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S. Boyd EE102Brief review of complex numbersThese notes collect some basic facts about complex numbers.1 RepresentationsImaginary and complex numbersWe start by introducing a symbol i that represents the squareroot of −1, i.e., i2= −1. Forsome strange reason, electrical engineers use the symbol j instead of i. (Maybe because icould be confused with current?) It’s a dumb tradition, but I’ll respect it here.We refer to a number of the form bj, where b is real, as imaginary. A complex numberhas the form a + bj, where a and b are real. Some examples are: j, 0, −0.0023 + 0.553j,−3.44, and −106j.Real and imaginary partsIf c = a+bj, where a and b are real, we refer to a as the real part of c, and b as the imaginarypart of c. The real and imaginary parts are denoted<(c) = a, =(c) = b.Two complex numbers are equal if and only if their real parts are equal and also theirimaginary parts are equal. Another way to say this is that a complex number has only onerepresentation in the form a + bj.We can think of complex numbers as vectors with two (real) components, i.e., real andimaginary. Thus we can plot complex numbers in the complex plane, with the horizontal(x-) axis denoting real part and the vertical (y-) axis showing imaginary part.ConjugateThe conjugate of a complex number c = a+bj is denoted c or c∗, and is defined by c = a−bj.Geometrically, c is constructed by reflecting c through the real (i.e., horizontal) axis.Absolute valueThe magnitude, absolute value, or length of a complex number is defined as|a + bj| =√a2+ b2.Note that the magnitude is always nonnegative, and equals zero only if the complex numberis zero (i.e., 0 + 0j). The magnitude of a complex number is the Euclidean distance fromthe origin in the complex plane to the point.1AngleThe angle or phase or argument of the complex number a + bj is the angle, measured inradians, from the point 1 + 0j to a + bj, with counterclockwise denoting positive angle. Theangle of a complex number c = a + bj is denoted6c:6c = arctan b/a.Several comments are in order. First, angles that differ by a multiple of 2π are consideredequal. Second, the formula above uses the four quadrant arctan, and is much better writtenas6c = arctan(b, a) (often expressed as atan2(b,a) in computer languages). The angle ofthe complex number 0 is undefined. As examples,61.23 = 0 (but we could just as well write61.23 = −4π),6j = π/2,6(1 + j) = π/4,6(1 − j) = −π/4.AdditionWe define addition of complex numbers as(a + bj) + (c + dj) = (a + c) + (b + d)j,i.e., the real and imaginary parts add separately. This corresponds to vector addition in thecomplex plane. Subtraction is defined similarly.We can express the real and imaginary parts in terms of addition and conjugates as<(c) =c +c2, =(c) =c −c2j.The triangle inequality states that the absolute value of a complex sum is no more thanthe sum of the absolute values, i.e.,|f + g| ≤ |f| + |g|where f and g are complex numbers. The distance between two complex numbers f andg is given by |f − g| (which coincides with the Euclidean distance between f and g in thecomplex plane).MultiplicationComplex multiplication is based on the identity j2= −1:(a + bj)(c + dj) = ac + adj + bcj + bdj2= (ac − bd) + (ad + bc)j.Division is defined similarly: if f, g, and h are complex numbers, then f = g/h meansfh = g.The inverse of a complex number f means 1/f. If f = a + bj, where a and b are real,the inverse can be expressed as1/f =aa2+ b2− jba2+ b22(we’ll see later where this formula comes from).The absolute value of a product is the product of the absolute values, i.e., |fg| = |f ||g|,where f and g are complex numbers. The angle of a product is the sum of the angles, i.e.,6(fg) =6f +6g.Complex addition, subtraction, multiplication, and division obey the same standard rulesas real numbers: commutative, distributive, associative, etc.We can express the magnitude using the conjugate as |c| =√cc. (In particular, theproduct of a complex number with its conjugate is always real and nonnegative.) Sincecc = |c|2, we can express the inverse explicitly as1/c =c|c|2.Complex exponentialWith addition and multiplication, we can define polynomials of complex numbers. Thus forexample p(x) = 1 + x −x2evaluated at x = 1 + j yieldsp(1 + j) = 1 + (1 + j) + (1 + j)2= 2 + 3j.We can define more general functions of complex numbers via power series expansions.Perhaps the most important case is the complex exponential:ea+bj= eacos b + (easin b)j.One important special case isejφ= cos φ + j sin φ,where φ is real. For example, we have ejπ= −1.The formula for the exponential shows that|ec| = e<c,6ec= =c.Thus, we define the logarithm of a (nonzero) complex number c as log c = log |c| + j6c, sothat elog c= c. Note that the imaginary component of log c is ambiguous; we can freely addany multiple of 2π. Thus we can say thatlog(1 − j) = log√2 − j(π/4 + 2πk)where k is any integer. The ambiguity in the (imaginary part of the) logarithm requires care.Exponential representationFrom the exponential formula we see that we can express the complex number c = a + bj interms of the complex exponential as c = rejθ, wherer = |c|, θ =6c.3The form c = rejθis called the exponential or polar of the complex number c (since (r, θ)are the polar coordinates of the point with rectangular coordinates (a, b)).Note that r and θ can be thought of as related to the real and imaginary parts of thecomplex logarithm of c. It’s not too surprising, therefore, that the exponential representationof a product is easily expressed:³rejθ´³qejφ´= (rq)ej(θ+φ).In words: the magnitudes of a product multiply; the angles add. The conjugate and inversecan be expressed asc = re−jθ, 1/c = (1/r)e−jθ,where c = rejθ.This gives us a geometric picture of complex multiplication. Multiplication by a realnumber scales or stretches the number (reversing orientation if the real number is negative).Multiplication by j rotates a number by π/2 = 90◦; more generally multiplication byejθ= cos θ + j sin θrotates complex numbers (counterclockwise) by the angle


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Stanford EE 102 - Lecture Notes

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