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

Some equations have no real solution because they involve the square roots of negative numbers.To address this, complex numbers are introduced, defining the square root of &#8722;1 as the imaginary unit i.This can be visualized on the complex plane, where the real and imaginary parts form perpendicular axes, placing each complex number as a point.The addition of complex numbers involves separately adding their real and imaginary parts.Multiplication of complex numbers follows the distributive property. Since i2=&#8722;1, any occurrence of i2 is replaced with &#8722;1 during simplification.Dividing complex numbers involves multiplying both the numerator and denominator by the denominator&#8217;s conjugate&#8212;which has the same real part and the opposite imaginary part&#8212;to eliminate the imaginary part.Just as every positive real number has two square roots, every negative real number also has two complex square roots, which are complex conjugates.Complex numbers are used in Magnetic Resonance Imaging, where the scanner collects complex signal data called k-space. This data is converted into spatial images using inverse Fourier transforms.

complex numbers

The real number system cannot represent the square root of a negative number, which restricts solutions for certain equations, such as quadratics with negative discriminants. To address this, the complex number system was developed, introducing the imaginary unit i, where i = &#8730;(-1). This extension allows for the representation of all roots, including those involving negative radicands.A complex number is written in the form x + yi, where x and y are real numbers. Here, x represents the real part, and y represents the imaginary part. The imaginary unit i has the fundamental property i2=-1. Using this definition, any quadratic equation can have a solution within the complex number system, as both positive and negative square roots can be represented.Arithmetic with Complex NumbersOperations on complex numbers follow conventional algebraic rules, modified slightly by the property of i.Addition To add two complex numbers, such as x + yi and u + vi, the real parts and the imaginary parts are combined separately:<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>Subtraction Similarly, subtraction is performed by subtracting the corresponding real and imaginary parts:<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>Multiplication Multiplying complex numbers involves applying the distributive property and simplifying using i2=-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>Complex Conjugates and DivisionThe conjugate of a complex number x + yi is x - yi. The product of a complex number and its conjugate is a real number:<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>This property is essential in division. To divide x + yi by u + vi, the numerator and denominator are multiplied by the conjugate of the denominator:<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 of Complex NumbersComplex numbers are fundamental in various scientific and engineering fields. In electrical engineering, for example, impedance in alternating current circuits is expressed as a complex quantity, with the imaginary part representing the reactance. In signal processing and control systems, complex numbers model oscillations, phase shifts, and frequency responses. Their ability to express both magnitude and direction makes them essential tools for analyzing dynamic systems.

Complex Numbers

The real number system cannot represent the square root of a negative number, which restricts solutions for certain equations, such as quadratics with ...

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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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exponents

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radicals

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algebraic expressions

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rational expressions

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linear equations

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quadratic equations

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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

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inequalities

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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.

math fundamentals

Complex Numbers in Solving Equations

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