The Central Idea
The real numbers are extremely powerful, but they do not contain a solution to every polynomial equation.
Consider
\[ x^2+1=0. \]Rearranging gives
\[ x^2=-1. \]No real number has a square equal to \(-1\). Complex numbers enlarge the number system by defining a new number \(i\) such that
\[ \boxed{i^2=-1}. \]Once \(i\) is available, the equation \(x^2=-1\) has the two solutions
\[ \boxed{x=i \qquad \text{or} \qquad x=-i}. \]Why Extend the Real Numbers?
Number systems have been extended many times when existing numbers were not sufficient for the operations mathematicians wanted to perform.
- Natural numbers support counting.
- Integers allow subtraction such as \(3-5\).
- Rational numbers allow quotients such as \(\frac{2}{3}\).
- Real numbers include irrational quantities such as \(\sqrt{2}\) and \(\pi\).
- Complex numbers allow square roots of negative real numbers.
Complex numbers do not replace the real numbers. They contain them as a special case.
The Imaginary Unit
The symbol
\[ i \]denotes the imaginary unit, defined by
\[ \boxed{i=\sqrt{-1}} \]with the fundamental property
\[ \boxed{i^2=-1}. \]The notation \(i=\sqrt{-1}\) is convenient, but the defining algebraic relationship \(i^2=-1\) is the more fundamental statement.
Powers of \(i\)
Powers of \(i\) repeat in a four-step cycle.
\[ i^1=i, \] \[ i^2=-1, \] \[ i^3=i^2i=-i, \] \[ i^4=i^2i^2=(-1)(-1)=1. \]Multiplying by \(i\) again restarts the pattern:
\[ i^5=i,\qquad i^6=-1,\qquad i^7=-i,\qquad i^8=1. \]Thus,
\[ \boxed{i,\,-1,\,-i,\,1} \]repeats indefinitely.
Reducing Large Powers of \(i\)
Because the powers repeat every four exponents, a large power can be reduced by dividing the exponent by \(4\) and examining the remainder.
For example, simplify
\[ i^{37}. \]Since
\[ 37=4(9)+1, \]we have
\[ i^{37}=(i^4)^9i=1^9i. \]Therefore,
\[ \boxed{i^{37}=i}. \]What Is a Complex Number?
A complex number is a number that can be written in the form
\[ \boxed{z=a+bi}, \]where \(a\) and \(b\) are real numbers.
In this form:
- \(a\) is the real part;
- \(b\) is the imaginary coefficient;
- \(bi\) is the imaginary part.
It is common to use \(z\), \(w\), or another lowercase letter to denote a complex number.
Real and Imaginary Parts
If
\[ z=a+bi, \]then the real and imaginary parts are written
\[ \boxed{\operatorname{Re}(z)=a} \]and
\[ \boxed{\operatorname{Im}(z)=b}. \]Notice that \(\operatorname{Im}(z)\) is the real coefficient \(b\), not the full term \(bi\).
For example, if
\[ z=4-7i, \]then
\[ \operatorname{Re}(z)=4 \]and
\[ \operatorname{Im}(z)=-7. \]Real Numbers Are Complex Numbers
Every real number can be written as a complex number with imaginary coefficient \(0\).
For example,
\[ 5=5+0i. \]Thus the real numbers sit inside the complex numbers:
\[ \boxed{\mathbb{R}\subset\mathbb{C}}. \]The symbol \(\mathbb{C}\) denotes the set of complex numbers.
Pure Imaginary Numbers
A complex number whose real part is \(0\) is called pure imaginary.
Examples include
\[ 3i,\qquad -8i,\qquad \frac{1}{2}i. \]Each has the form
\[ 0+bi. \]Examples of Complex Numbers
Each of the following is a complex number:
\[ 3+4i, \] \[ -2+7i, \] \[ 5, \] \[ -6i, \] \[ \sqrt{2}-\pi i. \]The coefficients \(a\) and \(b\) may be any real numbers.
When Are Two Complex Numbers Equal?
Two complex numbers are equal exactly when their real parts are equal and their imaginary parts are equal.
If
\[ a+bi=c+di, \]then
\[ \boxed{a=c \qquad \text{and} \qquad b=d}. \]This gives a useful way to solve equations containing complex numbers.
Example: Matching Real and Imaginary Parts
Suppose
\[ (2x-1)+(y+3)i=7-4i. \]Match the real parts:
\[ 2x-1=7. \]Thus,
\[ x=4. \]Match the imaginary parts:
\[ y+3=-4. \]Thus,
\[ y=-7. \]Therefore,
\[ \boxed{x=4,\qquad y=-7}. \]Adding Complex Numbers
Add real parts to real parts and imaginary parts to imaginary parts.
\[ (a+bi)+(c+di) = (a+c)+(b+d)i. \]For example,
\[ (3+4i)+(5-2i) \]becomes
\[ (3+5)+(4-2)i. \]Therefore,
\[ \boxed{(3+4i)+(5-2i)=8+2i}. \]Subtracting Complex Numbers
Subtraction also combines corresponding real and imaginary components.
\[ (a+bi)-(c+di) = (a-c)+(b-d)i. \]For example,
\[ (7+3i)-(2+8i) \]gives
\[ (7-2)+(3-8)i. \]Therefore,
\[ \boxed{(7+3i)-(2+8i)=5-5i}. \]Multiplying Complex Numbers
Complex numbers multiply using the distributive property, with the additional rule \(i^2=-1\).
Consider
\[ (2+3i)(4-i). \]Distribute:
\[ 8-2i+12i-3i^2. \]Since \(i^2=-1\),
\[ -3i^2=3. \]Thus,
\[ 8+10i+3. \]Therefore,
\[ \boxed{(2+3i)(4-i)=11+10i}. \]The General Product Formula
Multiplying
\[ (a+bi)(c+di) \]gives
\[ ac+adi+bci+bd i^2. \]Because \(i^2=-1\),
\[ ac-bd+(ad+bc)i. \]Therefore,
\[ \boxed{(a+bi)(c+di)=(ac-bd)+(ad+bc)i}. \]Squaring a Complex Number
Squaring is a special case of complex multiplication.
For
\[ z=a+bi, \]we have
\[ z^2=(a+bi)^2. \]Expand:
\[ a^2+2abi+b^2i^2. \]Therefore,
\[ \boxed{(a+bi)^2=(a^2-b^2)+2abi}. \]The Complex Conjugate
If
\[ z=a+bi, \]then its complex conjugate is
\[ \boxed{\overline{z}=a-bi}. \]The conjugate keeps the real part unchanged and reverses the sign of the imaginary part.
For example, the conjugate of
\[ 5-3i \]is
\[ \boxed{5+3i}. \]A Number Times Its Conjugate
Multiplying a complex number by its conjugate eliminates the imaginary term.
\[ (a+bi)(a-bi). \]This is a difference of squares:
\[ a^2-(bi)^2. \]Since \(i^2=-1\),
\[ a^2-b^2(-1). \]Therefore,
\[ \boxed{(a+bi)(a-bi)=a^2+b^2}. \]The result is always a nonnegative real number.
Dividing Complex Numbers
To divide by a complex number, multiply the numerator and denominator by the conjugate of the denominator.
Consider
\[ \frac{3+2i}{1-i}. \]Multiply by
\[ \frac{1+i}{1+i}. \]Then
\[ \frac{(3+2i)(1+i)}{(1-i)(1+i)}. \]The numerator is
\[ 3+3i+2i+2i^2=1+5i. \]The denominator is
\[ 1^2+1^2=2. \]Therefore,
\[ \boxed{\frac{3+2i}{1-i}=\frac12+\frac52 i}. \]Why the Conjugate Works in Division
For a denominator
\[ c+di, \]multiplying by its conjugate gives
\[ (c+di)(c-di)=c^2+d^2. \]The denominator becomes real, allowing the quotient to be rewritten in standard form \(a+bi\).
The Complex Plane
A complex number can be represented geometrically using a coordinate plane. This representation is called the complex plane or Argand plane.
For
\[ z=a+bi, \]plot the point
\[ \boxed{(a,b)}. \]The horizontal axis represents real values, and the vertical axis represents imaginary values.
The Real and Imaginary Axes
The horizontal axis of the complex plane is the real axis.
The vertical axis is the imaginary axis.
Thus:
- real numbers lie on the horizontal axis;
- pure imaginary numbers lie on the vertical axis;
- general complex numbers lie elsewhere in the plane.
Example: Plotting a Complex Number
Consider
\[ z=3+2i. \]Its real part is \(3\), and its imaginary part is \(2\).
Therefore, \(z\) corresponds to the point
\[ \boxed{(3,2)} \]in the complex plane.
Example: Negative Real and Imaginary Parts
Consider
\[ z=-4-3i. \]Then
\[ \operatorname{Re}(z)=-4 \]and
\[ \operatorname{Im}(z)=-3. \]The corresponding point is
\[ \boxed{(-4,-3)}. \]Complex Numbers as Position Vectors
The point \(a+bi\) may also be viewed as a vector from the origin to \((a,b)\).
\[ a+bi \quad \longleftrightarrow \quad (a,b). \]This viewpoint explains why complex addition resembles vector addition.
If
\[ z_1=a+bi \]and
\[ z_2=c+di, \]then
\[ z_1+z_2=(a+c)+(b+d)i, \]corresponding to
\[ (a,b)+(c,d)=(a+c,b+d). \]Geometric Meaning of Addition
Adding complex numbers adds their horizontal and vertical components.
For example,
\[ (2+i)+(1+3i)=3+4i. \]On the complex plane, this is the same componentwise rule used to add two-dimensional vectors.
The Modulus of a Complex Number
The modulus of
\[ z=a+bi \]is the distance from the origin to the point \((a,b)\).
By the Pythagorean theorem,
\[ \boxed{|z|=\sqrt{a^2+b^2}}. \]The same vertical bars used for absolute value are used for complex modulus.
Example: Finding a Modulus
Find the modulus of
\[ z=3+4i. \]Then
\[ |z|=\sqrt{3^2+4^2}. \]Therefore,
\[ |z|=\sqrt{9+16}=\sqrt{25}. \]So
\[ \boxed{|3+4i|=5}. \]Modulus and Conjugation
Because
\[ z\overline{z} = (a+bi)(a-bi) = a^2+b^2, \]and
\[ |z|^2=a^2+b^2, \]we obtain the important identity
\[ \boxed{z\overline{z}=|z|^2}. \]For \(z\neq0\), this also leads to
\[ \boxed{\frac{1}{z}=\frac{\overline{z}}{|z|^2}}. \]Distance Between Two Complex Numbers
The distance between two points in the complex plane can be written using complex subtraction.
For complex numbers \(z_1\) and \(z_2\), the distance between them is
\[ \boxed{|z_1-z_2|}. \]If
\[ z_1=a+bi \]and
\[ z_2=c+di, \]then
\[ |z_1-z_2| = \sqrt{(a-c)^2+(b-d)^2}. \]This is exactly the ordinary distance formula in the coordinate plane.
Example: Distance in the Complex Plane
Find the distance between
\[ z_1=1+2i \]and
\[ z_2=4+6i. \]Subtract:
\[ z_2-z_1=(4+6i)-(1+2i)=3+4i. \]Therefore,
\[ |z_2-z_1|=|3+4i|=5. \]The distance is
\[ \boxed{5}. \]Circles in the Complex Plane
The equation
\[ |z-z_0|=r \]describes all complex numbers whose distance from \(z_0\) is \(r\).
Therefore,
\[ \boxed{|z-z_0|=r} \]represents a circle centered at \(z_0\) with radius \(r\).
Example: Reading a Complex Circle
Consider
\[ |z-(2+i)|=3. \]The center is the complex number
\[ 2+i, \]corresponding to the point \((2,1)\), and the radius is \(3\).
Thus the equation describes the circle centered at
\[ \boxed{(2,1)} \]with radius
\[ \boxed{3}. \]Disks and Regions
Modulus inequalities describe regions as naturally as modulus equations describe circles.
The inequality
\[ |z-z_0|The inequality
\[ |z-z_0|\le r \]includes the boundary circle and is called a closed disk.
Conjugation as Reflection
If
\[ z=a+bi, \]then
\[ \overline{z}=a-bi. \]Geometrically, the point \((a,b)\) becomes \((a,-b)\).
Therefore, taking the complex conjugate reflects a point across the real axis.
Negation in the Complex Plane
Negating
\[ z=a+bi \]gives
\[ -z=-a-bi. \]The point \((a,b)\) moves to \((-a,-b)\).
Geometrically, this is a half-turn, or a rotation by \(180^\circ\), about the origin.
Multiplication by \(i\) as a Rotation
Let
\[ z=a+bi. \]Multiplying by \(i\) gives
\[ iz=ai+b i^2. \]Since \(i^2=-1\),
\[ iz=-b+ai. \]Thus the point
\[ (a,b) \]becomes
\[ (-b,a). \]This is exactly a counterclockwise rotation of \(90^\circ\) about the origin.
\[ \boxed{z\mapsto iz \text{ rotates the plane by }90^\circ}. \]Example: Rotate by Multiplying by \(i\)
Start with
\[ z=2+i. \]Multiply by \(i\):
\[ iz=i(2+i)=2i+i^2. \]Therefore,
\[ iz=-1+2i. \]The original point \((2,1)\) becomes \((-1,2)\), confirming the \(90^\circ\) counterclockwise rotation.
Multiplication by \(-1\)
Multiplication by \(-1\) sends
\[ a+bi \]to
\[ -a-bi. \]Geometrically, this rotates the point by \(180^\circ\).
Since
\[ i^2=-1, \]multiplying twice by \(i\) also produces a \(180^\circ\) rotation.
The Four Powers of \(i\) Geometrically
The cycle of powers of \(i\) has a direct geometric meaning.
- Multiplication by \(1\): no rotation.
- Multiplication by \(i\): rotate \(90^\circ\).
- Multiplication by \(-1\): rotate \(180^\circ\).
- Multiplication by \(-i\): rotate \(270^\circ\), or \(90^\circ\) clockwise.
This is the first indication that multiplication of complex numbers encodes geometry, not merely arithmetic.
Rectangular Form
The expression
\[ z=a+bi \]is often called the rectangular form or Cartesian form of a complex number.
The name comes from its direct correspondence with the Cartesian point
\[ (a,b). \]Later, we will describe the same complex number by its distance from the origin and its direction from the positive real axis.
Example: A Quadratic with Complex Solutions
Solve
\[ x^2+4=0. \]Rearrange:
\[ x^2=-4. \]Since
\[ \sqrt{-4}=2i, \]the solutions are
\[ \boxed{x=\pm2i}. \]Complex Numbers and the Quadratic Formula
The quadratic formula
\[ x=\frac{-b\pm\sqrt{b^2-4ac}}{2a} \]may produce a negative discriminant.
For example, solve
\[ x^2-2x+5=0. \]Here,
\[ a=1,\qquad b=-2,\qquad c=5. \]The discriminant is
\[ b^2-4ac=4-20=-16. \]Therefore,
\[ x=\frac{2\pm\sqrt{-16}}{2}. \]Since \(\sqrt{-16}=4i\),
\[ x=\frac{2\pm4i}{2}. \]Thus,
\[ \boxed{x=1\pm2i}. \]Complex Roots Often Occur in Conjugate Pairs
A polynomial with real coefficients has the important property that nonreal complex roots occur in conjugate pairs.
For example, if
\[ 1+2i \]is a root of a polynomial with real coefficients, then
\[ 1-2i \]is also a root.
This fact will become useful when factoring real polynomials and studying analytic functions.
Why Complex Numbers Matter for Polynomial Equations
Complex numbers provide a setting in which polynomial equations behave in a remarkably complete way.
The Fundamental Theorem of Algebra states that every nonconstant polynomial with complex coefficients has at least one complex root.
Equivalently, a polynomial of degree \(n\) factors into \(n\) linear factors over the complex numbers when roots are counted with multiplicity.
This is one reason \(\mathbb{C}\) is a natural setting for advanced analysis.
A First Look at Algebraic Closure
A number system is called algebraically closed when every nonconstant polynomial formed with numbers from that system has a root in the system.
The real numbers are not algebraically closed because
\[ x^2+1=0 \]has no real solution.
The complex numbers are algebraically closed.
We will not prove this result here, but it helps explain why complex numbers occupy such a central position in higher mathematics.
Complex Numbers Form a Number System Closed Under Arithmetic
If \(z\) and \(w\) are complex numbers, then the following are also complex numbers:
- \(z+w\);
- \(z-w\);
- \(zw\);
- \(\frac{z}{w}\), provided \(w\neq0\).
Thus the familiar arithmetic operations remain available after the number system is extended from \(\mathbb{R}\) to \(\mathbb{C}\).
Complex Numbers Are Not Ordered Like Real Numbers
Real numbers can be arranged on a line using relations such as
\[ xTherefore, statements such as
\[ 2+3i<4+i \]are not meaningful in the usual complex-number arithmetic.
Instead, complex numbers are compared using quantities such as modulus, real part, imaginary part, or geometric location.
Real Absolute Value and Complex Modulus
For a real number \(x\), absolute value measures distance from \(0\) on the number line:
\[ |x|. \]For a complex number \(z=a+bi\), modulus measures distance from \(0\) in the plane:
\[ |z|=\sqrt{a^2+b^2}. \]Thus modulus is the two-dimensional extension of ordinary absolute value.
Basic Properties of the Modulus
For complex numbers \(z\) and \(w\),
\[ |z|\ge0, \] \[ |z|=0 \iff z=0, \] \[ |zw|=|z||w|, \] \[ \left|\frac{z}{w}\right| = \frac{|z|}{|w|} \qquad (w\neq0). \]The triangle inequality also holds:
\[ \boxed{|z+w|\le |z|+|w|}. \]Geometrically, this says that the direct distance from the origin to a sum cannot exceed the total distance traveled along two component vectors.
Basic Properties of Conjugation
Conjugation interacts cleanly with arithmetic:
\[ \overline{z+w} = \overline{z}+\overline{w}, \] \[ \overline{zw} = \overline{z}\,\overline{w}, \] \[ \overline{\overline{z}}=z, \] \[ |\overline{z}|=|z|. \]These relationships will be used repeatedly in complex algebra and complex analysis.
Recovering Real and Imaginary Parts with Conjugation
Let
\[ z=a+bi. \]Then
\[ z+\overline{z} = (a+bi)+(a-bi) = 2a. \]Therefore,
\[ \boxed{\operatorname{Re}(z)=\frac{z+\overline{z}}{2}}. \]Also,
\[ z-\overline{z} = 2bi. \]Therefore,
\[ \boxed{\operatorname{Im}(z)=\frac{z-\overline{z}}{2i}}. \]The Reciprocal of a Complex Number
For
\[ z=a+bi\neq0, \]the reciprocal can be found by multiplying by the conjugate:
\[ \frac{1}{a+bi} \cdot \frac{a-bi}{a-bi}. \]This gives
\[ \boxed{ \frac{1}{a+bi} = \frac{a-bi}{a^2+b^2} }. \]Equivalently,
\[ \boxed{ \frac{1}{z} = \frac{\overline{z}}{|z|^2} }. \]A Preview of the Geometry of Reciprocals
The formula
\[ \frac{1}{z} = \frac{\overline{z}}{|z|^2} \]shows two geometric effects:
- conjugation reflects the point across the real axis;
- division by \(|z|^2\) rescales its distance from the origin.
Once polar form is introduced, this geometric interpretation becomes even clearer.
Why Complex Multiplication Looks Different from Vector Multiplication
Complex addition is identical to ordinary componentwise vector addition. Complex multiplication is different.
The formula
\[ (a+bi)(c+di) = (ac-bd)+(ad+bc)i \]mixes the coordinates.
This mixing is not arbitrary. In polar form, complex multiplication will become a combination of scaling and rotation.
That geometric interpretation is one of the central ideas of complex analysis.
One Complex Dimension, Two Real Dimensions
A single complex number
\[ z=a+bi \]requires two real coordinates:
\[ (a,b). \]For this reason, the complex plane can be identified with
\[ \mathbb{R}^2. \]But complex multiplication gives \(\mathbb{C}\) additional algebraic structure that an ordinary pair of real coordinates does not automatically possess.
Why the Complex Plane Matters for Analysis
In real calculus, a function is often studied along a line.
In complex analysis, a function
\[ f:\mathbb{C}\to\mathbb{C} \]takes points from the complex plane and sends them to other points in the complex plane.
This changes how we think about:
- limits;
- continuity;
- differentiation;
- integration;
- series;
- geometry and mapping.
A complex function must behave consistently as its input approaches a point from every direction in the plane.
Common Error: Treating \(i\) as an Ordinary Real Variable
The symbol \(i\) is not an arbitrary variable.
It satisfies the fixed identity
\[ i^2=-1. \]Therefore, after multiplying complex expressions, every occurrence of \(i^2\) should be replaced by \(-1\).
Common Error: Misusing Square-Root Rules
Over the real numbers, the identity
\[ \sqrt{ab}=\sqrt{a}\sqrt{b} \]requires conditions on \(a\) and \(b\).
Extending this rule carelessly to negative quantities can produce contradictions.
For example, writing
\[ \sqrt{(-1)(-1)} = \sqrt{-1}\sqrt{-1} \]would give
\[ 1=i^2=-1, \]which is impossible.
In complex arithmetic, manipulate square roots of negative numbers with care and reduce expressions through the defining relation \(i^2=-1\).
Common Error: Misidentifying the Imaginary Part
If
\[ z=5-7i, \]then
\[ \operatorname{Im}(z)=-7, \]not \(-7i\).
The imaginary part function returns the real coefficient multiplying \(i\).
Common Error: Changing Both Signs in a Conjugate
The conjugate of
\[ a+bi \]is
\[ a-bi. \]Only the sign of the imaginary term changes.
The expression
\[ -a-bi \]is \(-z\), not \(\overline{z}\).
Common Error: Leaving a Complex Number in the Denominator
A quotient such as
\[ \frac{2+i}{3-4i} \]is valid, but when the goal is standard rectangular form, the denominator should be made real.
Multiply by the conjugate
\[ 3+4i. \]This converts the denominator to
\[ 3^2+4^2=25. \]Common Error: Trying to Order Complex Numbers
The complex plane provides positions, but not a real-number style ordering.
It is meaningful to say
\[ |z_1|<|z_2|, \]because moduli are real numbers.
It is generally not meaningful to write
\[ z_1A Reliable Procedure for Complex Arithmetic
- Write each number in the form \(a+bi\).
- Combine real parts and imaginary parts separately for addition or subtraction.
- Use ordinary distribution for multiplication.
- Replace every \(i^2\) by \(-1\).
- For division, multiply by the conjugate of the denominator.
- Collect the result into standard form \(a+bi\).
- Check the result numerically or by reversing the operation when practical.
A Reliable Procedure for Using the Complex Plane
- Write the complex number as \(a+bi\).
- Identify \(a=\operatorname{Re}(z)\).
- Identify \(b=\operatorname{Im}(z)\).
- Plot the point \((a,b)\).
- Use \(|z|=\sqrt{a^2+b^2}\) for distance from the origin.
- Use \(|z-z_0|\) for distance from another complex number \(z_0\).
- Interpret conjugation as reflection across the real axis.
Worked Example: Add, Subtract, and Multiply
Let
\[ z=3-2i \]and
\[ w=-1+4i. \]Add
\[ z+w=(3-1)+(-2+4)i. \] \[ \boxed{z+w=2+2i}. \]Subtract
\[ z-w=(3-(-1))+(-2-4)i. \] \[ \boxed{z-w=4-6i}. \]Multiply
\[ zw=(3-2i)(-1+4i). \] \[ zw=-3+12i+2i-8i^2. \] \[ zw=-3+14i+8. \] \[ \boxed{zw=5+14i}. \]Worked Example: Divide and Verify
Compute
\[ \frac{4+3i}{2-i}. \]Multiply by the conjugate of the denominator:
\[ \frac{4+3i}{2-i} \cdot \frac{2+i}{2+i}. \]The numerator is
\[ (4+3i)(2+i) = 8+4i+6i+3i^2. \] \[ =5+10i. \]The denominator is
\[ (2-i)(2+i)=4+1=5. \]Therefore,
\[ \boxed{\frac{4+3i}{2-i}=1+2i}. \]Check by multiplication:
\[ (1+2i)(2-i) = 2-i+4i-2i^2 = 4+3i. \]The original numerator returns.
Worked Example: Geometry and Modulus
Let
\[ z=-3+4i. \]The point in the complex plane is
\[ (-3,4). \]Its modulus is
\[ |z| = \sqrt{(-3)^2+4^2} = \sqrt{9+16} = 5. \]Its conjugate is
\[ \overline{z}=-3-4i, \]corresponding to the reflected point
\[ (-3,-4). \]Worked Example: A Complex Locus
Describe the set
\[ |z-(1-2i)|=4. \]The quantity
\[ |z-(1-2i)| \]is the distance from \(z\) to the point
\[ 1-2i. \]Therefore, the set is a circle centered at
\[ \boxed{(1,-2)} \]with radius
\[ \boxed{4}. \]Practice 1: Powers of \(i\)
Simplify:
\[ i^{58}. \]Practice 2: Real and Imaginary Parts
For
\[ z=-5+12i, \]find \(\operatorname{Re}(z)\) and \(\operatorname{Im}(z)\).
Practice 3: Addition
Simplify:
\[ (4-7i)+(-3+2i). \]Practice 4: Multiplication
Simplify:
\[ (2+5i)(3-2i). \]Practice 5: Conjugate
Find the conjugate of
\[ -6-9i. \]Practice 6: Division
Write in rectangular form:
\[ \frac{5+i}{2+i}. \]Practice 7: Modulus
Find
\[ |-8+6i|. \]Practice 8: Distance
Find the distance between
\[ z_1=2-i \]and
\[ z_2=-1+3i. \]Practice 9: Complex Plane
Identify the center and radius of
\[ |z+3-2i|=5. \]Practice 10: Quadratic Equation
Solve:
\[ x^2+6x+13=0. \]Practice Answers
Practice 1
Since
\[ 58=4(14)+2, \]we have
\[ i^{58}=i^2. \]Therefore,
\[ \boxed{i^{58}=-1}. \]Practice 2
\[ \boxed{\operatorname{Re}(z)=-5} \]and
\[ \boxed{\operatorname{Im}(z)=12}. \]Practice 3
\[ (4-7i)+(-3+2i) = 1-5i. \] \[ \boxed{1-5i}. \]Practice 4
\[ (2+5i)(3-2i) = 6-4i+15i-10i^2. \] \[ =16+11i. \] \[ \boxed{16+11i}. \]Practice 5
\[ \boxed{-6+9i}. \]Practice 6
\[ \frac{5+i}{2+i} \cdot \frac{2-i}{2-i} = \frac{(5+i)(2-i)}{5}. \] \[ (5+i)(2-i) = 10-5i+2i-i^2 = 11-3i. \]Therefore,
\[ \boxed{ \frac{5+i}{2+i} = \frac{11}{5}-\frac{3}{5}i }. \]Practice 7
\[ |-8+6i| = \sqrt{(-8)^2+6^2} = \sqrt{64+36} = 10. \] \[ \boxed{10}. \]Practice 8
\[ z_2-z_1 = (-1+3i)-(2-i) = -3+4i. \] \[ |z_2-z_1| = \sqrt{(-3)^2+4^2} = 5. \] \[ \boxed{5}. \]Practice 9
Rewrite
\[ |z+3-2i| = |z-(-3+2i)|. \]Therefore, the center is
\[ \boxed{-3+2i} \]or the point
\[ \boxed{(-3,2)}, \]and the radius is
\[ \boxed{5}. \]Practice 10
Use the quadratic formula:
\[ x = \frac{-6\pm\sqrt{36-52}}{2}. \] \[ x = \frac{-6\pm\sqrt{-16}}{2}. \] \[ x = \frac{-6\pm4i}{2}. \]Therefore,
\[ \boxed{x=-3\pm2i}. \]Checking Strategies
Complex-number calculations can often be checked quickly.
- After multiplication, verify that no \(i^2\) terms remain.
- After division, multiply the quotient by the original denominator and verify the numerator returns.
- For a conjugate product, check that the result is real and nonnegative.
- For modulus, verify that the result is a nonnegative real number.
- For geometric interpretations, translate \(a+bi\) back to the point \((a,b)\).
Connections to Earlier Mathematics
Complex numbers bring several earlier ideas together:
- Algebra: distribution, factoring, polynomial equations, and conjugates.
- Coordinate geometry: ordered pairs, distance, circles, and reflection.
- Vectors: componentwise addition and position vectors.
- Trigonometry: rotation and angular description, developed in the next lesson.
- Analysis: limits, continuity, differentiation, and integration in a two-dimensional input space.
What to Remember
The imaginary unit is defined by
\[ \boxed{i^2=-1}. \]Every complex number can be written
\[ \boxed{z=a+bi}. \]Its real and imaginary parts are
\[ \boxed{\operatorname{Re}(z)=a}, \] \[ \boxed{\operatorname{Im}(z)=b}. \]Its conjugate is
\[ \boxed{\overline{z}=a-bi}, \]and its modulus is
\[ \boxed{|z|=\sqrt{a^2+b^2}}. \]Geometrically,
\[ \boxed{a+bi \longleftrightarrow (a,b)}. \]This correspondence turns complex arithmetic into geometry and lays the foundation for the rest of complex analysis.
Where We Go Next
Rectangular form describes a complex number by its horizontal and vertical coordinates. But the complex plane also suggests another description: how far the number lies from the origin and the direction in which it points.
The next lesson, Polar and Exponential Forms, develops modulus, argument, polar coordinates, Euler's formula, and the exponential representation \[ z=re^{i\theta}, \] revealing complex multiplication as a direct combination of scaling and rotation.