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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. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
Roots of Unity
cos iy
the vector (a -b)
Euler Formula

2. Rotates anticlockwise by p/2
Liouville's Theorem -
Rules of Complex Arithmetic
ln z
Polar Coordinates - Multiplication by i

3. The product of an imaginary number and its conjugate is
Euler Formula
cos iy
a real number: (a + bi)(a - bi) = a² + b²
i²

4. Multiply moduli and add arguments
Real and Imaginary Parts
Polar Coordinates - Multiplication
ln z
(a + bi) = (c + bi) = (a + c) + ( b + d)i

5. The field of all rational and irrational numbers.
Real Numbers
How to solve (2i+3)/(9-i)
Any polynomial O(xn) - (n > 0)
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

6. x / r
Polar Coordinates - cos?
How to solve (2i+3)/(9-i)
non-integers
|z-w|

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
(a + bi) = (c + bi) = (a + c) + ( b + d)i
How to multiply complex nubers(2+i)(2i-3)
sin z
We say that c+di and c-di are complex conjugates.

8. 1st. Rule of Complex Arithmetic
subtracting complex numbers
transcendental
Imaginary Unit
i^2 = -1

9. (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 for some real a and b.
non-integers
How to solve (2i+3)/(9-i)
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

10. Starts at 1 - does not include 0
standard form of complex numbers
natural
|z| = mod(z)
i^4

11. We can also think of the point z= a+ ib as
Real Numbers
adding complex numbers
the vector (a -b)
(a + bi) = (c + bi) = (a + c) + ( b + d)i

12. All numbers
has a solution.
Imaginary number
complex
Complex Division

13. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
Complex Division
rational
multiplying complex numbers
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

14. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
(a + bi) = (c + bi) = (a + c) + ( b + d)i
adding complex numbers
How to add and subtract complex numbers (2-3i)-(4+6i)
i²

15. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
|z| = mod(z)
adding complex numbers
Integers
How to find any Power

16. To simplify a complex fraction
multiply the numerator and the denominator by the complex conjugate of the denominator.
natural
i^3
How to add and subtract complex numbers (2-3i)-(4+6i)

17. A plot of complex numbers as points.
the distance from z to the origin in the complex plane
Argand diagram
Complex numbers are points in the plane
a + bi for some real a and b.

18. V(zz*) = v(a² + b²)
|z| = mod(z)
Real Numbers
How to find any Power
multiplying complex numbers

19. z1z2* / |z2|²
zz*
Polar Coordinates - z?¹
z1 / z2
can't get out of the complex numbers by adding (or subtracting) or multiplying two

20. Any number not rational
0 if and only if a = b = 0
x-axis in the complex plane
rational
irrational

21. A subset within a field.
rational
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Subfield
transcendental

22. 2a
Complex Number
Complex Numbers: Add & subtract
the vector (a -b)
z + z*

23. xpressions such as ``the complex number z'' - and ``the point z'' are now
four different numbers: i - -i - 1 - and -1.
Every complex number has the 'Standard Form': a + bi for some real a and b.
Liouville's Theorem -
interchangeable

24. I = imaginary unit - i² = -1 or i = v-1
For real a and b - a + bi = 0 if and only if a = b = 0
Any polynomial O(xn) - (n > 0)
Polar Coordinates - z
Imaginary Numbers

25. The reals are just the
x-axis in the complex plane
Polar Coordinates - cos?
Argand diagram
rational

26. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
z - z*
Real Numbers
Absolute Value of a Complex Number
Complex Multiplication

27. (e^(iz) - e^(-iz)) / 2i
Polar Coordinates - r
sin z
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
z - z*

28. 1
imaginary
i^0
i^3
x-axis in the complex plane

29. Numbers on a numberline
integers
|z-w|
Euler's Formula
Square Root

30. For real a and b - a + bi =
i^0
0 if and only if a = b = 0
How to find any Power
Polar Coordinates - sin?

31. Cos n? + i sin n? (for all n integers)
subtracting complex numbers
Polar Coordinates - Multiplication
(cos? +isin?)n
z1 ^ (z2)

32. (a + bi)(c + bi) =
i^3
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Rational Number
Complex numbers are points in the plane

33. V(x² + y²) = |z|
the complex numbers
i^0
Polar Coordinates - r
|z| = mod(z)

34. The complex number z representing a+bi.
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Affix
Polar Coordinates - Division
v(-1)

35. The square root of -1.
Absolute Value of a Complex Number
Imaginary Unit
Polar Coordinates - Multiplication
How to solve (2i+3)/(9-i)

36. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
Euler's Formula
multiply the numerator and the denominator by the complex conjugate of the denominator.
has a solution.
Field

37. R?¹(cos? - isin?)
Polar Coordinates - Multiplication by i
Polar Coordinates - z?¹
We say that c+di and c-di are complex conjugates.
Complex numbers are points in the plane

38. To simplify the square root of a negative number
four different numbers: i - -i - 1 - and -1.
interchangeable
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Field

39. 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
Rules of Complex Arithmetic
i^3
Real and Imaginary Parts
standard form of complex numbers

40. 3
i^3
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Polar Coordinates - Multiplication by i
four different numbers: i - -i - 1 - and -1.

41. 2nd. Rule of Complex Arithmetic

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42. y / r
can't get out of the complex numbers by adding (or subtracting) or multiplying two
Polar Coordinates - sin?
Affix
The Complex Numbers

43. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
standard form of complex numbers
multiplying complex numbers
The Complex Numbers
Complex Number

44. Given (4-2i) the complex conjugate would be (4+2i)
Complex Conjugate
Imaginary number
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
i^0

45. 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 - z?¹
i^3
(a + c) + ( b + d)i
How to find any Power

46. I
v(-1)
Polar Coordinates - z
irrational
multiplying complex numbers

47. 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
(cos? +isin?)n
cosh²y - sinh²y
Polar Coordinates - sin?
Complex Numbers: Add & subtract

48. A + bi
How to multiply complex nubers(2+i)(2i-3)
Rational Number
i^4
standard form of complex numbers

49. 2ib
The Complex Numbers
z - z*
radicals
(a + c) + ( b + d)i

50. The modulus of the complex number z= a + ib now can be interpreted as
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
cos z
point of inflection
the distance from z to the origin in the complex plane