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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. y / r
Polar Coordinates - sin?
How to add and subtract complex numbers (2-3i)-(4+6i)
natural
Complex Numbers: Multiply

2. All numbers
complex
Real Numbers
Absolute Value of a Complex Number
e^(ln z)

3. 1
Polar Coordinates - Multiplication by i
How to solve (2i+3)/(9-i)
z1 / z2
i^4

4. In this amazing number field every algebraic equation in z with complex coefficients
has a solution.
irrational
cosh²y - sinh²y
cos iy

5. 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.
Polar Coordinates - Arg(z*)
Euler Formula
How to find any Power
natural

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

7. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
ln z
Complex Numbers: Multiply
i²
De Moivre's Theorem

8. Rotates anticlockwise by p/2
Polar Coordinates - Multiplication by i
transcendental
Polar Coordinates - z?¹
has a solution.

9. To simplify a complex fraction
multiply the numerator and the denominator by the complex conjugate of the denominator.
Integers
Liouville's Theorem -
Imaginary Unit

10. A + bi
i^3
non-integers
How to find any Power
standard form of complex numbers

11. To simplify the square root of a negative number
cos iy
Complex Subtraction
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
multiplying complex numbers

12. E^(ln r) e^(i?) e^(2pin)
Polar Coordinates - Multiplication
Imaginary Numbers
e^(ln z)
The Complex Numbers

13. The modulus of the complex number z= a + ib now can be interpreted as
Imaginary number
subtracting complex numbers
the distance from z to the origin in the complex plane
Liouville's Theorem -

14. x + iy = r(cos? + isin?) = re^(i?)
Polar Coordinates - z
rational
v(-1)
Polar Coordinates - Division

15. No i
How to find any Power
real
(a + c) + ( b + d)i
Complex Subtraction

16. 3
Complex Conjugate
i^3
Real Numbers
multiplying complex numbers

17. When two complex numbers are multipiled together.
z - z*
i^2
Complex Multiplication
i^3

18. 1
z - z*
i^0
conjugate pairs
Polar Coordinates - Multiplication by i

19. Cos n? + i sin n? (for all n integers)
Polar Coordinates - Arg(z*)
Imaginary Numbers
(cos? +isin?)n
sin iy

20. All the powers of i can be written as
four different numbers: i - -i - 1 - and -1.
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
non-integers
The Complex Numbers

21. Divide moduli and subtract arguments
Polar Coordinates - Division
complex
Complex Number
a + bi for some real a and b.

22. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
multiplying complex numbers
cos z
the complex numbers
Polar Coordinates - sin?

23. A+bi
Complex Addition
z1 / z2
Complex Number Formula
z1 ^ (z2)

24. E ^ (z2 ln z1)
(a + c) + ( b + d)i
i^3
Rational Number
z1 ^ (z2)

25. 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
Euler's Formula
We say that c+di and c-di are complex conjugates.
x-axis in the complex plane
a + bi for some real a and b.

26. 4th. Rule of Complex Arithmetic
four different numbers: i - -i - 1 - and -1.
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Real and Imaginary Parts
point of inflection

27. 1st. Rule of Complex Arithmetic
i^2 = -1
ln z
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Complex Division

28. 5th. Rule of Complex Arithmetic
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Irrational Number
complex numbers
Roots of Unity

29. Where the curvature of the graph changes
four different numbers: i - -i - 1 - and -1.
complex
multiplying complex numbers
point of inflection

30. We can also think of the point z= a+ ib as
Liouville's Theorem -
the vector (a -b)
adding complex numbers
Euler Formula

31. 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
Polar Coordinates - Multiplication by i
Complex Numbers: Multiply
Rules of Complex Arithmetic
irrational

32. A subset within a field.
Subfield
Roots of Unity
How to add and subtract complex numbers (2-3i)-(4+6i)
x-axis in the complex plane

33. 3rd. Rule of Complex Arithmetic
For real a and b - a + bi = 0 if and only if a = b = 0
Polar Coordinates - sin?
Complex Number Formula
Complex Division

34. A number that can be expressed as a fraction p/q where q is not equal to 0.
a real number: (a + bi)(a - bi) = a² + b²
z + z*
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Rational Number

35. (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
conjugate pairs
How to multiply complex nubers(2+i)(2i-3)
multiply the numerator and the denominator by the complex conjugate of the denominator.
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

36. Real and imaginary numbers
sin iy
Polar Coordinates - z
complex numbers
Argand diagram

37. 2ib
Every complex number has the 'Standard Form': a + bi for some real a and b.
ln z
Euler Formula
z - z*

38. When two complex numbers are subtracted from one another.
conjugate pairs
Square Root
the distance from z to the origin in the complex plane
Complex Subtraction

39. A number that cannot be expressed as a fraction for any integer.
multiplying complex numbers
Irrational Number
Imaginary Numbers
Integers

40. 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
Complex Exponentiation
Real Numbers
De Moivre's Theorem
adding complex numbers

41. I
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
(a + bi) = (c + bi) = (a + c) + ( b + d)i
the vector (a -b)
v(-1)

42. The square root of -1.
a + bi for some real a and b.
interchangeable
Imaginary Unit
real

43. A complex number may be taken to the power of another complex number.
Complex Exponentiation
For real a and b - a + bi = 0 if and only if a = b = 0
i^2 = -1
the complex numbers

44. Starts at 1 - does not include 0
Complex Numbers: Add & subtract
Any polynomial O(xn) - (n > 0)
Argand diagram
natural

45. V(zz*) = v(a² + b²)
Imaginary Numbers
|z| = mod(z)
complex
How to multiply complex nubers(2+i)(2i-3)

46. The complex number z representing a+bi.
Complex Subtraction
Complex Number
Affix
Real Numbers

47. Derives z = a+bi
transcendental
Euler Formula
Irrational Number
Complex Subtraction

48. 2nd. Rule of Complex Arithmetic

49. 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
Complex Numbers: Multiply
Complex Number
Real and Imaginary Parts
interchangeable

50. 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.
x-axis in the complex plane
Complex numbers are points in the plane
multiply the numerator and the denominator by the complex conjugate of the denominator.
i^4