to be from an Anglicization of an Arab mathematician who invented a form of algebra. These processes typically have their asymptotic performance expressed using big O notation. Djikstra
[DIKE
struh] names one of these instructions
that finds the shortest path between two vertices of a graph. Quicksort is an example of one for sorting.
For the point, name this set of cookie-cutter instructions given to a computer to solve a problem.
Algorithm
According to Goldbachs conjecture, all integers with this property can be written as the sum of two primes. Functions described by this term are symmetric about the y-axis. Performing an
exponential on a negative real number with a natural number that has this property is always positive.
The smallest prime number is the only prime with this property. For the point, name this property present
in numbers divisible by two.
Even
This unit was recently redefined through measurements of Planck’s constant, until then this was the only SI unit defined by a physical device. The standard units for moment of inertia are derived
by multiplying this unit by meters squared. Balances measure this SI unit, typically by one thousandths
of it. For the point, name this SI unit for mass approximately equal to 2.2 pounds.
Kilogram
In languages like C++, this sort of data structure is similar to an array but has dynamic size. In linear algebra, this word describes mathematical objects that can be manipulated with dot and
cross products. In biology, this term refers to organisms which carry and transmit disease. Velocity can
be described by this term, unlike speed, which is scalar. For the point, name this term which can describe mathematical objects with magnitude and direction.
Vectors
Every natural number greater than 1 can be factored into these numbers according to the fundamental theorem of arithmetic. If two numbers have a greatest common factor of 1, they are said to be co- this property. 2 is the only even number with this property, and they are contrasted with composite numbers. For the point, name these numbers whose only factors are one and themselves.
Prime numbers
Dijkstras algorithm is used on weighted examples of these objects. If one of these
mathematical objects has no cycles of odd length, it is called bipartite and can be colored with two colors.
If a connected one of these objects has no cycles it is called a tree. For the point, name these objects made up of vertices connected by edges.
Graphs
The constellation Sagittarius is identified by this number in the Messier catalogue.
This number is the smallest even two-digit number whose digits sum to 10. This number is the atomic number of Nickel, as well as the number of days in two fortnights. For the point, identify this number, the product of 4 times 7.
28
Eigenvalues can be found by performing this operation on the original matrix
minus lambda times the identity and setting the result equal to (+) zero. This
operation on a matrix is zero if and only if the matrix is noninvertible, and taking a
cross product is equivalent to performing this operation on a 3 by (*) 3 matrix. For the points, identify this operation, which for a 2 by 2 matrix equals a times d minus b times c.
Determinant
One type of these entities can be iteratively solved using Runge-Kutta
methods. Guessing solutions composed of exponential functions or using (+) Euler’s
[[OY-lers]] method are simple methods for explicitly solving the ordinary type of
these entities. Euler and Lagrange name a system for these equations, and a first step in dealing with these equations is to employ separation of (*) variables. For the points, what type of equations contain a variable and its derivatives?
Differential Equations
Thales’ [[THAY-LEEZ]] theorem applies when this entity is the diameter of an
excircle. It’s not 1, but this entity is the numerator in the cosecant [[koh-SEE-kehnt]] (+) function. The length of this entity equals the square root of two times one of its smaller counterparts in a 45-45-90 triangle. This term is represented by the “c” in the Pythagorean (*) theorem. For the points, give this term for the longest side of a right triangle.
Hypotenuse
The lack of this construct differentiates a pseudo-ring from a ring. For
matrices, the multiplicative type of this construct consists of a square (+) matrix with 1’s on the main diagonal and 0’s elsewhere. For addition on the real numbers, this construct equals 0, and for multiplication, it equals 1 except for the (*) zero case. For the points, what construct for a binary operation does not change the element it acts on?
Identity
This operation’s epsilon-delta definition was first formulated by Cauchy [[koh
SHEE]]. This operation can be evaluated by L’Hopital’s [[loh-pee-TAHLS] rule. (+) The
formal definition of a derivative is this operation applied to a difference quotient. The two-sided type of this operation does not exist at jump discontinuities. (*) For the points, name this operation of finding where a function “approaches” as an input goes toward a value.
Limit
A theorem named for this man states that the order of any element of a group
divides the order of that group. This man names (+) “multipliers” which are used to
find local maxima and minima of a function under equality constraints. This
mathematician names “points” in which the centrifugal force balances (*) gravity. For the points, name this Italian-French mathematician known for his contributions to number theory.
Lagrange
This mathematician found six proofs of quadratic reciprocity. Complex
numbers that have both an integer real and imaginary part are known as this man’s (+) “integers.” This mathematician names a probability distribution that resembles a “bell curve.” According to legend, this man stunned his teacher by calculating the sum of the first 100 positive (*) integers rapidly. For the points, give this mathematician, the namesake of the normal distribution.
Gauss (accept Gaussian Distribution)
The Heawood [[HAY-wood]] conjecture outlines the optimal way of doing this
process on different surfaces. A theorem about this process was proven in 1976 by
Kenneth (+) Appel and Wolfgang Haken, making it the first proof to use a computer. In graph theory, the minimum number required to perform this process is the (*) chromatic number. For the points, name this process of labeling a graph requiring only four of the namesake properties.
Graph coloring
Numbers that have this property but satisfy Fermat’s little theorem are known
as Carmichael numbers. It’s not pronic, but “rectangular numbers” are (+) numbers
with this property. The Sieve of Eratosthenes crosses these numbers out in its algorithm. Numbers with this property have at least one other (*) divisor between one and itself. For the points, give this property of a number which can be factored into the product of two smaller numbers, the opposite of prime.
Composite Number
A population’s standard deviation is divided by the square root of this
quantity to give the standard error of the mean. The distribution of sample means
approaches a (+) normal distribution as this quantity increases according to the
central limit theorem. The mean of a set equals the sum of the values (*) divided by
this quantity. For the points, name this quantity symbolized n, which is the number of data points in a set.
sample size
This man names a rule for differentiating under the integral sign. The
alternating series test in series (+) convergence is alternately named for this
mathematician. This man, whose philosophical works include Discourse on
Metaphysics, also published a text which presented a formulation of a subject that
another mathematician explained with (*) fluxions. For the points, name this
mathematician who invented calculus independently of Isaac Newton.
Gottfried Leibniz or Leibniz
This man pioneered the method of infinite descent in proofs and frequently
applied that method to Diophantine [[dee-oh-FAHN-teen]] (+) equations. This
mathematician proved that a prime number must be 1 mod 4 to be the sum of two perfect squares, and his “Little Theorem” is one of the pillars of (*) number theory. For the points, name this French mathematician whose “Last Theorem” was proved by Andrew
Wiles in 1995.
Pierre de Fermat
This mathematical concept has a double type that only involves odd numbers, and a hyper type that form the discriminants of Hermite polynomials. This mathematical function increases faster than exponential growth, but slower than a double exponential function. For the point, name this mathematical function that is the product of all positive integers less than or equal to the input of the function.
Factorial
The quotient of these entities can be expressed as a solution to a system of linear equations via Cramer’s Rule. These entities, which come in Singleton and Jacobian varieties, are non-commutative when multiplied. The determinant of a two-by-two variety of these entities is ad minus bc. For the point, name these rectangular mathematical entities that can
be arranged in rows and columns.
matrix
This set of numbers is not bounded above in the reals by the Archimedean property.
Peano’s axioms are a set of rules governing these numbers. The sum of the reciprocal of these numbers is equivalent to the harmonic series. This set of numbers is the intersection of the positive numbers with the integers. For the point, what set of numbers includes 1, 2, 3, and so on?
Natural numbers
If the p-value is less than significance level alpha, the null type of this statement is
rejected in favor of the alternative type. In the scientific method, these statements are often written in an “if/then” format. For the point, name these proposed statements that can be tested by experimentation.
Hypothesis
This operation describes the runtime for the fastest known algorithm that solves the traveling salesman problem. This operation is performed on “cos(x) + i sin(x)” in De Moivre’s [[deh MWAHVS]] formula. This operation’s second input is nt in the compound interest formula. When this operation has a base of two, it is the same as doubling. For the point, name this operation that multiplies a number by itself.
Exponent
This Wilson prime is also the third Catalan number and the fifth of the Fibonacci
numbers. This number appears under the square root in the numerator of the golden number, and this number is also the value of the hypotenuse of the smallest Pythagorean triple. For the point, identify this number, the atomic number of boron, as well as the total sides in a pentagon.
Five
Arc length can be calculated by multiplying this quantity by theta over 360. The
constant tau is equal to the ratio between this quantity and the radius, and this quantity can be estimated by multiplying the diameter by 22 over 7. For the point, give this term for the distance around a circle.
Circumference
One algorithm for this task relies on partitioning data around a pivot and then
calling itself recursively. The “insertion” and “selection” forms of algorithms for this task typically have the worst runtime. The concept of stability is most often applied to algorithms for this task, which the “quick” form notably lacks. For the point, name this task of putting elements of a list in order.
Sorting
Cramer’s rule can be used to find a solution to a system of these types of equations, which can also be solved via Gauss-Jordan elimination or substitution. For the point, name this term that describes polynomials with degree one that have constant slope, examples of
which include the line y equals three x plus one.
linear (accept linear equations)
According to the Fundamental Theorem of Arithmetic, every positive integer greater
than one can be expressed as the unique product of these numbers, which can be computed using the Sieve of Eratosthenes [[ehr-uh-TOSS-thuh-neez]]. 2, 3, and 5 are examples of, for the point, what numbers, often contrasted with composite numbers, that are only divisible by 1 and themself?
Prime Numbers
This mathematician solved the Basel problem by utilizing a power series. This man names a formula relating the exponential of complex numbers to trigonometric functions. The base of the natural logarithm is, for the point, a number named for which mathematician, symbolized e?
Leonhard Euler (accept Euler or Euler’s formula)
The distance formula in Cartesian coordinates is derived from this formula,
solutions for which include namesake triples such as 6, 8, and 10. The hypotenuse of a triangle can be computed by, for the point, what formula named for a Greek mathematician that states that a-squared plus b-squared equals c-squared?
Pythagorean Theorem
This shape’s namesake numbers are generated by the formula [n times 3n minus 1 all over 2]. This shape contains as many diagonals as it does sides. Each internal angle of a regular one of this polygon measures 108 degrees. For the point, name this type of polygon
which has five sides.
Pentagon
A program called Stockfish has been applied to analyze agents on one of these
objects. This object, which can be traversed using only L-shaped jumps, is the subject of a discrete mathematics problem about arranging eight of a certain piece such that they do not
attack each other. For the point, name this eight-by-eight board, the subject of the eight queens puzzle.
Chessboard
This number is the order of the smallest non-abelian group and appears in the
denominator of the solution to the Basel problem. This is the smallest perfect number, and also the number of possible handshakes among 4 people. 3 factorial is equal to, for the point, what number that gives the number of sides in a hexagon?
Six
The Abel-Ruffini theorem says it is impossible to find algebraic solutions to these objects when their degree is greater than four. The power rule can be used to differentiate these objects, which can be divided using synthetic division. For the point, name these functions consisting of sums of the powers of variables.
Polynomials
Conflicts have arisen over whether the binary or decimal system should be used for this unit of measurement. The most common type of this unit is referred to by the Internet Protocol as an octet. For the point, name this unit of measurement for data size most commonly consisting of eight bits and expressed in kilo, mega and giga variants.
Byte
A “space” named after this man is the fundamental space of geometry. This man
proved that there are infinitely many primes by constructing their product plus one. The parallel postulate is found in one set of writings by this man. For the point, name this mathematician from Alexandria who wrote the landmark math treatise Elements.
Euclid
Applications of this function include determining the length of vectors and the
geometric mean. This function appears above 5 in the numerator of the Golden Ratio and can be expressed as the one-half power in exponential notation. For the point, name this function, which when applied to negative one is i and when applied to 9 is 3.
square root
Hilbert’s paradox of the Grand Hotel is a thought experiment that illustrates the
counterintuitivity of sets with this property. Georg Cantor denoted a “countable” variety of this property aleph-null. Euclid proved that there are this many prime numbers, while irrational numbers have this many digits after the decimal point. For the point, name this
mathematical property denoting a number that is uncountably large.
Infinity
The Minkowski inequality is a generalization of a statement named for this shape. The area of this shape can be found by multiplying the semiperimeter and inradius, or it can be calculated using Heron’s formula. For the point, name this geometric shape that comes in “obtuse” and “acute” varieties and has three sides.
Triangle
This function names a law that is a generalized form of the Pythagorean theorem. This trigonometric function is obtained by dividing the adjacent side of a right triangle by its hypotenuse. The secant function is the reciprocal of, for the point, what mathematical function, the co-function of sine?
Cosine
This set of numbers was shown to be uncountable by Cantor’s diagonalization
argument. The rational numbers are dense in this set of numbers, which also includes the irrational numbers. For the point, what set contains any number that can be expressed on a number line, contrasted with complex numbers?
Real numbers
A method named for Euler repeats this operation to find the limit of a series. This
operation is applied to trapezoids in a method for approximating integrals. Doing this operation on two vectors applies the parallelogram law by aligning them, and, with multiplication, this is the only commutative arithmetic operation. Zero is the identity of, for the point, what operation represented by a plus sign?
Addition
In a controversy surrounding this concept, a hash code was sent between the
opponents, which, when deciphered, contained the mention of fluxions. The publishing of the work Nova Methodus pro Maximis et Minimis furthered one man’s claim to invent this subject often described as the “study of infinitesimals.” For the point, name this subject that was invented independently by Gottfried Leibnitz and Isaac Newton.
Calculus
The only stand-alone group of Hilbert axioms concerns an axiom about this
property, which is equivalent to Playfair’s axiom. Two vectors that have a cross product of zero have this property. Euclid’s fifth postulate is named for this property and implies that all angles of a rectangle are right angles. Two lines with this property have the same slope. For the point, name this property of two lines in the same plane that will never intersect.
Parallel
If the polar angle is this quantity, the resulting point will be on the negative side of the x-axis. The Madhava-Leibniz series approaches this number, and it’s not e, but this was the first number proven to be transcendental. This number times four-thirds times the radius cubed gives the volume of a sphere. For the point, give this term for the ratio between a circle’s circumference and its diameter, approximately 3.14.
Pi
One algorithm for this task that uses paired comparisons is named after
garden gnomes. Exchange-based algorithms for this task have a worst-case runtime of “Big O [[OH]] of n (+) squared.” Common algorithms like bubble, quick, and merge name types of, for the point, what task of organizing an array or list into a desired order?
Sorting
This man’s proof of the fundamental theorem of algebra wascreatedtorefute
D’Alembert’s [[dah-lem-BEHRS]] false proof. This man names a theorem relating the electric field to the charge (+) distribution, and this man also names a distribution shaped like a bell curve. The Normal Distribution (*) is named for, for the point, what German mathematician?
Gauss
This value represents the Turing degree of the partial computable functions.
This value represents the bottom element ofaboundlatticeandisthe(+)cardinality
of the empty set. Creating an undefined (*) expression as the denominator of a fraction, for the point, what is this number?
Zero
