Below you will find example sentences with "complex numbers". The examples show how this phrase is used in natural context and which words often surround it.
Complex Numbers in a sentence
Corpus data
- Displayed example sentences: 20
- Discovered as a combination around: numbers
- Corpus frequency in the collocation scan: 14
- Phrase length: 2 words
- Average sentence length: 25.8 words
Sentence profile
- Phrase position: 7 start, 7 middle, 6 end
- Sentence types: 20 statements, 0 questions, 0 exclamations
Corpus analysis
- The phrase "complex numbers" has 2 words and usually appears near the start in these examples. The average sentence has 25.8 words and is mostly made up of statements.
- Around this phrase, patterns and context words such as in the complex numbers, amplitudes as complex numbers addition and, real, two and multiplication stand out.
- In the phrase index, this combination connects with real numbers, apartment complex, natural numbers, real numbers and security numbers, linking the page to nearby combinations.
Example types with complex numbers
This selection groups the examples by length and sentence type, making usage of the full phrase easier to scan:
Quaternions as pairs of complex numbers main Quaternions can be represented as pairs of complex numbers. (16 words)
The complex numbers consist of all numbers of the form : where a and b are real numbers. (17 words)
Even though H contains copies of the complex numbers, it is not an associative algebra over the complex numbers. (19 words)
Properties Square roots The two square roots of i in the complex plane The three cube roots of i in the complex plane i has two square roots, just like all complex numbers (except zero, which has a double root). (40 words)
Equivalently, it is a function space whose elements are functions from the natural numbers to the field K, where K is either the field of real numbers or the field of complex numbers. (33 words)
Addition of probability amplitudes as complex numbers Multiplication of probability amplitudes as complex numbers Addition and multiplication are familiar operations in the theory of complex numbers and are given in the figures. (32 words)
Example sentences (20)
Addition of probability amplitudes as complex numbers Multiplication of probability amplitudes as complex numbers Addition and multiplication are familiar operations in the theory of complex numbers and are given in the figures.
Hypercomplex numbers main Some number systems that are not included in the complex numbers may be constructed from the real numbers in a way that generalize the construction of the complex numbers.
In the theory of numbers and complex quantities, he was the first to define complex numbers as pairs of real numbers.
Matrix representation of complex numbers Complex numbers a + bi can also be represented by 2 × 2 matrices that have the following form: : Here the entries a and b are real numbers.
Complex functions A complex function is one in which the independent variable and the dependent variable are both complex numbers.
Hyperbolic functions for complex numbers Since the exponential function can be defined for any complex argument, we can extend the definitions of the hyperbolic functions also to complex arguments.
Properties Square roots The two square roots of i in the complex plane The three cube roots of i in the complex plane i has two square roots, just like all complex numbers (except zero, which has a double root).
Even though H contains copies of the complex numbers, it is not an associative algebra over the complex numbers.
In symbols: : Ordering Because complex numbers are naturally thought of as existing on a two-dimensional plane, there is no natural linear ordering on the set of complex numbers.
More generally the absolute value of the difference of two complex numbers is equal to the distance between those two complex numbers.
Quaternions as pairs of complex numbers main Quaternions can be represented as pairs of complex numbers.
Since the positive reals form a subgroup of the complex numbers under multiplication, we may think of absolute value as an endomorphism of the multiplicative group of the complex numbers. citation.
The geometric description of the multiplication of complex numbers can also be expressed in terms of rotation matrices by using this correspondence between complex numbers and such matrices.
Equivalently, it is a function space whose elements are functions from the natural numbers to the field K, where K is either the field of real numbers or the field of complex numbers.
In this way, the complex numbers contain the ordinary real numbers while extending them in order to solve problems that cannot be solved with real numbers alone.
Order properties of the real numbers The real numbers have several important lattice-theoretic properties that are absent in the complex numbers.
Subclasses of the complex numbers Algebraic, irrational and transcendental numbers Algebraic numbers are those that are a solution to a polynomial equation with integer coefficients.
The complex numbers consist of all numbers of the form : where a and b are real numbers.
The set of the p-adic numbers contains the rational numbers, but is not contained in the complex numbers.
According to the fundamental theorem of algebra all solutions of equations in one unknown with complex coefficients are complex numbers, regardless of degree.