Complex Number Magnitude
The magnitude of a complex number, also known as the modulus, is a measure of its distance from the origin in the complex plane. It is denoted by the letter 'r' and is calculated using the formula \( r = \sqrt{a^2 + b^2} \), where 'a' and 'b' are the real and imaginary parts of the complex number, respectively.
For instance, if you have a complex number \(-5 - 2i\), the real part 'a' is -5 and the imaginary part 'b' is -2. To find the magnitude, you square each part, sum them up, and then take the square root: \( r = \sqrt{(-5)^2 + (-2)^2} = \sqrt{25 + 4} = \sqrt{29} \).
Rounding to three significant digits, as prescribed in the exercise, the magnitude of our complex number is approximately \( r \approx 5.39 \), since \( \sqrt{29} \) is approximately 5.385, and rounding retains the first three significant digits.
Complex Number Argument
The argument of a complex number is the angle \( \theta \) it makes with the positive real axis, and it's an essential part of representing a complex number in polar form. For the complex number with negative real and imaginary parts, the function atan2 is used to determine the correct quadrant for the angle.
In our example, for the complex number \(-5 - 2i\), the argument is calculated using \( \theta = \text{atan2}(-2, -5) \). This typically yields an angle in the fourth quadrant. To adjust this to the standard convention of measuring angles counter-clockwise from the positive real axis, we need to add \( \pi \) radians to this result to represent the angle in the second quadrant where both coordinates are negative.
Using a calculator, we find that the angle in the fourth quadrant for \(-5 - 2i\) is approximately -0.3805 radians. After adjusting to the second quadrant: \( \theta \approx -0.3805 + \pi \approx 2.761 \) radians. Again, we round to three significant digits, as the exercise suggests.
Polar Coordinates
Polar coordinates offer another way to locate a point in the plane, using an angle and a distance from the origin, rather than x and y coordinates. In the context of complex numbers, expressing a number in polar form highlights its geometric interpretation.
To convert a complex number into polar coordinates, you need the magnitude 'r' and the argument '\( \theta \)'. The complex number is then represented as \( r(\cos \theta + i\sin \theta) \), where \( \cos \) and \( \sin \) are the cosine and sine of the argument, respectively.
For our example \(-5 - 2i\), its polar coordinates, using the calculated magnitude and argument, become \( 5.39(\cos(2.761) + i\sin(2.761)) \), with all values rounded to three significant digits. This highlights the relationship between algebraic and geometric perspectives of complex numbers.
Significant Digits
Significant digits, also known as significant figures, are a way of expressing the precision of a number. They include all the digits that are known reliably, plus the first uncertain digit. Rounding to a specific number of significant digits is common in scientific and mathematical contexts to reflect the accuracy of calculations and measurements.
When an exercise specifies that the result should be given to 'three significant digits', it means that only the three most important digits in a number should be retained, with the third digit being rounded up if the fourth digit is 5 or greater. For our calculations above, starting from exact values, we rounded to three significant digits at each stage to comply with the instructions, leading to our final polar form expression being accurate to the level of precision requested by the exercise.
