Why exactally does the absolute value of -3+4i = 5. I know (-3) squared plus (4i) squared = 5? Could you include a graphic approach as well? Thank you.
i is an imaginary number. YOu can graph it only on the complex number plane.
Consider a rectangular graph, the x axis is the real numbers, and the y axis is imaginary numbers. THen you plot the point (-3,4). Find the distance from the origin, it is 5.
To understand why the absolute value of -3+4i is 5, let's break it down step by step.
The absolute value of a complex number z = a + bi is defined as √(a^2 + b^2), where a and b represent the real and imaginary parts of the complex number, respectively.
In this case, for the complex number -3+4i, we have:
a = -3 (real part)
b = 4 (imaginary part)
Plugging in these values into the absolute value formula, we get:
| -3+4i | = √((-3)^2 + 4^2) = √(9 + 16) = √25 = 5.
So, based on the formula, the absolute value of -3+4i is indeed 5.
Now, let's explore a graphical approach to understand this result.
Geometrically, the absolute value of a complex number represents its distance from the origin (0,0) on the complex plane. In this case, we can plot -3+4i as a point on the complex plane.
The point corresponding to -3+4i can be plotted by moving 3 units to the left (in the negative x-direction) and 4 units up (in the positive y-direction) from the origin. The resulting point is (-3, 4), as shown below:
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___________|__________
/|(-3, 4)
/ |
/ |
Now, we can measure the distance from this point (-3, 4) to the origin (0,0). Using Pythagorean's theorem, the distance is given by:
√((-3)^2 + 4^2) = √(9 + 16) = √25 = 5.
This distance represents the absolute value of the complex number -3+4i, which we obtained earlier using the formula.
Hence, both algebraically and graphically, we confirm that the absolute value of -3+4i is indeed 5.