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

Certaines équations n’ont pas de véritable solution car elles impliquent les racines carrées des nombres négatifs.Pour résoudre ce problème, des nombres complexes sont introduits, définissant la racine carrée de −1 comme l’unité imaginaire i.Cela peut être visualisé sur le plan complexe, où les parties réelles et imaginaires forment des axes perpendiculaires, plaçant chaque nombre complexe comme un point.L’addition de nombres complexes implique l’addition séparée de leurs parties réelles et imaginaires.La multiplication des nombres complexes suit la propriété distributive. Puisque i2=−1, toute occurrence de i2 est remplacée par −1 lors de la simplification.La division des nombres complexes consiste à multiplier à la fois le numérateur et le dénominateur par le conjugué du dénominateur, qui a la même partie réelle et la partie imaginaire opposée, pour éliminer la partie imaginaire.Tout comme tout nombre réel positif a deux racines carrées, tout nombre réel négatif a également deux racines carrées complexes, qui sont des conjugués complexes.Les nombres complexes sont utilisés en imagerie par résonance magnétique, où le scanner collecte des données de signal complexes appelées k-space. Ces données sont converties en images spatiales à l’aide de transformées de Fourier inverses.

nombres complexes

Le syst&#232;me des nombres r&#233;els ne permet pas de repr&#233;senter la racine carr&#233;e d&#8217;un nombre n&#233;gatif, ce qui restreint la r&#233;solution de certaines &#233;quations, notamment celles du second degr&#233; &#224; discriminant n&#233;gatif. Pour pallier cette limitation, le syst&#232;me des nombres complexes a &#233;t&#233; &#233;labor&#233;, introduisant l&#8217;unit&#233; imaginaire i, d&#233;finie par i = &#8730;(-1). Cette extension rend possible la repr&#233;sentation de toutes les racines, y compris celles comportant des radicandes n&#233;gatifs.Un nombre complexe s&#8217;&#233;crit sous la forme x + yi, o&#249; x et y sont des nombres r&#233;els. Le terme x d&#233;signe la partie r&#233;elle et y la partie imaginaire. L&#8217;unit&#233; imaginaire i poss&#232;de la propri&#233;t&#233; fondamentale i^2 = -1. Gr&#226;ce &#224; cette d&#233;finition, toute &#233;quation du second degr&#233; peut admettre une solution dans le syst&#232;me des nombres complexes, puisque les racines carr&#233;es positives et n&#233;gatives peuvent y &#234;tre repr&#233;sent&#233;es.Arithm&#233;tique des nombres complexesLes op&#233;rations sur les nombres complexes ob&#233;issent aux r&#232;gles alg&#233;briques usuelles, l&#233;g&#232;rement adapt&#233;es en raison de la propri&#233;t&#233; de l&#8217;unit&#233; imaginaire i.AdditionPour additionner deux nombres complexes, tels que x + yi et u + vi, on additionne s&#233;par&#233;ment les parties r&#233;elles et les parties imaginaires&#160;:<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>SoustractionDe m&#234;me, la soustraction s&#8217;effectue en retranchant les parties r&#233;elles et imaginaires correspondantes :<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>MultiplicationLa multiplication de nombres complexes consiste &#224; appliquer la propri&#233;t&#233; distributive, puis &#224; simplifier en utilisant i^2 = -1 &#160;:<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>Conjugu&#233;s complexes et divisionLe conjugu&#233; d&#8217;un nombre complexe x + yi est x &#8722; yi. Le produit d&#8217;un nombre complexe par son conjugu&#233; est un nombre r&#233;el&#160;:<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>Cette propri&#233;t&#233; est essentielle pour la division. Pour diviser x + yi par u + vi, on multiplie le num&#233;rateur et le d&#233;nominateur par le conjugu&#233; du d&#233;nominateur :<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>Applications des nombres complexesLes nombres complexes occupent une place fondamentale dans de nombreux domaines scientifiques et techniques. En g&#233;nie &#233;lectrique, par exemple, l&#8217;imp&#233;dance d&#8217;un circuit &#224; courant alternatif s&#8217;exprime sous forme complexe, la partie imaginaire repr&#233;sentant la r&#233;actance. En traitement du signal et en automatique, les nombres complexes permettent de mod&#233;liser les oscillations, les d&#233;calages de phase et les r&#233;ponses fr&#233;quentielles. Leur capacit&#233; &#224; exprimer simultan&#233;ment le module et la phase en fait des outils indispensables &#224; l&#8217;analyse des syst&#232;mes dynamiques.

Nombres complexes

Le syst&#232;me des nombres r&#233;els ne permet pas de repr&#233;senter la racine carr&#233;e d&#8217;un nombre n&#233;gatif, ce qui restreint la ...

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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

Real Numbers on the Number Line

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

Exponent Rules and Scientific Notation

Exponents show repeated multiplication in a short form. They are important in algebra and in many other math topics, including scientific computation, scaling laws, and dimensional analysis. For a nonzero real number a and an integer n, a^n means a multiplied by itself n times. Exponent rules help simplify expressions with powers. These properties make it easier to work with exponential expressions in both symbolic and numerical form. They also support common calculations that use exponents in...

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

Solving Linear Equations

Linear equations are a basic algebra tool for modeling constant rates and everyday relationships. They use constants and a single variable, and each term is either a constant or a product of a constant and that variable. When graphed on a Cartesian coordinate plane, they form a straight line that shows a constant rate of change between two quantities. A linear equation in one variable often has the form ax + b = c, where a, b, and c are constants and x is the variable. Solving the equation...

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

Solving Inequalities and Interval Notation

Inequalities show how two values compare when they are not equal. They use symbols such as , ≤, and ≥. These statements often describe a range of possible answers instead of one exact value. Interval notation is a shorter way to write these solution sets. An open interval, written as (a, b), leaves out the endpoints. A closed interval, written as [a, b], includes both endpoints. Half-open intervals, such as (a, b] and [a, b), include only one endpoint. Intervals can also extend forever in one...

absolute value inequalities

Absolute Value Inequalities on a Number Line

Absolute value inequalities show how far a number can be from zero or from a target value. Absolute value means distance on the number line, so the sign of the number does not matter. In inequalities, this distance helps define a valid range of values for a variable. An inequality of the form |x| ≤ a, where a ≥ 0, includes every x between −a and a. This can also be written as a compound inequality. On a number line, the solution is shown as a solid segment from −a to a with closed circles at...

mathematical modeling: problem solving

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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