In mathematics, an algebraic number is a number that is a root of a non-zero polynomial in one variable with rational coefficients ( or equivalently — by clearing denominators — with integer coefficients ).

The geometric interpretation of curl as rotation corresponds to identifying bivectors ( 2-vectors ) in 3 dimensions with the special orthogonal Lie algebra so ( 3 ) of infinitesimal rotations ( in coordinates, skew-symmetric 3 × 3 matrices ), while representing rotations by vectors corresponds to identifying 1-vectors ( equivalently, 2-vectors ) and so ( 3 ), these all being 3-dimensional spaces.

While other forms of church government are more likely to define " tyranny " as " the imposition of unjust rule ", a congregationally-governed church would more likely define tyranny as " transgression of liberty " or equivalently, " rule by one man ".

To establish that a function is computable by Turing machine, it is usually considered sufficient to give an informal English description of how the function can be effectively computed, and then conclude " By the Church – Turing thesis " that the function is Turing computable ( equivalently partial recursive ).

One can use the same principle to specify the position of any point in three-dimensional space by three Cartesian coordinates, its signed distances to three mutually perpendicular planes ( or, equivalently, by its perpendicular projection onto three mutually perpendicular lines ).

The depth of the study of general Diophantine equations is shown by the characterisation of Diophantine sets as equivalently described as recursively enumerable.

Note that this derivation does assume that ƒ ( x ) is sufficiently differentiable and well-behaved ; specifically, that ƒ may be approximated by polynomials ; equivalently, that ƒ is a real analytic function of exponential type.

* The defining property of unary idempotence, for x in the domain of f, can equivalently be rewritten as, using the binary operation of function composition denoted by ∘.

* Uniformizability, the property of a topological space of being homeomorphic to a uniform space, or equivalently the topology being defined by a family of pseudometrics

Or, equivalently, in terms of the wave's angular frequency ω, which specifies the number of oscillations per unit of time, and wavenumber k, which specifies the number of oscillations per unit of space, by

Refraction is described by Snell's law, which states that for a given pair of media and a wave with a single frequency, the ratio of the sines of the angle of incidence θ < sub > 1 </ sub > and angle of refraction θ < sub > 2 </ sub > is equivalent to the ratio of phase velocities ( v < sub > 1 </ sub > / v < sub > 2 </ sub >) in the two media, or equivalently, to the opposite ratio of the indices of refraction ( n < sub > 2 </ sub > / n < sub > 1 </ sub >):

Alternate definitions are, " the entropy of all systems and of all states of a system is smallest at absolute zero ," or equivalently " it is impossible to reach the absolute zero of temperature by any finite number of processes ".

We can work the other way and start by choosing < as a transitive trichotomous binary relation ; then a total order ≤ can equivalently be defined in two ways:

Let ρ be the initial topology on X induced by C < sub > τ </ sub >( X ) or, equivalently, the topology generated by the basis of cozero sets in ( X, τ ).

In effect, helium is a faster gas to saturate and desaturate, which is a distinct advantage in saturation diving, but less so in bounce diving, where the increased rate of off-gassing is largely counterbalanced by the equivalently increased rate of on-gassing.

A digital filter is characterized by its transfer function, or equivalently, its difference equation.

Right: The reduction in flux passing through a surface can be visualized by reduction in F or dS equivalently ( resolved into Euclidean vector # Decomposition | components, θ is angle to normal n ).

The elementary charge, the charge of a proton ( equivalently, the negative of the charge of an electron ), is approximately .< ref name = CODATA > The inverse value ( the number of elementary charges in 1C ) is given by 1 / = 6. 24150965 ( 16 )× 10 < sup > 18 </ sup >.</ ref > In SI, the elementary charge in coulombs is an approximate value: no experiment can be infinitely accurate.

if is divisible by ( or equivalently if and have the same remainder when divided by ).

since is a multiple of 10, or equivalently since both and have a remainder of when divided by.

An ideal low-pass filter can be realized mathematically ( theoretically ) by multiplying a signal by the rectangular function in the frequency domain or, equivalently, convolution with its impulse response, a sinc function, in the time domain.

is an Icelandic ( and equivalently Old Norse ) term consisting of two parts.

* The number of partitions of n in which all parts are 1 or 2 ( or, equivalently, the number of partitions of n into 1 or 2 parts ) is

* The number of partitions of n in which all parts are 1, 2 or 3 ( or, equivalently, the number of partitions of n into 1, 2 or 3 parts ) is the nearest integer to ( n + 3 )< sup > 2 </ sup > / 12.

A partition of a set S is a pairwise disjoint set of non-empty subsets, called " parts " or " blocks ", whose union is all of S. Consider a finite set that is linearly ordered, or ( equivalently, for purposes of this definition ) arranged in a cyclic order like the vertices of a regular n-gon.

The course has two entrances known in typical British understatement as ' top ' and ' junction ' respectively ; and two parts, or banks, known as ' upper ' and ' lower ' or equivalently, ' bottom '.

or, equivalently, the Wirtinger derivative of ƒ with respect to the complex conjugate of z is zero:

Let ƒ be the fraction of the circumference subtended by the golden angle, or equivalently, the golden angle divided by the angular measurement of the circle.

One reason for the popularity of the dissociation constant in biochemistry and pharmacology is that in the frequently encountered case where x = y = 1, K < sub > d </ sub > has a simple physical interpretation: when = K < sub > d </ sub >, = or equivalently /(+)= 1 / 2.

Matching the theory's prediction to observational results for planetary orbits ( or, equivalently, assuring that the weak-gravity, low-speed limit is Newtonian mechanics ), the proportionality constant can be fixed as κ = 8πG / c < sup > 4 </ sup >, with G the gravitational constant and c the speed of light.

Under Galilean transformations, the time t < sub > 2 </ sub > − t < sub > 1 </ sub > between two events is the same for all inertial reference frames and the distance between two simultaneous events ( or, equivalently, the length of any object, | r < sub > 2 </ sub > − r < sub > 1 </ sub >|) is also the same.

Chemistry textbooks quote the electron configuration of nickel as 4s < sup > 2 </ sup > 3d < sup > 8 </ sup >, or equivalently as 3d < sup > 8 </ sup > 4s < sup > 2 </ sup >.

For a preorder "", a relation "<" can be defined as a < b if and only if ( a b and not b a ), or equivalently, using the equivalence relation introduced above, ( a b and not a ~ b ).

The claim is negated to assume that a + b ≤ c. Squaring both sides results in ( a + b )< sup > 2 </ sup > ≤ c < sup > 2 </ sup > or, equivalently, a < sup > 2 </ sup > + 2ab + b < sup > 2 </ sup > ≤ c < sup > 2 </ sup >.

The electric field is a vector field with SI units of newtons per coulomb ( N C < sup >− 1 </ sup >) or, equivalently, volts per metre ( V m < sup >− 1 </ sup >).

The condition Ext < sup > 1 </ sup >( A, Z ) = 0 can be equivalently formulated as follows: whenever B is an abelian group and f: B → A is a surjective group homomorphism whose kernel is isomorphic to the group of integers Z, then there exists a group homomorphism g: A → B with fg = id < sub > A </ sub >.

We can also reparametrize geodesics to be unit speed, so equivalently we can define exp < sub > p </ sub >( v ) = β (| v |) where β is the unit-speed geodesic ( geodesic parameterized by arc length ) going in the direction of v. As we vary the tangent vector v we will get, when applying exp < sub > p </ sub >, different points on M which are within some distance from the base point p — this is perhaps one of the most concrete ways of demonstrating that the tangent space to a manifold is a kind of " linearization " of the manifold.

* a global function field, i. e., the function field of an algebraic curve over a finite field, equivalently, a finite extension of F < sub > q </ sub >( T ), the field of rational functions in one variable over the finite field with q elements.

If derivatives f < sup >( n )</ sup > exist for all positive integers n, the function is smooth or, equivalently, of class C < sup >∞</ sup >.

and showed that the coefficient ring π < sub >*</ sub >( MU ) ( equal to the complex cobordism of a point, or equivalently the ring of cobordism classes of stably complex manifolds ) is a polynomial ring Zx < sub > 2 </ sub >,... on infinitely many generators x < sub > i </ sub > ∈ π < sub > 2i </ sub >( MU ) of positive even degrees.

where I ranges through the non-zero ideals of the ring of integers O < sub > K </ sub > of K and N < sub > K / Q </ sub >( I ) denotes the absolute norm of I ( which is equal to both the index of I in O < sub > K </ sub > or equivalently the cardinality of quotient ring O < sub > K </ sub > / I ).