Name: complex numbers in solving equations
Uploaded: 2025-11-26T05:44:30.000Z
Duration: 1 min 29 s
Channel: JoVE

Einige Gleichungen haben keine wirkliche Lösung, weil sie die Quadratwurzeln negativer Zahlen beinhalten.Um dies zu adressieren, werden komplexe Zahlen eingeführt, die die Quadratwurzel von −1 als imaginäre Einheit i definieren.Dies kann auf der komplexen Ebene visualisiert werden, wo die realen und imaginären Teile senkrechte Achsen bilden, wobei jede komplexe Zahl als Punkt platziert wird.Bei der Addition komplexer Zahlen werden ihre reellen und imaginären Teile getrennt addiert.Die Multiplikation komplexer Zahlen folgt der distributiven Eigenschaft. Da i2= −1 ist, wird jedes Vorkommen von i2 bei der Vereinfachung durch −1 ersetzt.Beim Dividieren komplexer Zahlen werden sowohl der Zähler als auch der Nenner mit dem Konjugat des Nenners multipliziert, das den gleichen Realteil und den entgegengesetzten Imaginärteil hat, um den Imaginärteil zu eliminieren.So wie jede positive reelle Zahl zwei Quadratwurzeln hat, hat auch jede negative reelle Zahl zwei komplexe Quadratwurzeln, die komplexe Konjugate sind.Komplexe Zahlen werden in der Magnetresonanztomographie verwendet, wo der Scanner komplexe Signaldaten sammelt, die als k-Raum bezeichnet werden. Diese Daten werden mit Hilfe von inversen Fourier-Transformationen in räumliche Bilder umgewandelt.

komplexe zahlen

Das reelle Zahlensystem kann die Quadratwurzel einer negativen Zahl nicht darstellen, was die L&#246;sung bestimmter Gleichungen, wie z. B. quadratischer Gleichungen mit negativen Diskriminanten, einschr&#228;nkt. Um dem zu begegnen, wurde das komplexe Zahlensystem entwickelt, das die Imagin&#228;reinheit i einf&#252;hrt, wobei i = &#8730;(-1) gilt. Diese Erweiterung erm&#246;glicht die Darstellung aller Wurzeln, einschlie&#223;lich solcher mit negativen Radikanden.Eine komplexe Zahl wird in der Form x + yi geschrieben, wobei x und y reelle Zahlen sind. Dabei bezeichnet x den Realteil und y den Imagin&#228;rteil. Die Imagin&#228;reinheit i besitzt die grundlegende Eigenschaft i^2 = -1. Mit dieser Definition kann jede quadratische Gleichung im komplexen Zahlensystem eine L&#246;sung besitzen, da sowohl positive als auch negative Quadratwurzeln dargestellt werden k&#246;nnen.Arithmetik mit komplexen ZahlenOperationen mit komplexen Zahlen folgen den konventionellen algebraischen Regeln, die durch die Eigenschaft von i leicht modifiziert werden.AdditionUm zwei komplexe Zahlen, etwa x + yi und u + vi, zu addieren, werden Real- und Imagin&#228;rteil jeweils getrennt addiert:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:50%;'><img style='aspect-ratio:1920/120;' src='https://cloudfront.jove.com/files/ftp_upload/40032/40032_Equation_1_1__editor_40032.svg' alt='40032_Equation_1_1_.svg' /></figure>SubtraktionAnalog erfolgt die Subtraktion durch Abziehen der entsprechenden Real- und Imagin&#228;rteile:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:50%;'><img style='aspect-ratio:1920/120;' src='https://cloudfront.jove.com/files/ftp_upload/40032/40032_Equation_2_1__editor_40032.svg' alt='40032_Equation_2_1_.svg' /></figure>MultiplikationDie Multiplikation komplexer Zahlen erfolgt mittels Distributivgesetz und anschlie&#223;ender Vereinfachung unter Verwendung von i^2 = -1:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:50%;'><img style='aspect-ratio:1920/250;' src='https://cloudfront.jove.com/files/ftp_upload/40032/40032_Equation_3_1__editor_40032.svg' alt='40032_Equation_3_1_.svg' /></figure>Komplexe Konjugate und DivisionDie Komplexkonjugierte einer komplexen Zahl x + yi ist x - yi. Das Produkt einer komplexen Zahl mit ihrer Komplexkonjugierten ist eine reelle Zahl:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:50%;'><img style='aspect-ratio:1920/120;' src='https://cloudfront.jove.com/files/ftp_upload/40032/40032_Equation_4_1__editor_40032.svg' alt='40032_Equation_4_1_.svg' /></figure>Diese Eigenschaft ist f&#252;r die Division wesentlich. Um x + yi durch u + vi zu dividieren, werden Z&#228;hler und Nenner mit der Komplexkonjugierten des Nenners multipliziert:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:50%;'><img style='aspect-ratio:1920/250;' src='https://cloudfront.jove.com/files/ftp_upload/40032/40032_Equation_5_1__editor_40032.svg' alt='40032_Equation_5_1_.svg' /></figure>Anwendungen komplexer ZahlenKomplexe Zahlen sind in verschiedenen wissenschaftlichen und technischen Disziplinen von grundlegender Bedeutung. In der Elektrotechnik wird beispielsweise die Impedanz in Wechselstromkreisen als komplexe Gr&#246;&#223;e dargestellt, wobei der Imagin&#228;rteil die Reaktanz repr&#228;sentiert. In der Signalverarbeitung und in Regelungssystemen modellieren komplexe Zahlen Schwingungen, Phasenverschiebungen und Frequenzg&#228;nge. Ihre F&#228;higkeit, sowohl Betrag als auch Richtung auszudr&#252;cken, macht sie zu unverzichtbaren Werkzeugen f&#252;r die Analyse dynamischer Systeme.

Komplexe Zahlen

Das reelle Zahlensystem kann die Quadratwurzel einer negativen Zahl nicht darstellen, was die L&#246;sung bestimmter Gleichungen, wie z. B. quadratischer ...

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Complex Numbers

complex numbers in solving equations

Complex numbers extend the real number system so equations with negative square roots can still be solved. The real numbers cannot represent the square root of a negative number. That limits some quadratic equations, especially those with negative discriminants. To solve this problem, mathematicians introduced the imaginary unit i, where i = √(-1).

A complex number is written as x + yi, where x and y are real numbers. In this form, x is the real part and y is the imaginary part. The key rule is that i2 = -1. With this definition, both positive and negative square roots can be represented, so quadratic equations can have solutions in the complex number system.

Arithmetic with complex numbers follows normal algebra rules, but it uses the property of i. To add complex numbers, combine the real parts and the imaginary parts separately. Subtraction works the same way, by subtracting matching parts. Multiplication uses the distributive property and then simplifies with i2 = -1.

Complex conjugates are important for division. The conjugate of x + yi is x - yi. When a complex number is multiplied by its conjugate, the result is a real number. To divide x + yi by u + vi, multiply the numerator and denominator by the conjugate of the denominator.

Complex numbers are useful in science and engineering. In electrical engineering, impedance in alternating current circuits is written as a complex quantity, and the imaginary part represents reactance. They also help in signal processing and control systems, where they model oscillations, phase shifts, and frequency responses. Because they can show both magnitude and direction, complex numbers are valuable for studying dynamic systems.

Complex Numbers in Solving Equations

real number system

Number Types on the Real Line

The real number system brings together the number types used for counting, measuring, and comparing quantities. It includes natural numbers, integers, rational numbers, and irrational numbers. Each set adds new kinds of numbers to the same system. Natural numbers are the basic counting numbers: 1, 2, 3, and so on. Integers extend that set by adding zero and negative whole numbers, such as ..., -3, -2, -1, 0, 1, 2, .... Rational numbers are numbers that can be written as a ratio of two...

real number operations

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Real numbers are the values that can be placed on a continuous number line. The system grew from counting numbers used for enumeration. It later expanded to include zero, which shows the absence of quantity, and negative numbers, which are opposites of counting numbers. The real number system also includes fractions and decimals that represent parts of a whole or equal divisions. When written in decimal form, these numbers may end or repeat in a regular pattern. Together, these values describe...

exponents

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radicals

Simplifying Radicals

Radicals show the root of a number and the exponent needed to get that value. A radical expression has two main parts. The radicand is the number inside the root symbol, and the index tells which root to take. The notation n√a means the principal nth root of a. The index changes the type of root. When n equals 2, the expression is a square root. When n equals 3, it is a cube root. The sign of the radicand also matters. If the index is even, a negative radicand does not give a real result. If...

algebraic expressions

Simplifying Algebraic Expressions

Algebraic expressions use variables, constants, and operations to show mathematical relationships. They help describe patterns and solve problems across many areas of mathematics. Knowing the parts of an expression makes it easier to simplify and work with it correctly. Each algebraic expression has separate pieces that work together. A coefficient is the number attached to a variable, and an exponent shows repeated multiplication. Some expressions have a single part, while others include...

rational expressions

Working with Rational Expression Domains

Rational expressions are algebraic fractions with polynomials in the numerator and denominator. They follow the same basic rules as numerical fractions, but variables make them more delicate to work with. A key step is finding values that make the expression undefined, usually because of division by zero or an undefined radical. To determine the domain, start by identifying the values excluded from the denominator. The domain includes all real numbers except those that make the denominator...

linear equations

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quadratic equations

Solving Quadratics by Factoring and Formula

Quadratic equations are algebraic equations with a variable squared, a first-power term, and a constant term. They are set equal to zero. In high school math, they often help model area, motion, and optimization problems. The general form of a quadratic equation uses real numbers a, b, and c, with a not equal to zero. The nonzero a value keeps the squared term in the equation. One common way to solve a quadratic is to rewrite it as a product of two linear expressions. In factored form, the...

complex numbers

Complex numbers extend the real number system so equations with negative square roots can still be solved. The real numbers cannot represent the square root of a negative number. That limits some quadratic equations, especially those with negative discriminants. To solve this problem, mathematicians introduced the imaginary unit i, where i = √(-1). A complex number is written as x + yi, where x and y are real numbers. In this form, x is the real part and y is the imaginary part. The key rule...

quadratic equations in the complex number system

quadratic-equations-in-the-complex-number-system

Complex Roots of Quadratic Equations

Quadratic equations can have real or complex roots, depending on the discriminant. For an equation in the form ax² + bx + c = 0, the coefficients a, b, and c are the numbers that multiply x², x, and the constant term. The discriminant, b² − 4ac, tells us what kind of solutions to expect. When the discriminant is negative, the equation has no real number solutions. In that case, complex numbers provide a way to continue solving the problem. The imaginary unit i is defined by i = √−1, so a...

radical equations

Solving Radical Equations with Checks

Radical equations are equations with a variable inside a radical, most often a square root or cube root. These equations appear in science, engineering, and real-world measurement problems that involve nonlinear relationships. Solving them starts by isolating the radical expression. Then the radical is removed by raising both sides to the power that matches the radical’s index. This process can create extraneous solutions, which are values that work in the transformed equation but not in the...

inequalities

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absolute value inequalities

Absolute Value Inequalities on a Number Line

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Mathematical Modeling for Loan Payments

Mathematical modeling turns a real-life situation into an equation that can be solved. It starts with defining the problem, choosing measurable quantities, and assigning variables to them. Then a matching model is selected and the equation is solved for the unknown value. This process is especially useful in finance, where models help evaluate investments, loans, and repayment plans. One common example is finding the fixed monthly payment on a loan. That calculation uses the standard annuity...

key terms and concepts

Learn how complex numbers use the imaginary unit i to solve quadratic equations, support arithmetic, and model circuits, signals, and control systems.

Complex Numbers in Solving Equations | JoVE K12

The JoVE Math Fundamentals collection is designed to build a deep and comprehensive understanding of essential mathematics. Covering the foundations of algebra, geometry, trigonometry, and analytic reasoning, this course establishes a strong base for future studies. The collection connects core mathematical concepts with real-world practice, equipping learners to solve complex problems, master problem-solving skills, and apply abstract ideas.

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Complex Numbers in Solving Equations

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