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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.
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. 2a
Affix
i²
z + z*
point of inflection

2. A² + b² - real and non negative
cosh²y - sinh²y
zz*
-1
i^3

3. Given (4-2i) the complex conjugate would be (4+2i)
Complex Conjugate
How to multiply complex nubers(2+i)(2i-3)
has a solution.
transcendental

4. 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
rational
Imaginary Unit
Complex Numbers: Add & subtract
Affix

5. A subset within a field.
Imaginary number
Subfield
Square Root
Polar Coordinates - cos?

6. Not on the numberline
Irrational Number
non-integers
i^3
sin z

7. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
Field
the distance from z to the origin in the complex plane
Affix
i^3

8. The complex number z representing a+bi.
multiply the numerator and the denominator by the complex conjugate of the denominator.
complex
Affix
radicals

9. When two complex numbers are divided.
Complex Division
Polar Coordinates - r
the vector (a -b)
subtracting complex numbers

10. x + iy = r(cos? + isin?) = re^(i?)
Polar Coordinates - z
Complex Number
ln z
Rules of Complex Arithmetic

11. E^(ln r) e^(i?) e^(2pin)
Complex Division
z1 / z2
e^(ln z)
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

12. xpressions such as ``the complex number z'' - and ``the point z'' are now
Roots of Unity
interchangeable
multiply the numerator and the denominator by the complex conjugate of the denominator.
Square Root

13. Like pi
the complex numbers
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
sin z
transcendental

14. ½(e^(iz) + e^(-iz))
cos z
z + z*
Rational Number
four different numbers: i - -i - 1 - and -1.

15. The product of an imaginary number and its conjugate is
a real number: (a + bi)(a - bi) = a² + b²
Euler's Formula
x-axis in the complex plane
Polar Coordinates - sin?

16. We can also think of the point z= a+ ib as
standard form of complex numbers
adding complex numbers
the vector (a -b)
radicals

17. 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)
|z-w|
Imaginary number
Roots of Unity

18. (a + bi) = (c + bi) =
Polar Coordinates - Arg(z*)
interchangeable
(a + c) + ( b + d)i
0 if and only if a = b = 0

19. I
v(-1)
imaginary
cosh²y - sinh²y
e^(ln z)

20. 1
Complex Addition
z - z*
i^0
Euler's Formula

21. E ^ (z2 ln z1)
Square Root
How to add and subtract complex numbers (2-3i)-(4+6i)
Irrational Number
z1 ^ (z2)

22. Starts at 1 - does not include 0
four different numbers: i - -i - 1 - and -1.
conjugate
Complex Number
natural

23. Rotates anticlockwise by p/2
Polar Coordinates - Multiplication by i
transcendental
Roots of Unity
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

24. (a + bi)(c + bi) =
transcendental
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
radicals
non-integers

25. 3
adding complex numbers
cos z
i^3
Argand diagram

26. I^2 =
-1
How to solve (2i+3)/(9-i)
Complex numbers are points in the plane
multiply the numerator and the denominator by the complex conjugate of the denominator.

27. 5th. Rule of Complex Arithmetic
the vector (a -b)
can't get out of the complex numbers by adding (or subtracting) or multiplying two
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Every complex number has the 'Standard Form': a + bi for some real a and b.

28. I
Liouville's Theorem -
i^1
cosh²y - sinh²y
For real a and b - a + bi = 0 if and only if a = b = 0

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

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30. Cos n? + i sin n? (for all n integers)
(cos? +isin?)n
Complex Division
Rules of Complex Arithmetic
x-axis in the complex plane

31. Real and imaginary numbers
Every complex number has the 'Standard Form': a + bi for some real a and b.
Real and Imaginary Parts
complex numbers
rational

32. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
Euler's Formula
the distance from z to the origin in the complex plane
Every complex number has the 'Standard Form': a + bi for some real a and b.
Integers

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
z - z*
conjugate
For real a and b - a + bi = 0 if and only if a = b = 0
How to solve (2i+3)/(9-i)

34. Where the curvature of the graph changes
point of inflection
Polar Coordinates - r
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Polar Coordinates - Division

35. The square root of -1.
Complex numbers are points in the plane
conjugate pairs
Subfield
Imaginary Unit

36. All numbers
complex
i^3
standard form of complex numbers
i²

37. V(x² + y²) = |z|
x-axis in the complex plane
radicals
Polar Coordinates - r
v(-1)

38. 1
x-axis in the complex plane
Liouville's Theorem -
a real number: (a + bi)(a - bi) = a² + b²
cosh²y - sinh²y

39. A number that can be expressed as a fraction p/q where q is not equal to 0.
Rational Number
Complex Numbers: Multiply
adding complex numbers
non-integers

40. 3rd. Rule of Complex Arithmetic
For real a and b - a + bi = 0 if and only if a = b = 0
Complex Number
standard form of complex numbers
e^(ln z)

41. A plot of complex numbers as points.
Polar Coordinates - sin?
transcendental
Argand diagram
'i'

42. A complex number and its conjugate
conjugate pairs
i^4
Complex Conjugate
Subfield

43. When you subtract two complex numbers a + bi and c + di - you get the difference of the real parts and the difference of the imaginary parts: (a + bi) - (c + di) = (a - c) + (b - d)i
Integers
conjugate pairs
subtracting complex numbers
Subfield

44. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
Integers
How to solve (2i+3)/(9-i)
standard form of complex numbers
The Complex Numbers

45. 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.
'i'
De Moivre's Theorem
Complex numbers are points in the plane
rational

46. I = imaginary unit - i² = -1 or i = v-1
We say that c+di and c-di are complex conjugates.
Imaginary Numbers
multiplying complex numbers
Absolute Value of a Complex Number

47. A complex number may be taken to the power of another complex number.
can't get out of the complex numbers by adding (or subtracting) or multiplying two
natural
How to solve (2i+3)/(9-i)
Complex Exponentiation

48. V(zz*) = v(a² + b²)
z + z*
|z| = mod(z)
the vector (a -b)
sin iy

49. Divide moduli and subtract arguments
zz*
Any polynomial O(xn) - (n > 0)
Polar Coordinates - Division
z1 ^ (z2)

50. 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
Roots of Unity
Liouville's Theorem -
Any polynomial O(xn) - (n > 0)