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CLEP General Mathematics: Complex Numbers

Subjects : clep, math

Instructions:

Answer 50 questions in 15 minutes.
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Match each statement with the correct term.
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This is a study tool. The 3 wrong answers for each question are randomly chosen from answers to other questions. So, you might find at times the answers obvious, but you will see it re-enforces your understanding as you take the test each time.

1. When two complex numbers are subtracted from one another.
Complex Subtraction
Real Numbers
Complex Division
the vector (a -b)

2. E^(i?) = cos? + isin? ; e^(ip) + 1 = 0

3. In the same way that we think of real numbers as being points on a line - it is natural to identify a complex number z=a+ib with the point (a -b) in the cartesian plane.
Complex numbers are points in the plane
Complex Subtraction
Complex Number Formula
rational

4. In this amazing number field every algebraic equation in z with complex coefficients
has a solution.
cos iy
point of inflection
Polar Coordinates - sin?

5. E^(ln r) e^(i?) e^(2pin)
point of inflection
complex numbers
e^(ln z)
multiplying complex numbers

6. The field of all rational and irrational numbers.
imaginary
0 if and only if a = b = 0
z1 ^ (z2)
Real Numbers

7. 1
Complex Numbers: Multiply
Every complex number has the 'Standard Form': a + bi for some real a and b.
(a + c) + ( b + d)i
i^2

8. When two complex numbers are multipiled together.
sin z
Complex Multiplication
i^0
Complex Number Formula

9. Have radical
Complex Multiplication
radicals
For real a and b - a + bi = 0 if and only if a = b = 0
i^2 = -1

10. To simplify the square root of a negative number
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
radicals
v(-1)
|z-w|

11. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
How to solve (2i+3)/(9-i)
i^4
How to add and subtract complex numbers (2-3i)-(4+6i)
point of inflection

12. Root negative - has letter i
i^1
e^(ln z)
imaginary
Polar Coordinates - z?¹

13. V(zz*) = v(a² + b²)
|z| = mod(z)
Imaginary Unit
Roots of Unity
complex

14. A² + b² - real and non negative
Polar Coordinates - cos?
Subfield
interchangeable
zz*

15. 2nd. Rule of Complex Arithmetic

16. What about dividing one complex number by another? Is the result another complex number? Let's ask the question in another way. If you are given four real numbers a -b -c and d - can you find two other real numbers x and y so that
We say that c+di and c-di are complex conjugates.
imaginary
the vector (a -b)
i^3

17. I
four different numbers: i - -i - 1 - and -1.
Polar Coordinates - Multiplication by i
i^1
Liouville's Theorem -

18. 1. i^2 = -1 2. Every complex number has the 'Standard Form': a + bi for some real a and b. 3. For real a and b - a + bi = 0 if and only if a = b = 0 4. (a + bi) = (c + bi) = (a + c) + ( b + d)i 5. (a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc
Rules of Complex Arithmetic
Complex Number
subtracting complex numbers
Polar Coordinates - sin?

19. 1
real
sin iy
cosh²y - sinh²y
cos iy

20. 2a
z + z*
Polar Coordinates - Division
Absolute Value of a Complex Number
-1

21. All numbers
irrational
Complex Exponentiation
We say that c+di and c-di are complex conjugates.
complex

22. zn = (cos? + isin?)n = cosn? + isinn? - For all integers n

23. z1z2* / |z2|²
The Complex Numbers
z1 / z2
can't get out of the complex numbers by adding (or subtracting) or multiplying two
'i'

24. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
Field
Polar Coordinates - z
has a solution.
Subfield

25. (a + bi) = (c + bi) =
Affix
Complex Number Formula
(a + c) + ( b + d)i
Any polynomial O(xn) - (n > 0)

26. 5th. Rule of Complex Arithmetic
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Polar Coordinates - z?¹
multiplying complex numbers
the distance from z to the origin in the complex plane

27. I
v(-1)
four different numbers: i - -i - 1 - and -1.
Complex Number
the vector (a -b)

28. (2+i)(2i-3) you would use the foil methom which is first outter inner last. (2x2i)(2x-3)(ix2i^2)(ix(-3) =i-8
Polar Coordinates - Division
How to multiply complex nubers(2+i)(2i-3)
z + z*
-1

29. All the powers of i can be written as
-1
z1 ^ (z2)
four different numbers: i - -i - 1 - and -1.
Polar Coordinates - Multiplication

30. Cos n? + i sin n? (for all n integers)
How to multiply complex nubers(2+i)(2i-3)
Subfield
(cos? +isin?)n
For real a and b - a + bi = 0 if and only if a = b = 0

31. We consider the a real number x to be the complex number x+ 0i and in this way we can think of the real numbers as a subset of
the complex numbers
z + z*
transcendental
Imaginary Numbers

32. Formula: z1 · z2 = (a + bi)(c + di) = ac +adi +cbi +bdi² = (ac - bd) + (ad +cb)i - when you multiply a complex number by its conjugate - you get a real number.
'i'
Complex Numbers: Multiply
i^3
zz*

33. A + bi = z1 c + di = z2 - addition: z1 + z2 = (a + bi) + (c + di) = (a + c) + (b + d)i subtraction: z1 - z2 = (a - c) + (b - d)i
i^2 = -1
Complex Numbers: Multiply
Complex Numbers: Add & subtract
z - z*

34. When two complex numbers are divided.
How to solve (2i+3)/(9-i)
Complex Division
radicals
i^2 = -1

35. Imaginary number

36. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
Complex Addition
Integers
Imaginary Numbers
Liouville's Theorem -

37. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
i²
|z| = mod(z)
ln z
Field

38. Multiply moduli and add arguments
multiply the numerator and the denominator by the complex conjugate of the denominator.
Any polynomial O(xn) - (n > 0)
Polar Coordinates - Multiplication
Complex Number

39. To simplify a complex fraction
multiply the numerator and the denominator by the complex conjugate of the denominator.
Complex Exponentiation
Polar Coordinates - z
For real a and b - a + bi = 0 if and only if a = b = 0

40. Equivalent to an Imaginary Unit.
the complex numbers
irrational
Argand diagram
Imaginary number

41. R?¹(cos? - isin?)
(a + bi) = (c + bi) = (a + c) + ( b + d)i
i²
Imaginary Unit
Polar Coordinates - z?¹

42. I^26/4= i^24 x i^2 =-1 so u divide the number by 4 and whatevers left over is the number that its equal to.
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
How to solve (2i+3)/(9-i)
How to find any Power
Imaginary number

43. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
non-integers
How to solve (2i+3)/(9-i)
The Complex Numbers
Roots of Unity

44. Has exactly n roots by the fundamental theorem of algebra
Polar Coordinates - z
Any polynomial O(xn) - (n > 0)
i^2
Complex Number Formula

45. If z= a+bi is a complex number and a and b are real - we say that a is the real part of z and that b is the imaginary part of z
conjugate
Polar Coordinates - z?¹
Real and Imaginary Parts
adding complex numbers

46. I^2 =
cosh²y - sinh²y
Every complex number has the 'Standard Form': a + bi for some real a and b.
the complex numbers
-1

47. (e^(iz) - e^(-iz)) / 2i
(a + c) + ( b + d)i
sin iy
conjugate
sin z

48. We see in this way that the distance between two points z and w in the complex plane is
|z-w|
interchangeable
We say that c+di and c-di are complex conjugates.
Euler Formula

49. (a + bi)(c + bi) =
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
interchangeable
0 if and only if a = b = 0
Imaginary number

50. (e^(-y) - e^(y)) / 2i = i sinh y
Integers
sin iy
integers
z + z*