Learn the structure of complex numbers in the standard form 'a + bi', how to identify their real and imaginary parts, and how to visualize them on the complex plane. This stack also covers the concept of the absolute value (modulus) of a complex number.
The standard form of a complex number is \(a + bi\), where \(a\) is the real part and \(b\) is the coefficient of the imaginary part. Both \(a\) and \(b\) are real numbers.
In the complex number \(z = 5 - 9i\), identify the real part and the imaginary part.
The real part is \(5\). The imaginary part is \(-9i\). The coefficient \(b\) of the imaginary part is \(-9\).
What is a 'purely imaginary number'?
A purely imaginary number is a complex number \(a + bi\) where the real part \(a\) is zero. Examples include \(3i\) and \(-2i\).
How is any real number, like \(7\), represented as a complex number in standard form?
A real number is a complex number \(a + bi\) where the imaginary part's coefficient \(b\) is zero. For example, \(7\) can be written as \(7 + 0i\).
What is the complex plane?
The complex plane is a two-dimensional coordinate system used to plot complex numbers. The horizontal axis is called the 'real axis' and the vertical axis is called the 'imaginary axis'.
Which point on the complex plane corresponds to the complex number \(z = -4 + 2i\)?
The point \((-4, 2)\). You move \(4\) units to the left on the real (horizontal) axis and \(2\) units up on the imaginary (vertical) axis.
What is the absolute value (or modulus) of a complex number \(z = a + bi\), and how is it calculated?
The absolute value, denoted \(|z|\) or \(|a + bi|\), represents the distance from the origin to the point \((a, b)\) on the complex plane. It is calculated using the formula: $$ |z| = \sqrt{a^2 + b^2} $$
Calculate the absolute value (modulus) of the complex number \(z = 3 + 4i\).
Using the formula \(\sqrt{a^2 + b^2}\), we get \(\sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5\).
Calculate the modulus of \(z = -6 - 8i\).
The modulus is \(|-6 - 8i| = \sqrt{(-6)^2 + (-8)^2} = \sqrt{36 + 64} = \sqrt{100} = 10\).
When are two complex numbers, \(a + bi\) and \(c + di\), considered equal?
Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal. So, \(a + bi = c + di\) only if \(a = c\) and \(b = d\).
Solve for \(x\) and \(y\): \( (2x - 1) + 5i = 7 + (y + 2)i \)
Equate the real parts: \(2x - 1 = 7\), which gives \(2x = 8\) and \(x = 4\). Equate the imaginary coefficients: \(5 = y + 2\), which gives \(y = 3\).
