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Problem in computer science
Unsolved problem in computer science What is the Turing run-time complexity of the square-root sum problem? More unsolved problems in computer science
Square-root_sum_problem
Number whose square is a given number
equations with continued fractions Square root algorithms Square-root sum problem Square-root method – Method of allocating voting weight by populationPages
Square_root
Figurate number
=n^{2}} with the sum being the square of the difference between the two (and thus the difference of the two being the square root of the sum): T n + T n −
Triangular_number
Linear combination of nth roots
computational complexity theory is the square-root sum problem, asking whether it is possible to determine the sign of a sum of square roots, with integer coefficients
Sum_of_radicals
Product of an integer with itself
prime is a sum of two squares Some identities involving several squares Integer square root – Greatest integer less than or equal to square root Methods
Square_number
Algorithms for calculating square roots
Square root algorithms compute the non-negative square root S {\displaystyle {\sqrt {S}}} of a positive real number S {\displaystyle S} . Since all square
Square_root_algorithms
List of unsolved computational problems
in polynomial time? Can the square-root sum problem be solved in polynomial time in the Turing machine model? Skolem problem: Is it decidable whether an
List of unsolved problems in computer science
List_of_unsolved_problems_in_computer_science
(Mathematical) decomposition into a product
guess a root of the polynomial. For example, for P ( x ) = x 3 − 3 x + 2 , {\displaystyle P(x)=x^{3}-3x+2,} one may easily see that the sum of its coefficients
Factorization
Approximation method in statistics
In regression analysis, least squares is a method to determine the best-fit model by minimizing the sum of the squared residuals—the differences between
Least_squares
Polynomial with no repeated root
In mathematics, a square-free polynomial is a univariate polynomial (over a field or an integral domain) that has no multiple root in an algebraically
Square-free_polynomial
Integer that is a perfect square modulo some integer
that finding a square root of a number modulo a large composite n is equivalent to factoring (which is widely believed to be a hard problem) has been used
Quadratic_residue
Condition under which an odd prime is a sum of two squares
In additive number theory, Fermat's theorem on 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}
Fermat's theorem on sums of two squares
Fermat's_theorem_on_sums_of_two_squares
Measure of distance between atoms of superimposed proteins
In bioinformatics, the root mean square deviation of atomic positions, or simply root mean square deviation (RMSD), is the measure of the average distance
Root mean square deviation of atomic positions
Root_mean_square_deviation_of_atomic_positions
Repeated sum of a number's digits
digital root (also repeated digital sum) of a natural number in a given radix is the (single digit) value obtained by an iterative process of summing digits
Digital_root
Set of problems solved by small circuits
best known bound for the square-root sum problem is in the fourth level of the counting hierarchy, and it is an unsolved problem whether better complexity
P/poly
Arithmetic operation, inverse of nth power
number x of which the root is taken is the radicand. A root of degree 2 is called a square root and a root of degree 3, a cube root. Roots of higher degree
Nth_root
Number with an integer power equal to 1
unity are not quadratic integers, but the sum of any root of unity with its complex conjugate (also an nth root of unity) is a quadratic integer. For n
Root_of_unity
Every natural number can be represented as the sum of four integer squares
Lagrange's four-square theorem, also known as Bachet's conjecture, states that every nonnegative integer can be represented as a sum of four non-negative
Lagrange's four-square theorem
Lagrange's_four-square_theorem
Product of a number by itself
distance Exponentiation by squaring Hilbert's seventeenth problem, for the representation of positive polynomials as a sum of squares of rational functions
Square_(algebra)
Geometric arrangements of points, foundational to Lie theory
{\displaystyle \Phi ^{+}} is called a simple root (also fundamental root) if it cannot be written as the sum of two elements of Φ + {\displaystyle \Phi
Root_system
Principal square root of minus 1
distinct complex-valued square roots, which are additive inverses of each other, while zero has only zero as its (double) square root. Historically, the imaginary
Imaginary_unit
Subfield of convex optimization
Reduce the size of the variable matrix. Square-root sum problem - a special case of an SDP feasibility problem. Gärtner, Bernd; Matoušek, Jiří (2012),
Semidefinite_programming
Square of numbers with equal row, column and diagonal totals
and recreational mathematics, a magic square is a square array of numbers, usually positive integers, where the sums of the numbers in each row, each column
Magic_square
Formula that provides the solutions to a quadratic equation
formula with the square root in the numerator or denominator depending on the sign of b {\displaystyle b} can avoid this problem. See § Numerical
Quadratic_formula
Economical computational problem
1/2 is at least as hard as the square-root sum problem, as well as a more general arithmetic circuit decision problem. Tsaknakis and Spirakis presented
Nash_equilibrium_computation
Divergent sum of positive unit fractions
infinite series formed by summing all positive unit fractions: ∑ n = 1 ∞ 1 n = 1 + 1 2 + 1 3 + 1 4 + 1 5 + ⋯ . {\displaystyle \sum _{n=1}^{\infty }{\frac
Harmonic_series_(mathematics)
Number of stacked spheres in a pyramid
part of a more general solution to the problem of finding formulas for sums of progressions of squares. The square pyramidal numbers were also one of the
Square_pyramidal_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
Integer that is both a perfect square and a triangular number
a square triangular number (or triangular square number) is a number which is both a triangular number and a square number, in other words, the sum of
Square_triangular_number
v\equiv 1{\bmod {4}}} and v {\displaystyle v} is an odd sum of two squares? Conway's 99-graph problem: does there exist a strongly regular graph with parameters
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
accuracy reasons. See Root Finding Methods for a summary of the existing methods available in each case. The root-finding problem of polynomials was first
Polynomial_root-finding
Length of a line segment
The two squared formulas inside the square root give the areas of squares on the horizontal and vertical sides, and the outer square root converts the
Euclidean_distance
Measure of variation in statistics
{\displaystyle s_{N}={\sqrt {{\frac {1}{N}}\sum _{i=1}^{N}\left(x_{i}-{\bar {x}}\right)^{2}}}.} Here taking the square root introduces further downward bias, by
Standard_deviation
equilibrium to any factor smaller than 1/2 is at least as hard as the square-root sum problem. Etessami, Kousha; Yannakakis, Mihalis (January 2010). "On the
FIXP
Relation between sides of a right triangle
includes a problem involving two squares whose areas sum to a third square, whose solution is the Pythagorean triple 6:8:10, but the problem does not mention
Pythagorean_theorem
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)
Mathematical relationships
\left({\frac {\sum _{i=1}^{n}x_{i}}{n}}\right)^{\!2}\leq {\frac {\sum _{i=1}^{n}x_{i}^{2}}{n}}} . For positive x i {\displaystyle x_{i}} the square root of this
QM–AM–GM–HM_inequalities
Number raised to the third power
Katherine (1 October 2004). "Proof without Words: The Sum of Cubes: An Extension of Archimedes' Sum of Squares". Mathematics Magazine. 77 (4): 298–299. doi:10
Cube_(algebra)
Hypotenuse of right triangle from its sides
implementations of this operation instead compute Pythagorean sums by reducing the problem to the square root function, they do so in a way that has been designed
Pythagorean_addition
theorem Hundred Fowls Problem 1729 Davenport–Schmidt theorem Irrational number Square root of two Quadratic irrational Integer square root Algebraic number
List_of_number_theory_topics
Curves whose limit does not preserve length
uniformly to the diagonal of the square. However, each staircase has length two, while the length of the diagonal is the square root of 2 (approximately 1.4142)
Staircase_paradox
Natural number
divisible by 9 if and only if the sum of its digits is divisible by 9. 9 is the only square number that is the sum of two consecutive, positive cubes:
9
Type of figurate number constructed by combining heptagons
{\displaystyle x={\frac {5n^{2}-3n}{2}}} for its unique positive root n. "Beyond the Basel Problem: Sums of Reciprocals of Figurate Numbers" (PDF). Archived from
Heptagonal_number
Statistical principle about ratio of effects to causes
is known as the square-root-of-the-sum-of-the-squares axiom. This states that the variation caused by the steepest slope must be squared, and then the result
Pareto_principle
Fixed number that has received a name
encounter during pre-college education in many countries. The square root of 2, often known as root 2 or Pythagoras' constant, and written as √2, is the unique
Mathematical_constant
Number divisible only by 1 and itself
first major result, the solution to the Basel problem. The problem asked for the value of the infinite sum 1 + 1 4 + 1 9 + 1 16 + … , {\displaystyle 1+{\tfrac
Prime_number
Methods of constructing magic squares
and column sums will be identical and result in a magic sum, whereas the diagonal sums will differ. The result will thus be a semimagic square and not a
Constructions of magic squares
Constructions_of_magic_squares
Result of multiplying six instances of a number
are simultaneously square and triangular, and the solutions to the cannonball problem, which are simultaneously square and square-pyramidal. Because of
Sixth_power
Indicator for how well data points fit a line or curve
goodness of fit. The norm of residuals is calculated as the square-root of the sum of squares of residuals (SSR): norm of residuals = S S res = ‖ e ‖ .
Coefficient_of_determination
Type of algorithm
calculating the optimal rotation matrix that minimizes the RMSD (root mean squared deviation) between two paired sets of points. It is useful for point-set
Kabsch_algorithm
Number equal to the sum of its proper divisors
particular the digital root of every even perfect number other than 6 is 1. The only square-free perfect number is 6. The sum of proper divisors gives
Perfect_number
1617 device for calculating products and quotients
above, 6839925 is 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
Napier's_bones
Correction for sample variance bias
variance, its square root, the sample standard deviation, is a biased estimate of the population standard deviation; because the square root is a concave
Bessel's_correction
Statistical measure of how far values spread from their average
the expected value of the squared deviation from the mean of a random variable. The standard deviation is the square root of the variance. Technically
Variance
Figurate number
pentagonal square triangular number problem can be reduced to solving the equation: x 2 − 6 y 2 = − 5 {\displaystyle x^{2}-6y^{2}=-5} This places the problem within
Pentagonal_number
Recursive integer sequence
n 2 2 n + 1 = 1. {\displaystyle \sum _{n=0}^{\infty }{\frac {C_{n}}{2^{2n+1}}}=1.} There are many counting problems in combinatorics whose solution is
Catalan_number
Existence of a prime number between each square and pronic number
Unsolved problem in mathematics Is every pair of a square number and a pronic number (both greater than one) separated by at least one prime? More unsolved
Oppermann's_conjecture
Algorithm in numerical analysis
worst-case error that grows proportional to n {\displaystyle n} , and a root mean square error that grows as n {\displaystyle {\sqrt {n}}} for random inputs
Kahan_summation_algorithm
Method for estimating the unknown parameters in a linear regression model
in a linear regression model by the principle of least squares: minimizing the sum of the squares of the differences between the observed dependent variable
Ordinary_least_squares
Irrational number irrationality of log23 irrationality of the square root of 2 Mathematical induction sum identity Power rule differential of xn Product and Quotient
List_of_mathematical_proofs
that the number of elements in a square-difference-free set of positive integers could only exceed the square root of its largest value by a polylogarithmic
List of conjectures by Paul Erdős
List_of_conjectures_by_Paul_Erdős
Babylonian clay tablet of numbers in Pythagorean triples
s^{2}+l^{2}=d^{2}} , the rule that equates the sum of the squares of the legs of a right triangle to the square of the hypotenuse. The era in which Plimpton
Plimpton_322
Sequence in computer science
functional programming languages. Prefix sums have also been much studied in parallel algorithms, both as a test problem to be solved and as a useful primitive
Prefix_sum
Positive integer of the form (2^(2^n))+1
multiply this by a number A, which is greater than the square root of P and is a primitive root modulo P (i.e., it is not a quadratic residue). Then take
Fermat_number
Modular arithmetic concept
is a primitive 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
Primitive_root_modulo_n
Deliberate process that transforms inputs to outputs with variable change
multiplying 7 by 6 is a simple algorithmic calculation. Extracting the square root or the cube root of a number using mathematical models is a more complex algorithmic
Calculation
Indian mathematician and astronomer (598–668)
23. The square-root of the sum of the two products of the sides and opposite sides of a non-unequal quadrilateral is the diagonal. The square of the diagonal
Brahmagupta
Mathematical expression with outer and inner radicals
a nested radical is a radical expression (one containing a square root sign, cube root sign, etc.) that contains (nests) another radical expression
Nested_radical
Method for solving quadratic equations
(x-h)^{2}} and taking the square root, a quadratic problem can be reduced to a linear problem. The name completing the square comes from a geometrical
Completing_the_square
Type of vector space in math
notion of magnitude, the complex modulus |z|, which is defined as the square root of the product of z with its complex conjugate: | z | 2 = z z ¯ . {\displaystyle
Hilbert_space
Number that remains the same when its digits are reversed
receive most attention in the realm of recreational mathematics. A typical problem asks for numbers that possess a certain property and are palindromic. For
Palindromic_number
Arithmetic operation
) 2 = b {\displaystyle (b^{1/2})^{2}=b} , which is the definition of square root: b 1 / 2 = b {\displaystyle b^{1/2}={\sqrt {b}}} . The definition of
Exponentiation
Algorithm to multiply two numbers
signals. In this application, the sum and difference of two input voltages are formed using operational amplifiers. The square of each of these is approximated
Multiplication_algorithm
that the area of the square on the side of a regular pentagon inscribed in a circle is equal to the sum of the areas of the squares on the sides of the
List of trigonometric identities
List_of_trigonometric_identities
Equation for radii of tangent circles
inverse radii) of the four circles: The sum of the squares of all four bends Is half the square of their sum Special cases of the theorem apply when one
Descartes'_theorem
Problem in statistical estimation
choosing from H possible outputs. This square root corresponds to half the digits. For example, in any base, the square root of a number with 100 digits is approximately
German_tank_problem
Polynomial equation of degree 3
only one root. This allows computing the multiple root, and the third root can be deduced from the sum of the roots, which is provided by Vieta's formulas
Cubic_equation
Shape with four equal sides and angles
term squaring to mean raising any number to the second power. Reversing this relation, the side length of a square of a given area is the square root of
Square
Natural number
cannot be expressed as a sum of two abundant numbers 20230 = pentagonal pyramidal number 20412 = Leyland number: 93 + 39 20540 = square pyramidal number 20569
20,000
Euler's sum of powers conjecture – disproved for exponents 4 and 5 during the 20th century; unsolved for higher exponents Euler's Graeco-Latin square conjecture
List of topics named after Leonhard Euler
List_of_topics_named_after_Leonhard_Euler
equation with integer coefficients. Evaluating the square of the sum P is connected with the problem of counting how many quadratic residues between 1
Gaussian_period
Calculating method used in ancient China
approximation of the square root 234567 ≈ 484 311 968 {\displaystyle {\sqrt {234567}}\approx 484{\tfrac {311}{968}}} from the algorithm in chap 2 problem 19 of Sunzi
Rod_calculus
Computational method
square root of the same value. The generalized math summary is: n = order of the squared polynomial being factored m = order of the extracted square root
Factorization_of_polynomials
Mathematical formula involving a given set of operations
Commonly, the basic functions that are allowed in closed forms are nth root, exponential function, logarithm, and trigonometric functions. However, the
Closed-form_expression
Type of equation involving matrix-valued functions
0 m λ i A i . {\displaystyle M(\lambda )=\sum _{i=0}^{m}\lambda ^{i}A_{i}.} The rational eigenvalue problem: M ( λ ) = ∑ i = 0 m 1 A i λ i + ∑ i = 1 m
Nonlinear_eigenproblem
Term in quantum mechanics
unique positive square root of ρ and | ψ ρ ⟩ = ∑ i = 1 n ( ρ 1 / 2 | e i ⟩ ) ⊗ | e i ⟩ ∈ C n ⊗ C n {\displaystyle |\psi _{\rho }\rangle =\sum _{i=1}^{n}(\rho
Fidelity_of_quantum_states
Type of function in database management
the square root of the difference between the average of the squares of the values and the square of the average value. STDDEV ( X ⊎ Y ) = SUM ( X
Aggregate_function
Lattice in 8-dimensional space with special properties
lattice of rank 8. The name derives from the fact that it is the root lattice of the E8 root system. The norm of the E8 lattice (divided by 2) is a positive
E8_lattice
Mathematical series with a finite sum
In mathematics, a series is the sum of the terms of an infinite sequence of numbers. More precisely, an infinite sequence ( a 1 , a 2 , a 3 , … ) {\displaystyle
Convergent_series
eighth problem of the second book of Arithmetica by Diophantus (c. 200/214 AD – c. 284/298 AD) is to divide a square into a sum of two squares. Diophantus
Diophantus_II.VIII
Multiplication algorithm
method to divide the problem into subproblems. c k = ∑ ( i , j ) : i + j ≡ k ( mod N ( n ) ) a i b j {\displaystyle c_{k}=\sum _{(i,j):i+j\equiv k{\pmod
Schönhage–Strassen_algorithm
Sum of elements on the main diagonal
In linear algebra, the trace of a square matrix A, denoted tr(A), is defined as a sum of the elements on its main diagonal, a 11 + a 22 + ⋯ + a n n {\displaystyle
Trace_(linear_algebra)
Mathematical operation that predicts future values of a discrete-time signal
e(n)=x(n)-{\widehat {x}}(n)=x(n)-\sum _{i=1}^{p}a_{i}x(n-i)=-\sum _{i=0}^{p}a_{i}x(n-i)} where the optimisation problem searching over all a i {\displaystyle
Linear_prediction
Numbers whose prime factors all divide the number more than once
powerful numbers in the interval [1,x]. Then k(x) is proportional to the square root of x. More precisely, c x 1 / 2 − 3 x 1 / 3 ≤ k ( x ) ≤ c x 1 / 2 , c
Powerful_number
Sum type in number theory
In number theory, quadratic Gauss sums are certain finite sums of roots of unity. A quadratic Gauss sum can be interpreted as a linear combination of
Quadratic_Gauss_sum
Negative integer two units from the origin in mathematics
positive; the inverse-square law; grid turbulence decay; and the Basel problem. The Basel problem states that the sum of the square reciprocals of natural
−2
Noncentral generalization of the chi-squared distribution
{\displaystyle \lambda } in other ways, such as half of the above sum, or its square root. This distribution arises in multivariate statistics as a derivative
Noncentral chi-squared distribution
Noncentral_chi-squared_distribution
In number theory, a kth root of unity modulo n for positive integers k, n ≥ 2, is a root of unity in the ring of integers modulo n; that is, a solution
Root_of_unity_modulo_n
Every graph has evenly many odd vertices
Seven Bridges of Königsberg Problem, which subsequently formalized Eulerian Tours, other applications of the degree sum formula include proofs of certain
Handshaking_lemma
Number used to approximate the square root of 2
comprise the denominators of the closest rational approximations to the square root of 2. This sequence of approximations begins 1/1, 3/2, 7/5, 17/12
Pell_number
SQUARE ROOT-SUM-PROBLEM
SQUARE ROOT-SUM-PROBLEM
Surname or Lastname
Dutch (also de Roos) and Swiss German
Dutch (also de Roos) and Swiss German : habitational name for someone living at a house distinguished by the sign of a rose.Dutch (also de Roos) : metonymic occupational name for someone who grew roses, from roos ‘rose’.Dutch : from the female personal name Rosa (Latin rosa ‘rose’).Dutch : nickname from roos ‘erysipelas’, an infection which causes reddening of the skin and scalp, applied presumably to someone with a ruddy complexion.Swiss German : from a personal name formed with hrÅd ‘renown’.Swedish and Danish (of German origin) : as 1.Swedish : variant of Ros.English and Scottish : variant of Ross 2.
Girl/Female
Indian, Kannada, Korean, Telugu
The Sun; Obedient
Surname or Lastname
English
English : patronymic from Squire.
Male
Swedish
Swedish name derived from Old Norse stúra, STURE means "obstinate."
Boy/Male
American, Australian, British, English
Shield Bearer; Knight's Companion
Boy/Male
Hindu, Indian, Indonesian, Kenyan
Root
Girl/Female
Australian, Danish, Swedish
Sun
Surname or Lastname
English
English : metonymic occupational name for a maker or seller of boots, from Middle English, Old French bote (of unknown origin).Dutch and North German : metonymic occupational name for a boatman, from Dutch boot ‘boat’.
Surname or Lastname
English
English : status name from Middle English squyer ‘esquire’, ‘a man belonging to the feudal rank immediately below that of knight’ (from Old French esquier ‘shield bearer’). At first it denoted a young man of good birth attendant on a knight, or by extension any attendant or servant, but by the 14th century the meaning had been generalized, and referred to social status rather than age. By the 17th century, the term denoted any member of the landed gentry, but this is unlikely to have influenced the development of the surname.
Male
English
French form of English Stewart, STUART means "house guard; steward." In use by the English and Scottish.
Girl/Female
African, Hindu, Indian, Sanskrit
First Root; Sun
Boy/Male
English American
Shieldbearer.
Surname or Lastname
English
English : nickname for a frugal person, from Middle English spare ‘sparing’, ‘frugal’.
Boy/Male
Egyptian
Root.
Boy/Male
Australian, Biblical, Danish, German, Swedish
Mame; Renown; Sun Child; Little Sun
Boy/Male
Hebrew American
Sun child; bright sun.
Surname or Lastname
English
English : patronymic from Root 1.
Boy/Male
Hindu, Indian, Marathi
Fragrance; Flower; Sum; Total
Surname or Lastname
English
English : nickname for a cheerful person, from Middle English rote ‘glad’ (Old English rÅt).English : metonymic occupational name for a player on the rote, an early medieval stringed instrument (Middle English, Old French rote, of uncertain origin but apparently ultimately akin to Welsh crwth).Dutch : topographic name for someone who lived by a retting place (Dutch root, a derivative of ro(o)ten ‘to ret’, akin to modern English rot), a place where flax is soaked in tubs of water until the stems rot to release the linen fibers.
Surname or Lastname
English
English : variant of Squire.
SQUARE ROOT-SUM-PROBLEM
SQUARE ROOT-SUM-PROBLEM
Girl/Female
Hindu, Indian, Marathi
Delightful; Beloved
Biblical
there is God;
Boy/Male
Muslim
Honorable, Outstanding
Boy/Male
Arabic, Australian, Muslim
Happy
Girl/Female
Hindu
Highly knowledge
Boy/Male
Indian, Telugu
Peaceful; Melodious; Good Sound; Musical; Music
Boy/Male
Arabic, Indian, Muslim
Unique
Boy/Male
Arabic, Muslim
Owner of the Two Horns; World Conqueror; Epithet of a Just King Mentioned in the Quran
Female
Finnish
Finnish form of Greek Eva, EEVI means "life."
Boy/Male
Hindu
SQUARE ROOT-SUM-PROBLEM
SQUARE ROOT-SUM-PROBLEM
SQUARE ROOT-SUM-PROBLEM
SQUARE ROOT-SUM-PROBLEM
SQUARE ROOT-SUM-PROBLEM
n.
A square piece or fragment.
n.
An edible or esculent root, especially of such plants as produce a single root, as the beet, carrot, etc.; as, the root crop.
n.
The principal points or thoughts when viewed together; the amount; the substance; compendium; as, this is the sum of all the evidence in the case; this is the sum and substance of his objections.
n.
Having the toe square.
n.
Hence, anything which is square, or nearly so
n.
An instrument having at least one right angle and two or more straight edges, used to lay out or test square work. It is of several forms, as the T square, the carpenter's square, the try-square., etc.
a.
Having four equal sides and four right angles; as, a square figure.
a.
Rendering equal justice; exact; fair; honest, as square dealing.
n.
A square. See 1st Squire.
n.
To place at right angles with the keel; as, to square the yards.
n.
To multiply by itself; as, to square a number or a quantity.
a.
Even; leaving no balance; as, to make or leave the accounts square.
a.
Full of roots; as, rooty ground.
a.
Forming a right angle; as, a square corner.
n.
A quantity of money or currency; any amount, indefinitely; as, a sum of money; a small sum, or a large sum.
imp. & p. p.
of Square
v. t.
To sum up, as the numbers in a column; -- sometimes with up; as, to foot (or foot up) an account.
n.
A square; a measure; a rule.
v. i.
To fix the root; to enter the earth, as roots; to take root and begin to grow.