Complex Numbers (DP IB Analysis & Approaches (AA))

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  • What is the imaginary number, straight i?

    The imaginary number straight i is solution to x squared equals negative 1. It is a square root of -1.

  • What is a complex number?

    A complex number can have both a real part and an imaginary part.

    For example, 2 plus 5 straight i is a complex number.

  • What is denoted by Re open parentheses z close parentheses?

    Re open parentheses z close parentheses is used to denote the real part of the complex number z.

    For example, Re open parentheses 2 plus 3 straight i close parentheses equals 2.

  • What is denoted by Im open parentheses z close parentheses?

    Im open parentheses z close parentheses is used to denote the imaginary part of the complex number z.

    For example, Im open parentheses 2 plus 3 straight i close parentheses equals 3.

  • True or False?

    2 plus 3 straight i and 4 plus 6 straight i represent the same complex number.

    False.

    2 plus 3 straight i and 4 plus 6 straight i are different complex numbers.

    Complex numbers are only equal if their real parts are equal and their imaginary parts are equal.

  • What is the Cartesian form of a complex number?

    The Cartesian form of a complex number is a plus b straight i where a and b are real numbers.

  • How do you add or subtract two complex numbers in Cartesian form?

    To add or subtract two complex numbers in Cartesian form, you add or subtract the real parts and imaginary parts separately.

    For example, open parentheses 2 plus 3 straight i close parentheses plus open parentheses 5 minus straight i close parentheses equals open parentheses 2 plus 5 close parentheses plus open parentheses 3 plus open parentheses negative 1 close parentheses close parentheses straight i equals 7 plus 2 straight i.

  • How do you multiply two complex numbers in Cartesian form?

    To multiply two complex numbers in Cartesian form, you expand brackets and use the fact that straight i squared equals negative 1.

    For example, open parentheses 2 plus 3 straight i close parentheses open parentheses 5 minus straight i close parentheses equals 10 plus 15 straight i minus 2 straight i minus 3 straight i squared equals 10 plus 13 straight i minus 3 open parentheses negative 1 close parentheses equals 13 plus 13 straight i.

  • If z is a complex number, then what is denoted by z asterisk times?

    If z is a complex number, then z asterisk times denotes the complex conjugate.

  • What is the complex conjugate of a complex number?

    The complex conjugate of the complex number is where the sign of the imaginary part has been changed.

    For example, the complex conjugate of 2 minus 3 straight i is 2 plus 3 straight i.

  • True or False?

    For any complex number z, z plus z asterisk times is a real number.

    True.

    For any complex number z, z plus z asterisk times is a real number.

  • If z is complex number, then is z minus z asterisk times a real number or an imaginary number?

    If z is complex number, then z minus z asterisk times is an imaginary number.

  • True or False?

    For any complex number z, z cross times z asterisk times is an imaginary number.

    False.

    For any complex number z, z cross times z asterisk times is a real number.

  • How do you divide a complex number by another complex number in Cartesian form?

    To divide a complex number by another complex number in Cartesian form:

    • write as a fraction,

    • multiply the numerator and denominator by the complex conjugate of the denominator,

    • simplify both parts of the fractions.

    For example, open parentheses 2 plus 3 straight i close parentheses divided by open parentheses 5 plus 4 straight i close parentheses equals fraction numerator open parentheses 2 plus 3 straight i close parentheses open parentheses 5 minus 4 straight i close parentheses over denominator open parentheses 5 plus 4 straight i close parentheses open parentheses 5 minus 4 straight i close parentheses end fraction equals fraction numerator 22 plus 7 straight i over denominator 41 end fraction.

  • What notation is used to denote the modulus of the complex number z?

    open vertical bar z close vertical bar is used to denote the modulus of the complex number z.

  • True or False?

    The modulus of a complex number is the distance from the origin to the complex number on an Argand diagram?

    True.

    The modulus of a complex number is the distance from the origin to the complex number on an Argand diagram.

  • If z equals a plus b straight i, what is the formula for open vertical bar z close vertical bar?

    If z equals a plus b straight i, then open vertical bar z close vertical bar equals square root of a squared plus b squared end root.

  • True or False?

    The modulus of 2 minus 3 straight i is square root of 2 squared minus 3 squared end root.

    False.

    The modulus of 2 minus 3 straight i is square root of 2 squared plus open parentheses negative 3 close parentheses squared end root, which is just square root of 2 squared plus 3 squared end root.

  • What is the formula that connects a complex number and its conjugate with its modulus?

    The formula that connects a complex number and its conjugate with its modulus is z cross times z asterisk times equals open vertical bar z close vertical bar squared.

  • True or False?

    If z subscript 1 and z subscript 2 are any two complex numbers then open vertical bar z subscript 1 plus z subscript 2 close vertical bar equals open vertical bar z subscript 1 close vertical bar plus open vertical bar z subscript 2 close vertical bar.

    False.

    If z subscript 1 and z subscript 2 are any two complex numbers then, in general, open vertical bar z subscript 1 plus z subscript 2 close vertical bar not equal to open vertical bar z subscript 1 close vertical bar plus open vertical bar z subscript 2 close vertical bar.

  • True or False?

    If z subscript 1 and z subscript 2 are any two complex numbers then open vertical bar z subscript 1 cross times z subscript 2 close vertical bar equals open vertical bar z subscript 1 close vertical bar cross times open vertical bar z subscript 2 close vertical bar.

    True.

    If z subscript 1 and z subscript 2 are any two complex numbers then open vertical bar z subscript 1 cross times z subscript 2 close vertical bar equals open vertical bar z subscript 1 close vertical bar cross times open vertical bar z subscript 2 close vertical bar.

  • True or False?

    If z subscript 1 and z subscript 2 are any two complex numbers then open vertical bar z subscript 1 over z subscript 2 close vertical bar equals fraction numerator open vertical bar z subscript 1 close vertical bar over denominator open vertical bar z subscript 2 close vertical bar end fraction.

    True.

    If z subscript 1 and z subscript 2 are any two complex numbers then open vertical bar z subscript 1 over z subscript 2 close vertical bar equals fraction numerator open vertical bar z subscript 1 close vertical bar over denominator open vertical bar z subscript 2 close vertical bar end fraction.

  • What is the argument of a complex number?

    The argument of a complex number is the angle that it makes when measured counter-clockwise from the positive real axis when plotted on an Argand diagram.

  • What notation is used to denote the argument of a complex number z?

    arg open parentheses z close parentheses is used to denote the argument of the complex number z.

  • True or False?

    arg open parentheses 0 close parentheses equals 0.

    False.

    arg open parentheses 0 close parentheses not equal to 0, arg open parentheses 0 close parentheses is undefined.

  • True or False?

    If z subscript 1 and z subscript 2 are any two complex numbers then arg open parentheses z subscript 1 cross times z subscript 2 close parentheses equals arg open parentheses straight z subscript 1 close parentheses cross times arg open parentheses straight z subscript 2 close parentheses.

    False.

    If z subscript 1 and z subscript 2 are any two complex numbers then arg open parentheses z subscript 1 cross times z subscript 2 close parentheses equals arg open parentheses straight z subscript 1 close parentheses plus arg open parentheses straight z subscript 2 close parentheses. If you multiply two complex numbers you add their arguments.

  • Write a formula for arg open parentheses z subscript 1 over z subscript 2 close parentheses in terms of arg open parentheses z subscript 1 close parentheses and arg open parentheses z subscript 2 close parentheses.

    arg open parentheses z subscript 1 over z subscript 2 close parentheses equals arg open parentheses z subscript 1 close parentheses minus arg open parentheses z subscript 2 close parentheses.

  • How do you find the argument of the complex number z equals a plus b straight i?

    The argument of the complex number z equals a plus b straight i is a solution to tan theta equals y over x.

    You might need to add or subtract straight pi from the principal angle, depending on where the complex number lies in the Argand diagram.

  • What is an Argand diagram?

    An Argand diagram is a two-dimensional plane on which complex numbers can be represented.

  • Which type of numbers are plotted on the horizontal axis of an Argand diagram, real or imaginary?

    Real numbers are plotted on the horizontal axis of an Argand diagram.

  • True or False?

    The complex number 2 minus 3 straight i can be thought of as the point with coordinates open parentheses 2 comma negative 3 close parentheses on an Argand diagram.

    True.

    The complex number 2 minus 3 straight i can be thought of as the point with coordinates open parentheses 2 comma negative 3 close parentheses on an Argand diagram.

    Just remember to interpret the coordinates open parentheses x comma space y close parentheses a a complex number x plus y straight i.