Isabella Carter
Complex numbers are the sum of a real number and an imaginary number. They are usually represented by the letter 'z' and are of the form 'a + ib', where 'a' is the real part and 'b' is the imaginary part. The real part of a complex number is denoted by Re(z) and the imaginary part by Im(z). Complex numbers are used in electrical engineering, control theory, and signal analysis, and they are particularly useful for solving equations with no real roots, such as x^2 + 1 = 0.
| Characteristics | Values | ||
|---|---|---|---|
| Definition | A complex number is the sum of a real number and an imaginary number | ||
| Formula | a + bi | ||
| Representation | z = a + ib | ||
| Real Part | a | ||
| Imaginary Part | b | ||
| Modulus | z | ||
| Coordinate Plane | Argand (or complex) plane | ||
| Other Names | Iota, imaginary number | ||
| Use Case | Used in signal analysis and other fields for a convenient description for periodically varying signals |
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What You'll Learn
Real and imaginary parts
A complex number is the sum of a real number and an imaginary number. It is usually represented by the letter 'z' and is of the form 'a + ib', where both 'a' and 'b' are real numbers. The value 'a' is called the real part and is denoted by Re(z), while 'b' is called the imaginary part and is denoted by Im(z). For example, in the complex number 3 + 4i, 3 is the real part and 4i is the imaginary part. The real part of a complex number can be regarded as a complex number a + 0i, whose imaginary part is 0. Conversely, a purely imaginary number bi is a complex number 0 + bi, whose real part is zero.
The real and imaginary parts of a complex number can be interpreted as coordinates of a point in a Euclidean plane, also known as the complex plane or Argand diagram. The horizontal axis, or x-axis, is used to display the real part, with increasing values moving to the right. The imaginary part marks the vertical axis, or y-axis, with increasing values moving upwards. This representation allows for the calculation of the modulus and argument of the complex number. The modulus is the linear distance from the origin (0, 0) to the point (a, ib) and is calculated as r = |√(a^2 + b^2)|. The complex number 3 + 4i, for example, can be represented as distance 5 and angle 0.927 radians.
The real and imaginary parts of a complex number can also be used to perform various arithmetic operations such as addition, subtraction, multiplication, and division. For instance, the division of two complex numbers can be achieved by multiplying the numerator and denominator by the conjugate value of the denominator and then applying the FOIL method. The product of two complex numbers and its conjugate value should result in a complex number and a positive value. Additionally, the n-th power of a complex number can be computed using de Moivre's formula, which involves raising the modulus to the n-th power and applying trigonometric functions to the argument.
In certain fields, such as electrical engineering, complex numbers have practical applications. For example, the Fourier transform is used to analyse varying electric currents and voltages by treating resistors, capacitors, and inductors as a single complex number called impedance. This approach, known as phasor calculus, utilises the imaginary unit 'j' to avoid confusion with the symbols commonly used for electric current. In control theory, systems are transformed from the time domain to the complex frequency domain using the Laplace transform, allowing for analysis in the complex plane.
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The complex plane
A complex number is the sum of a real number and an imaginary number. It is usually represented by the letter 'z' and is of the form
In electrical engineering, the complex plane is used to analyse varying electric currents and voltages using the Fourier transform. By introducing imaginary, frequency-dependent resistances, the treatment of resistors, capacitors, and inductors can be unified, and all three can be combined into a single complex number called impedance. This approach is known as phasor calculus.
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Division of complex numbers
Dividing complex numbers is a little more complicated than addition, subtraction, and multiplication of complex numbers. This is because it is difficult to divide a number by an imaginary number, and each fraction must have a real-number denominator.
To divide two complex numbers, we need to find a term by which we can multiply the numerator and denominator that will eliminate the imaginary part of the denominator so that we end up with a real number in the denominator. This technique is often called rationalizing the denominator.
The term we are looking for is called the complex conjugate of the denominator, which is found by altering the sign of the imaginary component of a complex integer. In other words, the complex conjugate of a+bi is a-bi. Note that complex conjugates have a reciprocal relationship: the complex conjugate of a+bi is a-bi, and the complex conjugate of a-bi is a+bi.
To divide a complex number a+ib by c+id, multiply the numerator and denominator of the fraction (a+ib)/(c+id) by c−id and simplify. The division of complex numbers z1=a+ib and z2=c+id is calculated using the following formula:
\[\frac{z_1}{z_2}=\frac{ac+bd}{c^2+d^2}+i\left(\frac{bc-ad}{c^2+d^2}\right)\]
Let the quotient be \\(a+ib)/\(c+id\). This can be written as \(\co: 3 frac{ac+bd}{c^2+d^2}+i\left(\frac{bc-ad}{c^2+d^2}\right). Divide the real part and the imaginary part of the complex number by that real number separately.
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The modulus of a complex number
A complex number is a representation of the addition of two numbers: a real number and an imaginary number. The real number is called the real part, and the imaginary number is called the imaginary part. The imaginary part of a complex number is denoted by the symbol 'i', which stands for iota (imaginary number).
If z = x + iy is a complex number, where x and y are real and i = √-1, then the non-negative value √(x^2 + y^2) is the modulus of the complex number. The modulus of a complex number is the square root of the sum of the squares of the real and imaginary parts of the complex number. It is denoted by |z|, and can be calculated using the conjugate of z:
|z| = √(z.z) = √(x^2 + y^2)
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Complex numbers in electrical engineering
Complex numbers are fundamental to modern electrical engineering. They are used to represent and calculate both AC signals and impedance. The two dimensions of complex numbers, length and angle, allow for the calculation of amplitude and phase together, ensuring consistency. Complex numbers are also used to express sinusoids in terms of complex exponentials.
In electrical engineering, the Fourier transform is used to analyse varying electric currents and voltages. The treatment of resistors, capacitors, and inductors can be unified by introducing imaginary, frequency-dependent resistances for the latter two and combining all three in a single complex number called the impedance. This approach is called phasor calculus.
In electrical engineering, the imaginary unit is denoted by 'j', to avoid confusion with 'i', which is generally used to denote electric current. Complex numbers can be expressed in polar coordinates, where locations in the plane can be expressed as the sum of vectors, with the vectors corresponding to the complex number. Complex numbers can also be expressed in rectangular form with real and imaginary components (a+jb). However, in electrical engineering, complex numbers are often represented in polar form (A@30deg).
Complex numbers are also used in control theory, where systems are transformed from the time domain to the complex frequency domain using the Laplace transform. The system's zeros and poles are then analysed in the complex plane. Techniques such as the root locus, Nyquist plot, and Nichols plot all utilise the complex plane.
Overall, complex numbers are a critical tool in electrical engineering, simplifying expressions and the number of equations, and providing a convenient way to represent and analyse electrical systems.
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Frequently asked questions
Complex numbers are the sum of a real number and an imaginary number. They are usually represented by the letter 'z' and are of the form 'a + ib' where both a and b are real numbers.
The value 'a' is called the real part and is denoted by Re(z), and 'b' is called the imaginary part and is denoted by Im(z).
The modulus of a complex number is the linear distance from the origin (0, 0) to the point (a, ib) and is calculated as r = |√(a^2 + b^2)|.
The complex number z = a + ib can be plotted on a graph with the real part 'a' on the x-axis and the imaginary part 'ib' on the y-axis.
