Test your basic knowledge |

CLEP General Mathematics: Complex Numbers

Subjects : clep, math

Instructions:

Answer 50 questions in 15 minutes.
If you are not ready to take this test, you can study here.
Match each statement with the correct term.
Don't refresh. All questions and answers are randomly picked and ordered every time you load a test.

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. 4th. Rule of Complex Arithmetic
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
complex
(a + bi) = (c + bi) = (a + c) + ( b + d)i
imaginary

2. 1
i^0
conjugate pairs
Complex Number
Polar Coordinates - z?¹

3. Divide moduli and subtract arguments
Complex Conjugate
cos z
Polar Coordinates - Division
Imaginary number

4. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
z1 / z2
Euler's Formula
The Complex Numbers
Liouville's Theorem -

5. Not on the numberline
non-integers
Field
cosh²y - sinh²y
|z-w|

6. 1
i^4
z1 ^ (z2)
sin iy
x-axis in the complex plane

7. ? = -tan?
cosh²y - sinh²y
|z-w|
z + z*
Polar Coordinates - Arg(z*)

8. 1
Any polynomial O(xn) - (n > 0)
i^2
Every complex number has the 'Standard Form': a + bi for some real a and b.
(a + c) + ( b + d)i

9. 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
non-integers
Real and Imaginary Parts
0 if and only if a = b = 0
How to multiply complex nubers(2+i)(2i-3)

10. The modulus of the complex number z= a + ib now can be interpreted as
x-axis in the complex plane
the distance from z to the origin in the complex plane
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Complex Multiplication

11. A plot of complex numbers as points.
Roots of Unity
Polar Coordinates - sin?
Argand diagram
Complex Number

12. Rotates anticlockwise by p/2
Polar Coordinates - Multiplication by i
transcendental
rational
Integers

13. I
i^2 = -1
v(-1)
Polar Coordinates - Multiplication by i
x-axis in the complex plane

14. V(zz*) = v(a² + b²)
v(-1)
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
|z| = mod(z)
Complex Numbers: Multiply

15. When two complex numbers are subtracted from one another.
Complex Subtraction
Irrational Number
Real Numbers
standard form of complex numbers

16. The field of numbers of the form - where and are real numbers and i is the imaginary unit equal to the square root of - . When a single letter is used to denote a complex number - it is sometimes called an 'affix.'
four different numbers: i - -i - 1 - and -1.
Any polynomial O(xn) - (n > 0)
Complex Number
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

17. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
adding complex numbers
Polar Coordinates - r
Field
Absolute Value of a Complex Number

18. 1st. Rule of Complex Arithmetic
i^2 = -1
Field
conjugate pairs
0 if and only if a = b = 0

19. When you add two complex numbers a + bi and c + di - you get the sum of the real parts and the sum of the imaginary parts: (a + bi) + (c + di) = (a + c) + (b + d)i
i^2
adding complex numbers
Polar Coordinates - Multiplication by i
Polar Coordinates - sin?

20. (e^(iz) - e^(-iz)) / 2i
sin z
z - z*
has a solution.
a + bi for some real a and b.

21. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
interchangeable
real
i^3
Roots of Unity

22. To simplify a complex fraction
Roots of Unity
Irrational Number
multiply the numerator and the denominator by the complex conjugate of the denominator.
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

23. No i
i^0
real
sin z
zz*

24. 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.
|z-w|
z1 / z2
i^1
Complex Numbers: Multiply

25. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
conjugate
De Moivre's Theorem
subtracting complex numbers
(a + bi) = (c + bi) = (a + c) + ( b + d)i

26. I
i^1
Liouville's Theorem -
(a + c) + ( b + d)i
Imaginary Numbers

27. V(x² + y²) = |z|
Real Numbers
cos iy
the complex numbers
Polar Coordinates - r

28. (a + bi) = (c + bi) =
(a + c) + ( b + d)i
Affix
z + z*
z1 / z2

29. The field of all rational and irrational numbers.
Complex Numbers: Multiply
x-axis in the complex plane
standard form of complex numbers
Real Numbers

30. The square root of -1.
Imaginary Unit
-1
How to multiply complex nubers(2+i)(2i-3)
a + bi for some real a and b.

31. We can also think of the point z= a+ ib as
Complex Numbers: Multiply
the vector (a -b)
Polar Coordinates - z?¹
non-integers

32. When two complex numbers are added together.
i^2
Complex Addition
cos z
Absolute Value of a Complex Number

33. (2i+3)/(9-i)for the denominator you multiply by the conjugate and what u do to the bottom u have to do to the top then you distribute the bottom then the top then add like terms then you simplify. 21i+25/17
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Argand diagram
cosh²y - sinh²y
How to solve (2i+3)/(9-i)

34. Where the curvature of the graph changes
Polar Coordinates - sin?
standard form of complex numbers
Real and Imaginary Parts
point of inflection

35. A subset within a field.
Subfield
Polar Coordinates - Arg(z*)
Affix
i^0

36. A+bi
Polar Coordinates - Multiplication
Complex Number Formula
'i'
point of inflection

37. ½(e^(iz) + e^(-iz))
i^4
Complex Division
cos z
|z| = mod(z)

38. A + bi
i²
standard form of complex numbers
Complex Addition
Absolute Value of a Complex Number

39. Any number not rational
the distance from z to the origin in the complex plane
irrational
(cos? +isin?)n
We say that c+di and c-di are complex conjugates.

40. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
subtracting complex numbers
transcendental
Absolute Value of a Complex Number
Rational Number

41. In this amazing number field every algebraic equation in z with complex coefficients
has a solution.
interchangeable
Polar Coordinates - z?¹
sin z

42. A complex number may be taken to the power of another complex number.
Complex Exponentiation
Absolute Value of a Complex Number
irrational
ln z

43. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
z1 ^ (z2)
imaginary
i^3
ln z

44. Imaginary number

45. A² + b² - real and non negative
four different numbers: i - -i - 1 - and -1.
zz*
Roots of Unity
sin iy

46. E^(ln r) e^(i?) e^(2pin)
Polar Coordinates - Multiplication by i
the complex numbers
e^(ln z)
The Complex Numbers

47. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
complex
Integers
complex numbers
Polar Coordinates - r

48. Numbers on a numberline
point of inflection
Irrational Number
integers
cos iy

49. Root negative - has letter i
imaginary
Complex Division
zz*
-1

50. 2ib
i^2
z - z*
0 if and only if a = b = 0
Rational Number