Historical Background of

Complex Numbers� Quadratic and Cubic equations� Quadratic and Cubic equations

� The Fundamental Theorem of Algebra

� The number i

Only Positive Numbers� 13th century� Works in mathematics were translated from the Arabic

into Latin allowing Western European scholars to learn about the medieval Arabic-language learn about the medieval Arabic-language mathematics and the older Greek mathematics,

� Only positive numbers were considered to be numbers.

� Negative numbers were not yet accepted as entities. (Some ancient cultures, including that of China and India, accepted negative numbers, but not the ones mentioned above.)

Solution of Quadratics, ax2 + bx + c = 0

15th CenturyPresent

� Quadratic equations were classified into four different kinds depending on the signs of the coefficients a, b, and c

� x = –b ±√(b2 – 4ac)2a

the coefficients a, b, and c� Move the negative terms to the

other side of the equation to get four forms� x2 = c� x2 + bx =c� x2 + c = bx� x2 = bx + c

� There are other forms, but either they have no solutions among the positive numbers or else they can be reduced to linear equations.

� There are� two distinct real solutions if

the discriminant b2 – 4ac is positive,

� one double real solution if the discriminant is 0,

� no real solutions if the discriminant is negative.

Solution of Cubics, x3 + bx2 + cx + d = 0

� As with the quadratic equation, there are several forms for the cubic when negative terms are moved to the other side of the equation and zero terms dropped.terms dropped.

Solution of Cubics, x3 + bx2 + cx + d = 0

� 16th century

� Great controversy in Italy between Cardano (1501-1576) and Tartaglia (1499-1557) over credit

� Negative numbers were becoming legitimatized,� Negative numbers were becoming legitimatized,

� A deeper insight into equations was developed,

� The first inkling of a complex number appeared.

� Symbolic algebra had not been developed, so all the equations were written in words instead of symbols!

Cardano’s “Ars Magna”� Finds negative solutions to equations� Calls these numbers "fictitious" � Noted an important fact connecting solutions of a cubic

equation to its coefficients:’-equation to its coefficients:’-� the sum of the solutions is the negation of b, the coefficient of

the x2 term.

� Mentions that the problem of dividing 10 into two parts so that their product is 40 would have to be � 5 + √(–15) and 5 – √(–15).

� Cardano did not go further into what later became to be called complex numbers than this observation.

Bombelli (1526-1572)� Gave examples to Cardano’s cubic formulas� One of Cardano's cubic formulas gives the solution to the equation x3 = cx + d as

� x = 3√(d/2 + √e) + 3√(d/2 – √e) where e = (d/2)2 – (c/3)3).

� Bombelli used this to solve the equation x3 = 15x + 4 to get the solution x = 3√(2 + √–121) + 3√(2 – √–121)

� Now, the square root of –121 is not a real number; � it's neither positive, negative, nor zero.

� Bombelli continued to work with this expression until he found equations that lead him to the solution 4. He determined that � √(2 + √–121) = 2 + √–1� √(2 – √–121) = 2 – √–1� and, therefore, the solution x = 4.

� This example is not given to show that Bombelli knew everything there is to know about complex numbers, rather to indicate that he was starting to understand them.

Fundamental Theorem of Algebra

- Girard (1595-1632)� A general relation between the n solutions to an nth degree

equation and its n coefficients.

� An nth degree equation can be written in modern notation as � xn + a1x

n–1 + ... + an–2x2 + an–1x + an = 0 � x + a1x + ... + an–2x + an–1x + an = 0

� where the coefficients a1, ..., an–2, an–1, and an are all constants.

� Girard said that an nth degree equation admits of n solutions, if you allow all roots and count roots with multiplicity, e.g.� the equation x2 + 1 = 0 has the two solutions √–1 and –√–1, and � the equation x2 – 2x + 1 = 0 has the two solutions 1 and 1.

� Girard wasn't particularly clear what form his solutions were to have, just that there be n of them: x1, x2, ..., xn–1, and xn.

Fundamental Theorem of Algebra

- Girard (1595-1632)

� Girard gave the relation between the n roots x1, x2, ..., xn, and xn and the n coefficients a1, ..., an–2, an–1, and an

that extends Cardano's remark. that extends Cardano's remark.

� the sum of the roots x1 + x2 + ..., + xn is –a1, the negation of the coefficient of xn–1 (Cardano's remark).

� the sum of all products of pairs of solutions is a2.

� the sum of all products of triples of solutions is –a3.

� And so on until the product of all n solutions is either an

(when n is even) or –an (when n is odd).

Girard’s principle of algebra:

An exampleThe 4th degree equation x4 – 6x3 + 3x2 + 26x – 24 = 0

has the four solutions –2, 1, 3, and 4.

� The sum of the solutions equals 6, � that is –2 + 1 + 3 + 4 = 6.

� The sum of all products of pairs (six of them) is 3 � (–2)(1) + (–2)(3) + (–2)(4) + (1)(3) + (1)(4) + (3)(4)

� The sum of all products of triples (four of them) is 26 � (–2)(1)(3) + (–2)(1)(4) + (–2)(3)(4) + (1)(3)(4)

� The product of all four solutions is –24.

Descartes (1596–1650)� Studied this relation between solutions and

coefficients, and showed more explicitly why the relationship holds.

� Called negative solutions "false" � Called negative solutions "false"

� treated other solutions (that is, complex numbers) "imaginary".

17th & 18th Century � Negative numbers became full–fledged numbers. � Complex numbers remained in limbo.

� They weren't considered to be real numbers, but they were useful in the theory of equations. were useful in the theory of equations.

� Complex numbers of the form a + b√–1 were sufficient to solve quadratic equations, but it wasn't clear they were enough to solve cubic and higher-degree equations.

� The part of the Fundamental Theorem of Algebra which stated there actually are n solutions of an nth degree equation was yet to be proved

The Number i� Euler (1707-1783) made the observation that

� eix = cos x + i sin x where i denotes √–1.

� This equation allows us to interpret the exponentiation of an imaginary number ix as having

Square Root of

Minus One, √–1

exponentiation of an imaginary number ix as having � a real part, cos x

� an imaginary part, i sin x.

� This was a useful observation in the solution of differential equations.

� Because of this and other uses of i, it became quite acceptable for use in mathematics.

x + yi� Used by research mathematicians, end 18th century

� Common to represent them as points in the plane.

� On the horizontal x-axis, � place the real numbers, x + 0i� place the real numbers, x + 0i

� with positive numbers to the right

� and negative ones to the left.

� On the vertical y-axis� imaginary numbers, 0 + yi,

� positive values of y are up,

� negative ones are down.

� Thus, i is located one unit above 0 (the origin, where the axes meet), and –i is located one unit below 0.

The Fundamental Theorem of

Algebra – proved!� Gauss published in 1799 his first proof that an nth

degree equation has n roots each of the form a + bi, for some real numbers a and b.

� Once he had done that, it was known that complex � Once he had done that, it was known that complex numbers (in the sense of solutions to algebraic equations) were the numbers a + bi, and

� It was appropriate to call the xy-plane the "complex plane".

The Mathematics of

Complex Numbers4. The complex plane, addition and subtraction Notation, arithmetic operations on C, parallelogram rule, addition as translation,

negation and subtraction

5. Absolute value The unit circle, the triangle inequality

6. Multiplication 6. Multiplication Multiplication done algebraically, multiplying a complex number by a real number,

multiplication and absolute value, powers of i, roots of unity, multiplying a complex number by i, a geometric interpretation of multiplication

7. Angles and polar coordinates

8. Reciprocals, conjugation, and division Reciprocals done geometrically, complex conjugates, division

9. Powers and roots Powers, roots, more roots of unity

The Complex Plane� Since Gauss proved the Fundamental Theorem of

Algebra, we know that all complex numbers are of the form x + yi, where x and y are real numbers, real numbers being all those numbers which are positive, negative, or zero. negative, or zero.

� Therefore, we can use the xy-plane to display complex numbers.

� We'll even call it the complex plane when we use the xy-plane that way.

� That gives us a second way to complex numbers, the first way being algebraically as in the expression x + yi.

The Complex plane

Notation�The standard symbol for the set of all complex

numbers is C, and the complex plane is C. � Real variables - use x and y

� Complex variables - use z and w� Complex variables - use z and w

� In general, the x part of a complex number z = x + yi is called the real part of z, while y is called the imaginary part of z. (Sometimes yi is called the imaginary part.)

The xy-plane� When we use the xy-plane for the complex plane C,

� the x-axis is called the real axis,

� the y-axis is called the imaginary axis.

Real numbers are special cases of complex numbers; � Real numbers are special cases of complex numbers;

� the numbers x + yi when y is 0, that is,

� they're the numbers on the real axis.

� For instance, the real number 2 is 2 + 0i.

� The numbers on the imaginary axis are sometimes called purely imaginary numbers.

Addition and Subtraction

Arithmetic Operations on C� To add or subtract two complex numbers, just add

or subtract the corresponding real and imaginary parts. � For instance, the sum of 5 + 3i and 4 + 2i is 9 + 5i.

� For another, the sum of 3 + i and –1 + 2i is 2 + 3i.� For another, the sum of 3 + i and –1 + 2i is 2 + 3i.

� Addition can be represented graphically on the complex plane C. � z = 3 + i is located 3 units to the right of the imaginary

axis and 1 unit above the real axis,

� w = –1 + 2i is located 1 unit left and 2 units up.

� So the sum z + w = 2 + 3i is 2 units right and 3 units up.

Parallelogram rule

0, z = 3 + i, w = –1 + 2i, and z + w = 2 + 3i

Parallelogram Rule Note in the last example that the

four complex numbers 0, z = 3 + i,

w = –1 + 2i, and z + w = 2 + 3i are

the corners of a parallelogram.

This is generally true.This is generally true.

To find where in the plane C the

sum z + w of two complex

numbers z and w is located, plot z

and w, draw lines from 0 to each

of them, and complete the

parallelogram. The fourth vertex

will be z + w.

Addition as Translation.

Addition as Translation� Addition by w is a

transformation of the plane C.

� Every point in C is moved the same direction and distance when w is added to it. when w is added to it.

� We can say that addition by w gives a translation the plane C in the direction and distance from 0 to w.

� The term "vector" is usually used in the description: "the plane is translated along the vector 0w

Adding w to 0 gives w, so 0 is moved to win this transformation. z is moved to z + w, so z is moved in the same direction the same distance.

Geometric interpretation

of negation

Geometric interpretation

of negation � The negation of a complex number will be located just

opposite 0 and the same distance away.same distance away.

� The negation of x + yi is –x – yi� For example, z = 2 + i is

located 2 units right and one unit up, so its negation –z = –2 – i is located 2 units left and one unit down.

Negation as Transformation

of plane C

� If you rotate the plane 180° around 0, then every point z is sent to its negation –z. Thus, negation gives a 180° rotation.

Geometric Rule For

Subtraction� From addition and negation, you can determine what the

geometric rule is for subtraction. geometric rule is for subtraction.

� To find where z – w will be, first negate w by finding the point opposite 0, then use the parallelogram rule.

� We can interpret subtraction of w as a transformation of C:

the plane is translated along the vector from 0 to –w.

� Another way of saying that is that the plane is translated

along the vector w0.

Concept of Absolute Value� The absolute value |x| of a real number x is

� itself, if it's positive or zero,

� but if x is negative, then its absolute value |x| is its negation –x, that is, the corresponding its negation –x, that is, the corresponding positive value.

� For example, |3| = 3, but |–4| = 4.

� The absolute value function strips a real number of its sign.

Absolute value for a

complex number z, |z|� For a complex number z = x

+ yi, we define the absolute value |z| as being the value |z| as being the distance from z to 0 in the complex plane C.

� |z|2 = x2 + y2

� since |x|2 = x2 and |y|2 = y2

� (x and y are real numbers)

� That gives us the formula

� |z| = √(x2 + y2)

The Unit Circle� 1 is the absolute value of both 1

and –1,� it's also the absolute value of

both i and –i� since they're both one unit away � since they're both one unit away

from 0 on the imaginary axis.� The unit circle is the circle of

radius 1 centered at 0. � It includes all complex numbers

of absolute value 1, so it has the equation |z| = 1.

� A complex number z = x + yi will lie on the unit circle when x2 + y2

= 1.

Multiplying Algebraically� (3 + 2i)(1 + 4i) = 3 + 12i + 2i + 8i2.

�Now the 12i + 2i simplifies to 14i,.

�What about the 8i2? �What about the 8i2? � Remember we introduced i as an abbreviation

for √–1, the square root of –1. In other words, iis something whose square is –1.

� Thus, 8i2 equals –8.

�Therefore, the product (3 + 2i)(1 + 4i) equals –5 + 14i.

Multiplying Algebraically� If you generalize this example, you'll get the general

rule for multiplication

�(x + yi)(u + vi) = (xu – yv) + (xv + yu)i

� Remember that (xu – yv), the real part of the product,

� is the product of the real parts minus the product of the imaginary parts,

� but (xv + yu), the imaginary part of the product,

� is the sum of the two products of one real part and the other imaginary part.

Multiplying a complex number

by a real number.

� (x + yi)(u + vi) = (xu – yv) + (xv + yu)i� if v is zero, then you get a formula for multiplying a

complex number x + yi and a real number u together:

�(x + yi) u = xu + yu i. � Multiply both parts of the complex number by the real number

Multiplication and absolute value,

|zw| = |z| |w| � To show that|zw|2 = |z|2|w|2.

� Let z be x + yi, and let w be u + vi.

� Then, according to the formula for multiplication, zw equals (xu – yv) + (xv + yu)i.zw equals (xu – yv) + (xv + yu)i.

� |z|2 = x2 + y2 and |w|2 = u2 + v2

� since zw = (xu – yv) + (xv + yu)i,

� |wz|2 = (xu – yv)2 + (xv + yu)2

� So, to show |zw|2 = |z|2|w|2, we show that

� (xu – yv)2 + (xv + yu)2 = (x2 + y2) (u2 + v2)

Powers of i�i2 = –1.

�i3 = i2 times I, i.e. i3 = –i.� Cube of i is its own negation.

�i4 .= 1� i4 is the square of i2, ie, square of –1.

� i is a fourth root of 1.

� –i is another fourth root of 1.

� –1 and 1 are square roots of 1,

� all 4 fourth roots of 1 are 1, i, –1, and –i.

� Fundamental Theorem of Algebra

� z4 = 1 is a fourth-degree equation so must have exactly four roots.

References� Dave's Short Course on Complex Numbers,

Retrieved from http://www.clarku.edu/~djoyce/complex/