| 单词 | complex number |
| 例句 | And while many mathematicians saw the complex numbers as a convenient fiction, others saw God. Zero: The Biography of a Dangerous Idea 2000-02-07T00:00:00Z Take any complex number and measure its angle. Zero: The Biography of a Dangerous Idea 2000-02-07T00:00:00Z Riemann merged projective geometry with the complex numbers, and all of a sudden lines became circles, circles became lines, and zero and infinity became the poles on a globe full of numbers. Zero: The Biography of a Dangerous Idea 2000-02-07T00:00:00Z In the 1830s Gauss realized that each complex number—numbers that have real and imaginary parts, like 1 – 2i—can be displayed on a Cartesian grid. Zero: The Biography of a Dangerous Idea 2000-02-07T00:00:00Z As early as the sixteenth century, mathematicians were using numbers with i included—the so-called complex numbers—to solve cubic and quartic polynomials. Zero: The Biography of a Dangerous Idea 2000-02-07T00:00:00Z The horizontal axis represents the real part of the complex number, while the vertical axis represents the imaginary part. Zero: The Biography of a Dangerous Idea 2000-02-07T00:00:00Z At seventeen, she won first place in the Westinghouse Science Talent Search, for a project on perfect Gaussian integers, a type of complex number. The Dark Energy of a Theoretical Physicist 2016-10-29T04:00:00Z It is a half-hour mystical work with a mathematical title that has something to do with complex numbers and four-dimensional space. Music ensembles Hocket and Sakura bathe Monk Space in an acoustical glow 2015-10-15T04:00:00Z The revamped "Spider-Man" will not just be a rewriting of the story, it will be stripping away some of the more complex numbers and cutting the shows costs in order to please the long-worried investors. Will anyone care about the revamped "Spider-Man" musical? 2011-04-15T16:24:00Z Tell Facebook you have solved the elusive Riemann hypothesis, proving conclusively that the zeta function has its zeros only at the negative even integers and complex numbers. Perspective | Gene Weingarten: What to do with that half-hour before a nuclear missile strikes 2018-02-13T05:00:00Z Although the band couldn’t reproduce every flourish of the recorded versions, it performed such complex numbers as “Have One on Me” as capably as it did the comparatively straightforward “Peach Plum Pear.” Joanna Newsom, unconventional in every way 2015-12-11T05:00:00Z Before the 20th century all such operations with complex numbers were simply considered a mathematical trick. Quantum Physics Falls Apart without Imaginary Numbers 2023-04-24T04:00:00Z Before multiplying, each square root of a negative number must be written as a complex number. Intermediate Algebra 2020-05-06T00:00:00Z Lines connected the dots to represent the gates, with each gate encoded in a tensor—a 2D or 4D grid of complex numbers. Ordinary computers can beat Google’s quantum computer after all 2022-08-02T04:00:00Z Let’s consider the number The real part of the complex number is and the imaginary part is We plot the ordered pair to represent the complex number as shown in Figure 1. Precalculus 2021-12-01T00:00:00Z The Linear Factorization Theorem tells us that a polynomial function will have the same number of factors as its degree, and that each factor will be in the form where is a complex number. Algebra and Trigonometry 2015-02-13T00:00:00Z Physical properties, such as the speed of a particle or its position, correspond to tables of complex numbers called operators. Quantum Physics Falls Apart without Imaginary Numbers 2023-04-24T04:00:00Z A complex number is in standard form when written as a + bi, where a and b are real numbers. Intermediate Algebra 2020-05-06T00:00:00Z Emphasizing problem-based learning, the course covers topics students need in sophomore engineering classes, including linear equations, quadratic equations, 2-D vectors and complex numbers. To Keep Students in STEM fields, Let’s Weed Out the Weed-out Math Classes 2022-03-15T04:00:00Z To divide complex numbers, multiply both the numerator and denominator by the complex conjugate of the denominator to eliminate the complex number from the denominator. Precalculus 2021-12-01T00:00:00Z When any complex number with an imaginary component is given as a zero of a polynomial with real coefficients, the conjugate must also be a zero of the polynomial. Algebra and Trigonometry 2015-02-13T00:00:00Z We had observed that, in such a scenario, which physicists call a quantum network, the Pál-Vértesi-McKague-Mosca-Gisin method could not reproduce the experimental outcomes predicted by complex number quantum theory. Quantum Physics Falls Apart without Imaginary Numbers 2023-04-24T04:00:00Z The diagram helps us visualize the complex number system. Intermediate Algebra 2020-05-06T00:00:00Z In societies with complex number systems, there were clues to how those systems developed. How Did Neanderthals and Other Ancient Humans Learn to Count? 2021-12-23T05:00:00Z To represent a complex number we need to address the two components of the number. Precalculus 2021-12-01T00:00:00Z TRY IT #6 Convert the complex number to rectangular form: Algebra and Trigonometry 2015-02-13T00:00:00Z Later, complex numbers, which are the sum of a real and an imaginary number, gained wide acceptance by mathematicians because of their usefulness for solving complicated mathematical problems. Quantum Physics Falls Apart without Imaginary Numbers 2023-04-24T04:00:00Z Adding and subtracting complex numbers is much like adding or subtracting like terms. Intermediate Algebra 2020-05-06T00:00:00Z Infinite-Degree Maps Start with a function, or f, defined in terms of a complex number, or z. The Art of Mathematics in Chalk 2021-04-26T04:00:00Z To add or subtract complex numbers, we combine the real parts and combine the imaginary parts. Precalculus 2021-12-01T00:00:00Z What does the absolute value of a complex number represent? Algebra and Trigonometry 2015-02-13T00:00:00Z The most common version of quantum theory relies on complex numbers. Quantum Physics Falls Apart without Imaginary Numbers 2023-04-24T04:00:00Z The complex number system includes the real numbers and the imaginary numbers. Intermediate Algebra 2020-05-06T00:00:00Z Ordinarily, but quantum vectors are often expressed as complex numbers, which contain imaginary numbers, based on the square root of negative one. Quantum Escapism 2020-08-22T04:00:00Z For the following exercises, solve the equations over the complex numbers. Precalculus 2021-12-01T00:00:00Z For the following exercises, plot the complex number in the complex plane. Algebra and Trigonometry 2015-02-13T00:00:00Z From the start, this deep reliance on complex numbers went against deeply held convictions that physical theories must be formulated in terms of real magnitudes. Quantum Physics Falls Apart without Imaginary Numbers 2023-04-24T04:00:00Z A complex number is of the form a + bi, where a, b are real numbers. Intermediate Algebra 2020-05-06T00:00:00Z Those who need only a computer and an internet connection to run simulations or crunch complex numbers continued their work from home. With laboratories shut, coronavirus forces scientists to 'stop cold' 2020-04-29T04:00:00Z No. A complex number is not necessarily imaginary. Precalculus 2021-12-01T00:00:00Z For the following exercises, use synthetic division to determine the quotient involving a complex number. Algebra and Trigonometry 2015-02-13T00:00:00Z At first, Schrödinger's uneasiness seemed simple to resolve: he rewrote the wave function, replacing a single vector of complex numbers with two real vectors. Quantum Physics Falls Apart without Imaginary Numbers 2023-04-24T04:00:00Z The square root of a negative number tells us the solutions are complex numbers. Intermediate Algebra 2020-05-06T00:00:00Z The decision was based on a complex number system Idaho and other states use to classify prisoners as minimum-, medium- or maximum-security risks. Prison death highlights challenges of cellmate assignments 2018-03-15T04:00:00Z Solve the following equations over the complex number system. Precalculus 2021-12-01T00:00:00Z As we will soon see, a polynomial of degree in the complex number system will have zeros. Algebra and Trigonometry 2015-02-13T00:00:00Z Complex here doesn’t mean complicated but refers to the complex plane, a two-dimensional plane with a real axis and an imaginary axis, where complex numbers like 2+3i live. Dusa McDuff's Favorite Theorem 2017-10-07T04:00:00Z The standard form of a complex number is a + bi, so this explains why the preferred form is = bi when b > 0. Intermediate Algebra 2020-05-06T00:00:00Z His eyes tend to glaze over when complex numbers come into play. The Daily 202: Want to know why Trump’s winning Ohio? Drink a beer with ‘the deplorables’ in Boehner’s old district. 2016-10-04T04:00:00Z Songs as early as 1981's "Controversy" focus on a quest for God, and his catalog overflows with complex number and color systems, prophetic statements and disquiet about the fallen state of humanity. Where music meets religion: What an L.A. Times writer learned spending a night with Prince in 2009 2009-01-11T05:00:00Z No. Real numbers are a subset of complex numbers, but not the other way around. Algebra and Trigonometry 2015-02-13T00:00:00Z “I like to challenge myself,” she said, reminiscing about learning complex number theory as a 13-year-old. Numbers, music and volunteerism all add up for math star 2015-12-13T05:00:00Z For a complex number of the form a + bi, its conjugate is a − bi. Intermediate Algebra 2020-05-06T00:00:00Z These are convoluted mathematical objects that can be analysed using complex numbers - i.e. numbers with real and imaginary parts. First female winner for maths medal 2014-08-12T04:00:00Z This is how some fractal mathematical objects, such as the Mandelbrot set — a set of complex numbers for which a particular iterated equation does not approach infinity — are generated. Iranian is first woman to nab highest prize in maths 2014-08-11T04:00:00Z Each factor will be in the form where is a complex number. Algebra and Trigonometry 2015-02-13T00:00:00Z The complex number 1+i, written in rectangular and polar coordinates. What Are Roots of Unity? 2013-02-07T21:15:00.597Z Explain the relationship between real numbers and complex numbers. Intermediate Algebra 2020-05-06T00:00:00Z Yet let’s remember that z can be any complex number, formed from a real part and an imaginary part: x+iy. Fractal Kitties Illustrate the Endless Possibilities for Julia Sets 2012-09-26T19:45:04.483Z These are complex numbers, although for simplicity we usually referred to them as probabilities. Readers Respond to "Flashy Fossils" 2012-08-24T04:45:00.420Z Find products of complex numbers in polar form. Algebra and Trigonometry 2015-02-13T00:00:00Z That complex number defines the "phase" of that spot in the orbital. Molecules Imaged Most Intimately 2011-08-23T22:52:34Z Explain how dividing complex numbers is similar to rationalizing a denominator. Intermediate Algebra 2020-05-06T00:00:00Z These complex numbers can be plotted on an ordinary graph, with the real part marked along the x axis and the imaginary part on the y axis. Fractal Kitties Illustrate the Endless Possibilities for Julia Sets 2012-09-26T19:45:04.483Z This proposition alone suffices to suggest the importance of complex numbers. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 3 "Frost" to "Fyzabad" 2011-08-15T02:00:28.473Z Find quotients of complex numbers in polar form. Algebra and Trigonometry 2015-02-13T00:00:00Z The author explains complex numbers in his usual lucid fashion. Kinematics of Mechanisms from the Time of Watt A complex number is in standard form when written as a + bi, where a, b are real numbers. Intermediate Algebra 2020-05-06T00:00:00Z The Mandelbrot set, one of the most famous fractals, is closely related to Julia sets of degree two polynomials: imagine the polynomial z2+c, where c can be any complex number. Fractal Kitties Illustrate the Endless Possibilities for Julia Sets 2012-09-26T19:45:04.483Z This term is called the complex conjugate of the denominator, which is found by changing the sign of the imaginary part of the complex number. Algebra and Trigonometry 2015-02-13T00:00:00Z Find roots of complex numbers in polar form. Algebra and Trigonometry 2015-02-13T00:00:00Z What is the basic principle in multiplication of complex numbers? Algebra and Trigonometry 2015-02-13T00:00:00Z In the previous example, our solutions were complex numbers. Intermediate Algebra 2020-05-06T00:00:00Z For the following exercises, plot the complex numbers on the complex plane. Algebra and Trigonometry 2015-02-13T00:00:00Z For the following exercises, perform the indicated operation and express the result as a simplified complex number. Algebra and Trigonometry 2015-02-13T00:00:00Z A complex number is the sum of a real number and an imaginary number. Algebra and Trigonometry 2015-02-13T00:00:00Z For the following exercises, evaluate the expressions, writing the result as a simplified complex number. Algebra and Trigonometry 2015-02-13T00:00:00Z To plot a complex number, we use two number lines, crossed to form the complex plane. Algebra and Trigonometry 2015-02-13T00:00:00Z To multiply complex numbers, distribute just as with polynomials. Algebra and Trigonometry 2015-02-13T00:00:00Z To divide complex numbers, multiply both numerator and denominator by the complex conjugate of the denominator to eliminate the complex number from the denominator. Algebra and Trigonometry 2015-02-13T00:00:00Z A complex number is expressed in standard form when written where is the real part and is the imaginary part. Algebra and Trigonometry 2015-02-13T00:00:00Z Find the absolute value of a complex number. Algebra and Trigonometry 2015-02-13T00:00:00Z Convert a complex number from polar to rectangular form. Algebra and Trigonometry 2015-02-13T00:00:00Z Find powers of complex numbers in polar form. Algebra and Trigonometry 2015-02-13T00:00:00Z This rather famous quote by nineteenth-century German mathematician Leopold Kronecker sets the stage for this section on the polar form of a complex number. Algebra and Trigonometry 2015-02-13T00:00:00Z To represent a complex number, we need to address the two components of the number. Algebra and Trigonometry 2015-02-13T00:00:00Z We first encountered complex numbers in Complex Numbers. Algebra and Trigonometry 2015-02-13T00:00:00Z The absolute value of a complex number is the same as its magnitude, or It measures the distance from the origin to a point in the plane. Algebra and Trigonometry 2015-02-13T00:00:00Z The absolute value of a complex number is the same as its magnitude. Algebra and Trigonometry 2015-02-13T00:00:00Z To write complex numbers in polar form, we use the formulas and Then, See Example 4 and Example 5. Algebra and Trigonometry 2015-02-13T00:00:00Z Let’s consider the number The real part of the complex number is and the imaginary part is 3. Algebra and Trigonometry 2015-02-13T00:00:00Z To find the product of two complex numbers, multiply the two moduli and add the two angles. Algebra and Trigonometry 2015-02-13T00:00:00Z To find the quotient of two complex numbers in polar form, find the quotient of the two moduli and the difference of the two angles. Algebra and Trigonometry 2015-02-13T00:00:00Z To find the power of a complex number raise to the power and multiply by See Example 10. Algebra and Trigonometry 2015-02-13T00:00:00Z Finding the roots of a complex number is the same as raising a complex number to a power, but using a rational exponent. Algebra and Trigonometry 2015-02-13T00:00:00Z We plot the ordered pair to represent the complex number as shown in Figure 2. Algebra and Trigonometry 2015-02-13T00:00:00Z These formulas have made working with products, quotients, powers, and roots of complex numbers much simpler than they appear. Algebra and Trigonometry 2015-02-13T00:00:00Z There are several ways to represent a formula for finding roots of complex numbers in polar form. Algebra and Trigonometry 2015-02-13T00:00:00Z How is a complex number converted to polar form? Algebra and Trigonometry 2015-02-13T00:00:00Z How do we find the product of two complex numbers? Algebra and Trigonometry 2015-02-13T00:00:00Z TRY IT #2 Plot the complex number on the complex plane. Algebra and Trigonometry 2015-02-13T00:00:00Z For the following exercises, find the absolute value of the given complex number. Algebra and Trigonometry 2015-02-13T00:00:00Z For the following exercises, convert the complex number from polar to rectangular form. Algebra and Trigonometry 2015-02-13T00:00:00Z For the following exercises, convert the complex number from polar to rectangular form. Algebra and Trigonometry 2015-02-13T00:00:00Z For the following exercises, find the powers of each complex number in polar form. Algebra and Trigonometry 2015-02-13T00:00:00Z To add or subtract complex numbers, we combine the real parts and then combine the imaginary parts. Algebra and Trigonometry 2015-02-13T00:00:00Z For the following exercises, plot the complex number in the complex plane. Algebra and Trigonometry 2015-02-13T00:00:00Z Convert the complex number from polar to rectangular form: Algebra and Trigonometry 2015-02-13T00:00:00Z |
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