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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. Any number not rational
Complex Conjugate
irrational
How to find any Power
sin z

2. A complex number and its conjugate
non-integers
Square Root
How to find any Power
conjugate pairs

3. A number that cannot be expressed as a fraction for any integer.
Rational Number
sin z
ln z
Irrational Number

4. Every complex number has the 'Standard Form':
interchangeable
a + bi for some real a and b.
Polar Coordinates - z?¹
zz*

5. Numbers on a numberline
integers
Real and Imaginary Parts
Complex Numbers: Add & subtract
the distance from z to the origin in the complex plane

6. 2ib
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Any polynomial O(xn) - (n > 0)
i^2
z - z*

7. (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
i^3
sin iy
How to multiply complex nubers(2+i)(2i-3)
non-integers

8. A + bi
How to solve (2i+3)/(9-i)
Roots of Unity
real
standard form of complex numbers

9. x + iy = r(cos? + isin?) = re^(i?)
Irrational Number
(cos? +isin?)n
Polar Coordinates - z
z1 / z2

10. (a + bi)(c + bi) =
Complex Numbers: Multiply
Complex Exponentiation
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
i^2 = -1

11. 1
Rational Number
How to add and subtract complex numbers (2-3i)-(4+6i)
ln z
i^4

12. V(x² + y²) = |z|
Polar Coordinates - z
Complex Division
Square Root
Polar Coordinates - r

13. 1
z + z*
i^0
a + bi for some real a and b.
How to multiply complex nubers(2+i)(2i-3)

14. y / r
Imaginary number
i²
Polar Coordinates - sin?
i^2 = -1

15. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
Polar Coordinates - z?¹
sin z
The Complex Numbers
irrational

16. ? = -tan?
Polar Coordinates - z?¹
conjugate
irrational
Polar Coordinates - Arg(z*)

17. A complex number may be taken to the power of another complex number.
Real and Imaginary Parts
the distance from z to the origin in the complex plane
Complex Exponentiation
cos z

18. The reals are just the
Complex numbers are points in the plane
x-axis in the complex plane
Polar Coordinates - cos?
imaginary

19. Multiply moduli and add arguments
Polar Coordinates - Division
a real number: (a + bi)(a - bi) = a² + b²
Polar Coordinates - Multiplication
Complex numbers are points in the plane

20. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
|z| = mod(z)
Roots of Unity
e^(ln z)
Polar Coordinates - z

21. 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
complex
Euler Formula
the complex numbers
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

22. Have radical
Polar Coordinates - Multiplication by i
radicals
Euler Formula
four different numbers: i - -i - 1 - and -1.

23. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
Polar Coordinates - z
Polar Coordinates - cos?
How to add and subtract complex numbers (2-3i)-(4+6i)
Rules of Complex Arithmetic

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.
Rules of Complex Arithmetic
Complex Numbers: Multiply
Euler's Formula
Irrational Number

25. The modulus of the complex number z= a + ib now can be interpreted as
real
a + bi for some real a and b.
natural
the distance from z to the origin in the complex plane

26. Starts at 1 - does not include 0
standard form of complex numbers
Polar Coordinates - z?¹
four different numbers: i - -i - 1 - and -1.
natural

27. Not on the numberline
Polar Coordinates - Multiplication by i
non-integers
Complex numbers are points in the plane
the distance from z to the origin in the complex plane

28. E ^ (z2 ln z1)
z1 ^ (z2)
cos iy
Complex Number Formula
multiply the numerator and the denominator by the complex conjugate of the denominator.

29. I = imaginary unit - i² = -1 or i = v-1
z1 ^ (z2)
Imaginary Numbers
Field
Complex Subtraction

30. I^2 =
Complex Numbers: Add & subtract
-1
Argand diagram
interchangeable

31. R?¹(cos? - isin?)
integers
How to find any Power
Polar Coordinates - z?¹
Polar Coordinates - Division

32. When two complex numbers are divided.
zz*
Complex Number
a real number: (a + bi)(a - bi) = a² + b²
Complex Division

33. A² + b² - real and non negative
How to multiply complex nubers(2+i)(2i-3)
zz*
sin iy
Polar Coordinates - sin?

34. 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
Real and Imaginary Parts
i^3
multiply the numerator and the denominator by the complex conjugate of the denominator.
multiplying complex numbers

35. For real a and b - a + bi =
ln z
multiplying complex numbers
i^2 = -1
0 if and only if a = b = 0

36. To prove that number field every algebraic equation in z with complex coefficients has a solution we need

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37. The field of all rational and irrational numbers.
i^2
Polar Coordinates - Arg(z*)
Real Numbers
conjugate pairs

38. Divide moduli and subtract arguments
a real number: (a + bi)(a - bi) = a² + b²
Polar Coordinates - Division
Rules of Complex Arithmetic
How to add and subtract complex numbers (2-3i)-(4+6i)

39. To simplify a complex fraction
Imaginary number
x-axis in the complex plane
Euler's Formula
multiply the numerator and the denominator by the complex conjugate of the denominator.

40. A subset within a field.
We say that c+di and c-di are complex conjugates.
Subfield
e^(ln z)
non-integers

41. 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.
'i'
Euler's Formula
Polar Coordinates - Multiplication by i

42. Root negative - has letter i
imaginary
Complex Number
Field
Complex Exponentiation

43. (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
We say that c+di and c-di are complex conjugates.
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Polar Coordinates - Arg(z*)
How to solve (2i+3)/(9-i)

44. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
The Complex Numbers
Integers
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Rules of Complex Arithmetic

45. A+bi
Complex Number Formula
Polar Coordinates - Multiplication
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
a + bi for some real a and b.

46. 1
integers
Complex Numbers: Multiply
i²
Polar Coordinates - Division

47. E^(ln r) e^(i?) e^(2pin)
the distance from z to the origin in the complex plane
e^(ln z)
z1 ^ (z2)
'i'

48. We can also think of the point z= a+ ib as
the vector (a -b)
z + z*
sin z
transcendental

49. 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
conjugate
Complex Number Formula
Square Root
subtracting complex numbers

50. The product of an imaginary number and its conjugate is
How to add and subtract complex numbers (2-3i)-(4+6i)
a real number: (a + bi)(a - bi) = a² + b²
sin z
-1