Download Review of complex number arithmetic

Review of complex number arithmetic:
addition, subtraction, and multiplication.
The imaginary number i is defined where i2 = -1; thus, i = √−1.
All complex numbers can be written in the form a + bi, where a and b are real numbers. Note that if
b=0, then the number is real so the set of real numbers is a subset of the complex numbers.
Arithmetic operations for the complex numbers parallel those of irrational numbers of the form a+b√𝒄.
Powers of i are cyclic:
i = i5 = i9 = … = √−1
i2 = i6 = i10 = … -1
i3 = i7 = i11 = … = −i 𝑜𝑟 − √−1
i4 = i8 = i12 = … = 1
1. Simplify, writing your answer in the form a + bi:
a) i 42 =
b) i 113 =
c) 4 + 5i – 3i2 +11i3 – 6i4 =
d) (3 - 5i) + (8 + 2i) =
e) (-1 + i) – 2(-4 + 3i) =
f) (4 + 5i)(3 - 5i) =
g) (-1 + 2i)3 =
2.
Is the set of complex numbers closed with respect to the operation:
a. Addition?
b. Subtraction?
c. Multiplication?
d. Division?
3. What is the additive identity for the complex numbers?
4. What is the additive inverse of the number a + bi ?
Division, Inverses, & Conjugates
When trying to simplify an irrational number of the form
1
𝑎+√𝑏
where a≠0 and b>0 are integers and b≠
𝑎2 , by removing the radical from the denominator (rationalization), we know we must multiply the
numerator and denominator by the factor 𝑎 − √𝑏 so the expression becomes
1
𝑎+√𝑏
=
1
𝑎+√𝑏
•
𝑎−√𝑏
𝑎−√𝑏
=
𝑎−√𝑏
𝑎 2 −𝑏
.
This expression now has a non-zero denominator which is an integer and the radical has now moved
into the numerator.
Similarly, we can do the same with a complex number of the form
1
.
𝑎+𝑏i
Given a number a + bi, the number a - bi is called the complex conjugate. It is used to rationalize the
denominator in the reciprocal of a complex number and in division of complex numbers, thus allowing
us to write the number again in the standard form.
5. What is the conjugate of the following numbers?
a) 3i ________
b) 4 + 5i ________
c) – 3 – 2i __________
6. What is the multiplicative identity for the complex numbers?
7. What is the multiplicative inverse of the complex number a + bi ?
8. Write in the form a + bi the following:
a)
c)
1
2+3𝑖
1 − i 2 + 3𝑖
(5 + 3i)2 2 − 𝑖
=
We recall the quadratic formula 𝑥 =
b)
1+i
5−3𝑖
d)
1−i
(5 + 3i)2
−𝑏±√𝑏2 −4𝑎𝑐
2𝑎
÷
2 + 3𝑖
2−𝑖
=
which gives us the solutions to the quadratic equation
ax2+bx+c=0, a≠0.
By the trichotomy principle, the discriminant could be positive, zero, or negative. We recall that the
graph of the quadratic equation y = ax2+bx+c is a parabola opening up if a>0 and downward if a<0.
If the graph has two x intercepts, this corresponds to the situation where the discriminant is positive.
If the graph has one x intercept, this corresponds to the situation where the discriminant is zero.
If the graph has NO x intercept, this corresponds to the situation where the discriminant is negative.
Thus, imaginary numbers occur “naturally” when using the quadratic formula.
9. Find the solution(s) to the following quadratic equations. Put the answers in the form a+bi:
a) x2 +x+1=0
b) x2 = -8
c) 2x2 -x+1=0
10. Find a real polynomial of degree 2 with x=2 and x=3 as roots.
11. Find a real polynomial of degree 3 with x=0, x=-1 and x=6 as roots.
Imaginary roots of polynomial equations with real coefficients occur in conjugate pairs; i.e. if a+bi is a
root, then so is a-bi.
12. Find a polynomial of degree 2 with real coefficients if x=3-i is a root.
13. Find a polynomial of degree 4 with real coefficients if x=-1 is a double root and x=2+3i is also a
root.