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Greatest integer less than or equal to square root
integer square root (isqrt) of a non-negative integer n is the non-negative integer m which is the greatest integer less than or equal to the square root
Integer_square_root
Algorithms for calculating square roots
In some applications, an integer square root is required, which is the square root rounded or truncated to the nearest integer (a modified procedure may
Square_root_algorithms
Number whose square is a given number
the square root of numbers having many digits. It was known to the ancient Greeks that square roots of positive integers that are not perfect squares are
Square_root
Unique positive real number which when multiplied by itself gives 2
A002193 in the On-Line Encyclopedia of Integer Sequences consists of the digits in the decimal expansion of the square root of 2, here truncated to 30 decimal
Square_root_of_2
Product of an integer with itself
real number system, square numbers are non-negative. A non-negative integer is a square number when its square root is again an integer. For example, 9=3
Square_number
Root-finding algorithm
approximation through integer operations by adding and subtracting the integer form of floating-point numbers, and taking a square root by dividing by two
Fast_inverse_square_root
Positive real number which when multiplied by itself gives 7
expansion of square root of 7)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Robert Nemiroff; Jerry Bonnell (2008). The square root of 7.
Square_root_of_7
Arithmetic operation, inverse of nth power
an nth root of a number x is the number r which, when n copies are multiplied together, yields x: rn=r×r×⋯×r⏟n factors=x. The positive integer n is called
Nth_root
Principal square root of minus 1
On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Nahin, Paul J. (1998). An Imaginary Tale: The story of i [the square root of minus one]. Chichester:
Imaginary_unit
Irrational algebraic number
In mathematics, the square root of 10 is the positive real number that, when multiplied by itself, gives the number 10. It is approximately equal to 3
Square_root_of_10
Decomposition of a number into a product
decomposition of a positive integer into a product of integers. Every positive integer greater than 1 is either the product of two or more integer factors greater
Integer_factorization
Complex number whose mapping on a coordinate plane produces a triangular lattice
cube root of unity. The Eisenstein integers form a triangular lattice in the complex plane, in contrast with the Gaussian integers, which form a square lattice
Eisenstein_integer
Complex number that solves a monic polynomial with integer coefficients
theory, an algebraic integer is a complex number that is integral over the integers. That is, an algebraic integer is a complex root of some polynomial
Algebraic_integer
System that regulates the formation of blocks on a blockchain
usage scenario. Here is a list of known proof-of-work functions: Integer square root modulo a large prime[dubious – discuss] Weaken Fiat–Shamir signatures
Proof_of_work
Number with an integer power equal to 1
In mathematics, a root of unity is any complex number that yields 1 when raised to some positive integer power n. Roots of unity are used in many branches
Root_of_unity
Two-dimensional packing problem
half-integer, the wasted space is at least proportional to its square root. The precise asymptotic growth rate of the wasted space, even for half-integer side
Square_packing
Lattice group in Euclidean space whose points are integer n-tuples
n-tuples of integers. The two-dimensional integer lattice is also called the square lattice (or grid lattice) and the three-dimensional integer lattice is
Integer_lattice
Modular arithmetic concept
root modulo n if every number a coprime to n is congruent to a power of g modulo n. In symbols, g is a primitive root modulo n if for every integer a
Primitive_root_modulo_n
Unique positive real number which when multiplied by itself gives 8
the square root of 8 can be represented with a simple repeating pattern of integers: Because the square root of nine is a rational number, the square root
Square_root_of_8
Positive real number which when multiplied by itself gives 6
(ed.). "Sequence A010464 (Decimal expansion of square root of 6)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Robert Nemiroff; Jerry
Square_root_of_6
(except possibly at 0). Integer-valued functions defined on the domain of non-negative real numbers include the integer square root function and the prime-counting
Integer-valued_function
Positive real number which when multiplied by itself gives 5
The square root of 5, denoted 5 {\displaystyle {\sqrt {5}}} , is the positive real number that, when multiplied by itself, gives the natural number
Square_root_of_5
Determines the points needed for rasterizing a circle
{\displaystyle d_{n}^{2}} and the squared radius r 2 {\displaystyle r^{2}} to avoid having to compute an expensive and non-integer square-root. This also slightly changes
Midpoint_circle_algorithm
Unique positive real number which when multiplied by itself gives 3
The square root of 3 is the positive real number that, when multiplied by itself, gives the number 3. It is denoted mathematically as 3 {\textstyle {\sqrt
Square_root_of_3
Matrix B such that B² equals a given matrix A
mathematics, the square root of a matrix extends the notion of square root from numbers to matrices. A matrix B is said to be a square root of A if the matrix
Square_root_of_a_matrix
Arithmetic operation
apply the square super-root twice: x = s s r t ( s s r t ( y x ) ) {\displaystyle x=\mathrm {ssrt} (\mathrm {ssrt} (y^{x}))} . For each integer n > 2, the
Tetration
Function that, applied twice, gives another function
In mathematics, a functional square root (sometimes called a half iterate) is a square root of a function with respect to the operation of function composition
Functional_square_root
Root of a quadratic polynomial with a unit leading coefficient
quadratic integers are a generalization of the usual integers to quadratic fields. A complex number is called a quadratic integer if it is a root of some
Quadratic_integer
Geometric arrangements of points, foundational to Lie theory
angle between two roots is constrained to be one-half of the square root of a positive integer. This is because ⟨ β , α ⟩ {\displaystyle \langle \beta ,\alpha
Root_system
Product of two distinct primes ≡ 3 (mod 4)
Given n = p × q a Blum integer, Qn the set of all quadratic residues modulo n and coprime to n and a ∈ Qn. Then: a has four square roots modulo n, exactly
Blum_integer
Mathematical concept
irrationals to quadruples of integers, so their cardinality is at most countable; since on the other hand every square root of a prime number is a distinct
Quadratic_irrational_number
Number in {..., –2, –1, 0, 1, 2, ...}
example, 21, 4, 0, and −2048 are integers, while 9.75, 5+1/2, 5/4, and the square root of 2 are not. The integers form the smallest group and the smallest
Integer
Figurate number
formula. So an integer x is triangular if and only if 8x + 1 is a square. Equivalently, if the positive triangular root n of x is an integer, then x is the
Triangular_number
Natural number
nine Heegner numbers, or square-free positive integers n that yield an imaginary quadratic field ℚ[−n] whose ring of integers has a unique factorization
9
Product of a number by itself
place of x2. The square of an integer may also be called a square number or a perfect square. In algebra, the operation of squaring is often generalized
Square_(algebra)
Davenport–Schmidt theorem Irrational number Square root of two Quadratic irrational Integer square root Algebraic number Pisot–Vijayaraghavan number
List_of_number_theory_topics
Arithmetic operation
are, say, square matrices of the same size, this formula cannot be used. It follows that in computer algebra, many algorithms involving integer exponents
Exponentiation
Computational method
has complex roots, implies that a polynomial with integer coefficients can be factored (with root-finding algorithms) into linear factors over the complex
Factorization_of_polynomials
Problem in computer science
Turing run-time complexity of the square-root sum problem? More unsolved problems in computer science The square-root sum problem (SRS) is a computational
Square-root_sum_problem
Natural number
number of ordered rooted trees with n edges having root of even degree)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Sloane, N. J. A. (ed
1,000,000
Rational number equal to an integer plus 1/2
{n}{2}}+1)}}R^{n}~.} The values of the gamma function on half-integers are rational multiples of the square root of pi: Γ ( 1 2 + n ) = ( 2 n − 1 ) ! ! 2 n π
Half-integer
Number that is not a ratio of integers
irrationality of the square root of two can be generalized using the fundamental theorem of arithmetic. This asserts that every integer has a unique factorization
Irrational_number
Algorithm for finding zeros of functions
Aitken's delta-squared process Bisection method Euler method Fast inverse square root Fisher scoring Gradient descent Integer square root Kantorovich theorem
Newton's_method
Integer that is both a perfect square and a triangular number
sum of all integers from 1 {\displaystyle 1} to n {\displaystyle n} has a square root that is an integer. There are infinitely many square triangular
Square_triangular_number
Natural number
normal magic square, called the Luoshu square. All integers n ≥ 34 {\displaystyle n\geq 34} can be expressed as the sum of five non-zero squares. There are
5
Type of complex number
mathematics, an algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For
Algebraic_number
numbers, and include the quadratic surds. Algebraic integer: A root of a monic polynomial with integer coefficients. Transfinite numbers: Numbers that are
List_of_types_of_numbers
Mathematical proof technique
jumping, also known as root flipping, is a proof technique. It is most often used for problems in which a relation between two integers is given, along with
Vieta_jumping
Mathematical formula
the multiplicative digital root of n {\displaystyle n} . The multiplicative digital root for the first few positive integers are: 0, 1, 2, 3, 4, 5, 6,
Multiplicative_digital_root
Natural number
The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Sloane, N. J. A. (ed.). "Sequence A122400 (Number of square (0,1)-matrices without zero
1,000,000,000,000
Algorithm for computing greatest common divisors
polynomials. The Gaussian integers are complex numbers of the form α = u + vi, where u and v are ordinary integers and i is the square root of negative one. By
Euclidean_algorithm
Square root of a non-positive real number
Alexandria is noted as the first to present a calculation involving the square root of a negative number, it was Rafael Bombelli who first set down the rules
Imaginary_number
Repeated sum of a number's digits
a divisibility rule. Generally, the digital root computed in integer base b ≥ 2 of a non-negative integer n satisfies d r b ( n ) = { 0 n = 0 1 + ( (
Digital_root
Natural number
Encyclopedia of Integer Sequences. OEIS Foundation. Sloane, N. J. A. (ed.). "Sequence A001913 (Full reptend primes: primes with primitive root 10.)". The On-Line
300_(number)
Integer side lengths of a right triangle
triple because the square root of 2 is not an integer. Moreover, 1 {\displaystyle 1} and 2 {\displaystyle {\sqrt {2}}} do not have an integer common multiple
Pythagorean_triple
Condition under which an odd prime is a sum of two squares
sums of two squares states that an odd prime p can be expressed as: p = x 2 + y 2 , {\displaystyle p=x^{2}+y^{2},} with x and y integers, if and only
Fermat's theorem on sums of two squares
Fermat's_theorem_on_sums_of_two_squares
Natural number
The chemical element with atomic number 2 is helium. Binary number Square root of 2 −2 Colman, Samuel (1912). Coan, C. Arthur (ed.). Nature's Harmonic
2
Decomposition of an integer as a sum of positive integers
partition of a non-negative integer n, also called an integer partition, is a way of writing n as a sum of positive integers. Two sums that differ only
Integer_partition
Methods for locating real roots of a polynomial
polynomials with integer coefficients, and intervals ending with rational numbers. Also, the polynomials are always supposed to be square free. There are
Real-root_isolation
quadratic Frobenius test (EQFT). Let n be a positive integer such that n is odd, and let b and c be integers such that ( b 2 + 4 c n ) = − 1 {\displaystyle
Quadratic_Frobenius_test
One of several equivalent definitions of a computable function
more complicated way, since they are all primitive recursive. The integer square root of x can be defined as the least z such that ( z + 1 ) 2 > x {\displaystyle
General_recursive_function
Family of related bitwise operations on machine words
Warren. Chapter 11-1: Integer Square Root. Schlegel, Benjamin; Gemulla, Rainer; Lehner, Wolfgang [in German] (June 2010). "Fast integer compression using
Find_first_set
Algebraic construction
ring of all algebraic integers contained in K {\displaystyle K} . An algebraic integer is a root of a monic polynomial with integer coefficients: x n +
Ring_of_integers
Natural number
square of six, and the eighth triangular number or the sum of the first eight non-zero positive integers, which makes 36 the first non-trivial square
36_(number)
Integer
concept of abstract algebra generalizing integers and real numbers. Although there are no real number square roots of −1, the complex number i satisfies
−1
Number whose cube is a given number
real cube roots of 8 and −8 are respectively 2 and −2. The real cube root of an integer or of a rational number is generally not a rational number, nor a
Cube_root
(Mathematical) decomposition into a product
this case, the factorization can be done with root-finding algorithms. The case of polynomials with integer coefficients is fundamental for computer algebra
Factorization
Natural number
of the fact that 41 is a factor of 99999. It is the smallest integer whose square root has a simple continued fraction with period 3. It is the smallest
41_(number)
Function in mathematical number theory
Carmichael function λ(n) of a positive integer n is the smallest positive integer m such that am≡1(modn) holds for every integer a coprime to n. In algebraic terms
Carmichael_function
Type of mathematical expression
addition, subtraction, multiplication and exponentiation to nonnegative integer powers, and has a finite number of terms. An example of a polynomial of
Polynomial
Square of numbers with equal row, column and diagonal totals
historical and recreational mathematics, a magic square is a square array of numbers, usually positive integers, where the sums of the numbers in each row,
Magic_square
Scale to estimate the roughness of the sea for navigation
is the optional rounding to the closest integer value. Without the rounding to integer, the root mean square error of this approximation is: . Wavelength
Douglas_sea_scale
Natural number
104 or 1 E+4 (equivalently 1 E4) in E notation. It is the square of 100 and the square root of 100,000,000. The value of a myriad to the power of itself
10,000
Replacing a number with a simpler value
value of a function with integer domain and range. For example, if an integer n is known to be a perfect square, its square root can be computed by converting
Rounding
Used to count, measure, and label
real numbers such as the square root of 2 (2), and pi (π), and complex numbers which extend the real numbers with a square root of −1 (i), and its combinations
Number
Natural number
nodes 107,890,609 = Wedderburn-Etherington number 111,111,111 = repunit, square root of 12345678987654321 111,111,113 = Chen prime, Sophie Germain prime,
100,000,000
Number divisible only by 1 and itself
a given integer n is called trial division. This method divides n by each integer from 2 up to the square root of n. Any such integer dividing
Prime_number
Negative integer two units from the origin in mathematics
In mathematics, negative two or minus two is an integer two units from the origin, denoted as −2 or −2. It is the additive inverse of 2, following −3
−2
Number used for counting
are 0 (if included), 1, 2, 3, and so on. The terms positive integers, non-negative integers, whole numbers, and counting numbers are also used. The set
Natural_number
Geometric shape
the square root of an integer, such as √2, √3, etc. The root-2 rectangle (ACDK in Fig. 10) is constructed by extending two opposite sides of a square to
Dynamic_rectangle
Number of partitions of an integer
number of possible partitions of a non-negative integer n. For instance, p(4) = 5 because the integer 4 has the five partitions 1 + 1 + 1 + 1, 1 + 1 +
Partition function (number theory)
Partition_function_(number_theory)
Measure of the size of the ring of integers
measures the size of the (ring of integers of the) algebraic number field. More specifically, it is proportional to the squared volume of the fundamental domain
Discriminant of an algebraic number field
Discriminant_of_an_algebraic_number_field
Theorem on polynomial roots modulo prime powers
infinity, it follows that a root or a factorization modulo p can be lifted to a root or a factorization over the p-adic integers. These results have been
Hensel's_lemma
Natural number
of "4". Lagrange's four-square theorem states that every positive integer can be written as the sum of at most four squares. Four is one of four all-Harshad
4
Probabilistic primality test
{\displaystyle s} is a positive integer and d {\displaystyle d} is an odd positive integer. Let’s consider an integer a {\displaystyle a} , called a
Miller–Rabin_primality_test
Lemma concerning the limit of subadditive sequences
In mathematics, and in particular in calculus, Fekete's lemma (also called Fekete's subadditive lemma) is a lemma concerning the limit of subadditive sequences
Fekete's_lemma
Number, approximately 1.618
the sum of any fifth root of unity and its complex conjugate, z+z¯, is a golden integer, a number of the form a+bφ with integer coefficients a and
Golden_ratio
Number representing a continuous quantity
showed that π is not the square root of a rational number. Liouville (1840) showed that neither e nor e2 can be a root of an integer quadratic equation, and
Real_number
Natural number
number of ordered rooted trees with n edges having root of even degree)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. "Curium | Cm (Element)
10,000,000
Irreducible polynomial whose roots are nth roots of unity
polynomial with integer coefficients that is the minimal polynomial over the field of the rational numbers of any primitive nth-root of unity ( e 2 i
Cyclotomic_polynomial
Representation of modular integers by "small" fractions
integer may be represented by a "modular fraction" such that the numerator and the denominator have absolute values not greater than the square root of
Thue's_lemma
Natural number
number of ordered rooted trees with n edges having root of even degree)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Sloane, N. J. A. (ed
100,000
Mathematical proof technique using contradiction
discovery that the square root of two is irrational, and, according to legend, Hippasus was murdered for divulging it. The square root of two is occasionally
Proof_by_infinite_descent
Irrational numbers which appear to be rational
ratio of two integers. Transcendental numbers like e and π, and noninteger surds such as square root of 2 are irrational.) Almost integer Normal number
Schizophrenic_number
Integer factorization algorithm
25 because the square of the next prime is 49, and below n = 25 just 2 and 3 are sufficient. Should the square root of n be an integer, then it is a factor
Trial_division
Natural number
number of ordered rooted trees with n edges having root of even degree)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-06-01
7000_(number)
1617 device for calculating products and quotients
less than 11669900, so the root needs to be rounded up to 6840.0. To find the square root of a number that isn't an integer, say 54782.917, everything
Napier's_bones
Number with a real and an imaginary part
nonzero real number, whether positive or negative, squares to a positive number, taking the square root of a negative number was historically considered
Complex_number
Number raised to the third power
cube, since the cube of a negative integer is negative. For example, (−4) × (−4) × (−4) = −64. Unlike perfect squares, perfect cubes do not have a small
Cube_(algebra)
Integer that is a perfect square modulo some integer
theory, an integer q is a quadratic residue modulo n if it is congruent to a perfect square modulo n; that is, if there exists an integer x such that
Quadratic_residue
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INTEGER SQUARE-ROOT
INTEGER SQUARE-ROOT
INTEGER SQUARE-ROOT
INTEGER SQUARE-ROOT
INTEGER SQUARE-ROOT
INTEGER SQUARE-ROOT
INTEGER SQUARE-ROOT
INTEGER SQUARE-ROOT
INTEGER SQUARE-ROOT
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