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

Algumas equações não têm solução real porque envolvem as raízes quadradas de números negativos.Para resolver isso, números complexos são introduzidos, definindo a raiz quadrada de −1 como a unidade imaginária i.Isso pode ser visualizado no plano complexo, onde as partes reais e imaginárias formam eixos perpendiculares, colocando cada número complexo como um ponto.A adição de números complexos envolve a adição separada de suas partes reais e imaginárias.A multiplicação de números complexos segue a propriedade distributiva. Como i2 = −1, qualquer ocorrência de i2 é substituída por −1 durante a simplificação.A divisão de números complexos envolve a multiplicação do numerador e do denominador pelo conjugado do denominador - que tem a mesma parte real e a parte imaginária oposta - para eliminar a parte imaginária.Assim como todo número real positivo tem duas raízes quadradas, todo número real negativo também tem duas raízes quadradas complexas, que são conjugados complexos.Números complexos são usados em imagens de ressonância magnética, onde o scanner coleta dados de sinais complexos chamados k-space. Esses dados são convertidos em imagens espaciais usando transformada inversa de Fourier.

números complexos

O sistema de n&#250;meros reais n&#227;o &#233; capaz de representar a raiz quadrada de um n&#250;mero negativo, o que limita as solu&#231;&#245;es de certas equa&#231;&#245;es, como equa&#231;&#245;es quadr&#225;ticas com discriminantes negativos. Para superar essa limita&#231;&#227;o, desenvolveu-se o sistema de n&#250;meros complexos, que introduz a unidade imagin&#225;ria i, definida como i = &#8730;(-1). Essa amplia&#231;&#227;o permite representar todas as ra&#237;zes, inclusive aquelas que envolvem radicandos negativos.Um n&#250;mero complexo &#233; expresso na forma x + yi, em que x e y s&#227;o n&#250;meros reais. Nesse caso, x representa a parte real e y representa a parte imagin&#225;ria. A unidade imagin&#225;ria i possui a propriedade fundamental i^2 = -1. Com base nessa defini&#231;&#227;o, qualquer equa&#231;&#227;o quadr&#225;tica pode ter solu&#231;&#227;o dentro do sistema de n&#250;meros complexos, j&#225; que tanto as ra&#237;zes positivas quanto as negativas podem ser representadas.Aritm&#233;tica com N&#250;meros ComplexosAs opera&#231;&#245;es com n&#250;meros complexos seguem as regras alg&#233;bricas convencionais, com pequenas adapta&#231;&#245;es em raz&#227;o da propriedade de i.Adi&#231;&#227;oPara somar dois n&#250;meros complexos, como x + yi e u + vi, combinam-se separadamente as partes reais e as partes imagin&#225;rias:<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>Subtra&#231;&#227;oDe modo an&#225;logo, a subtra&#231;&#227;o &#233; realizada subtraindo-se as partes reais e imagin&#225;rias correspondentes:<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>Multiplica&#231;&#227;oA multiplica&#231;&#227;o entre n&#250;meros complexos envolve aplicar a propriedade distributiva e simplificar com base em 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>Conjugados Complexos e Divis&#227;oO conjugado de um n&#250;mero complexo x + yi &#233; x - yi. O produto de um n&#250;mero complexo e seu conjugado resulta em um n&#250;mero real:<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>Essa propriedade &#233; fundamental para a divis&#227;o. Para dividir x + yi por u + vi, multiplicam-se numerador e denominador pelo conjugado do denominador:<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>Aplica&#231;&#245;es dos N&#250;meros ComplexosOs n&#250;meros complexos s&#227;o essenciais em diversos campos da ci&#234;ncia e da engenharia. Na engenharia el&#233;trica, por exemplo, a imped&#226;ncia em circuitos de corrente alternada &#233; expressa como uma grandeza complexa, cuja parte imagin&#225;ria representa a reat&#226;ncia. Em sistemas de processamento e controle de sinais, os n&#250;meros complexos s&#227;o usados para modelar oscila&#231;&#245;es, deslocamentos de fase e respostas em frequ&#234;ncia. Sua capacidade de expressar simultaneamente magnitude e dire&#231;&#227;o os torna ferramentas indispens&#225;veis para a an&#225;lise de sistemas din&#226;micos.

Números Complexos

O sistema de n&#250;meros reais n&#227;o &#233; capaz de representar a raiz quadrada de um n&#250;mero negativo, o que limita as solu&#231;&#245;es de ...

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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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