Complex Numbers and Operations - Unit 2

Numbers that can be expressed as a fraction where both the numerator and denominator are integers.

Numbers that cannot be expressed as a fraction of two integers.

A number that can be written as the product of an integer multiplied by itself.

A number that can be written as the product of an integer multiplied by itself three times.

A number that, when multiplied by itself, equals the given number.

A number that, when multiplied by itself three times, equals the given number.

A number that can be expressed as a fraction where both the numerator and denominator are integers.

The form where we use a radical symbol to represent a root.

A number that can be expressed as a fraction where both the numerator and denominator are integers.

A number that cannot be expressed as a fraction of two integers.

A number that can be written as the product of an integer multiplied by itself.

A number that can be written as the product of an integer multiplied by itself three times.

A number that, when multiplied by itself, equals the given number.

A number that, when multiplied by itself three times, equals the given number.

A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, which satisfies the equation i² = -1.

The process of adding or subtracting complex numbers involves combining the real and imaginary parts separately.

A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit. The imaginary unit i is defined as the square root of -1.

The real part of a complex number is the coefficient of the term without the imaginary unit (i); it's the plain number.

The imaginary part of a complex number is the coefficient of the term with the imaginary unit (i).

The powers of i are defined by the following pattern: i^0 = 1 i^1 = i i^2 = -1 i^3 = -i i^4 = 1 This pattern repeats itself every four terms, meaning that i^5 = i, i^6 = -1, i^7 = -i, and so on.

To determine the value of a higher power of i, divide the exponent by 4. The remainder of the division determines which value of i it equals:

Addition and subtraction of complex numbers are performed by adding/subtracting the real parts and the imaginary parts separately.

Multiplication of complex numbers is performed using the distributive property. Remember that i^2 = -1 when simplifying the result.

Division of complex numbers involves multiplying both the numerator and the denominator by the conjugate of the denominator. The conjugate of a complex number is obtained by changing the sign of the imaginary part.

To multiply complex numbers, you distribute just like you would with binomials. Remember that i² = -1.

Division of complex numbers involves rationalizing the denominator, which means eliminating any 'i' terms from the denominator. You do this by multiplying both the numerator and denominator by the complex conjugate of the denominator.

The complex conjugate of a complex number a + bi is a - bi. Multiplying a complex number by its conjugate always results in a real number.

An Argand diagram is a graphical representation of complex numbers, where the horizontal axis represents the real part and the vertical axis represents the imaginary part.

The parallelogram rule helps visualize complex number addition. If you draw two complex numbers as vectors, their sum is represented by the diagonal of the parallelogram formed by those vectors.

When a complex number is represented as a point on an Argand diagram, the distance from the origin to the point represents the modulus of the complex number.

The angle between the positive real axis and the line connecting the origin to the point representing the complex number on an Argand diagram is called the argument of the complex number.

Subtracting complex numbers involves subtracting the real parts and the imaginary parts separately. For example, to subtract (3 + 4i) from (-2 + 2i), you would subtract the real parts (-2 - 3 = -5) and the imaginary parts (2 - 4 = -2). This gives us -5 - 2i as the result.

To add complex numbers graphically, represent each complex number as a point on a graph. Then, create a parallelogram with the two complex numbers as adjacent sides. The diagonal of the parallelogram starting from the origin represents the sum of the complex numbers.

To subtract complex numbers graphically, simply reverse the direction of the subtrahend and add the numbers graphically as before. Create a parallelogram with the minuend and the negated subtrahend as adjacent sides (where the negated subtrahend is the subtrahend with a reversed direction). The diagonal of this parallelogram, starting from the minuend point, represents the difference.

Adding or subtracting complex numbers involves combining their real and imaginary parts separately, treating each part as a separate number. The result is another complex number in the form a + bi, where 'a' is the combined real part and 'b' is the combined imaginary part.