3. Complex numbers

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Complex numbers are an extension of the set of real numbers and are very important to mathematics and its applications, including physics and engineering. They are often used in high school and ENEM questions, so understanding this topic is crucial to performing well on the test.

A complex number is a number of the form a + bi, where a and b are real numbers and i is the imaginary unit, which has the property that i² = -1. Part a is called the real part of the complex number, and part bi is called the imaginary part. For example, if we have the complex number 3 + 4i, then 3 is the real part and 4 is the imaginary part.

Complex numbers can be graphed on the complex plane, which is similar to the Cartesian plane we use to represent real numbers. The difference is that the x-axis represents the real part of the complex number and the y-axis represents the imaginary part. Therefore, the complex number a + bi is represented as a point in the plane whose coordinates are (a, b).

There are several operations we can perform with complex numbers, including addition, subtraction, multiplication, and division. Addition and subtraction are performed by adding or subtracting respectively the real parts and the imaginary parts. For example, if we have the complex numbers z1 = a + bi and z2 = c + di, then z1 + z2 = (a + c) + (b + d)i and z1 - z2 = (a - c) + (b - d)i.

Multiplication is a little more complicated, but it's still pretty straightforward. If we have the complex numbers z1 = a + bi and z2 = c + di, then z1 * z2 = (ac - bd) + (ad + bc)i. Note that the product of two complex numbers is another complex number.

Dividing complex numbers is a bit more complicated, but it's also possible. If we have the complex numbers z1 = a + bi and z2 = c + di, then z1 / z2 = (ac + bd) / (c² + d²) + (bc - ad) / (c² + d²)i. Note that the denominator is always a real number, so the result is a complex number.

Complex numbers also have a concept of modulus and argument, which are analogous to the magnitude and angle of a vector in the plane. The modulus of a complex number z = a + bi is given by |z| = sqrt(a² + b²), which is the distance from the point representing z to the origin in the complex plane. The argument of z is the angle that the line from the point to the origin makes with the positive x axis, measured counterclockwise.

Complex numbers also have a concept of conjugate, which is obtained by changing the sign of the imaginary part. The conjugate of a complex number z = a + bi is denoted by z̅ and is given by z̅ = a - bi. The conjugate has the property that z * z̅ = |z|².

In summary, complex numbers are an extension of real numbers that allow you to perform calculations that would not be possible with just real numbers. They are essential for many areas of mathematics and their applications, and a good understanding of them is crucial for ENEM and other mathematics exams.

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