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How Do You Find the Absolute Value of a Complex Number?

Find the absolute value of 5 + 6i

Summary

  1. The imaginary unit 'i' is equal to the square root of -1
  2. 'a' and 'b' are variables that stand for any real number
  3. The absolute value of the complex number a+bi is the square root of the sum a2+b2
  4. 'a' is 5 and 'b' is 6
  5. 61 is a prime number so the square root can't be simplified
  6. The absolute value is the distance from the point 5+6i to the origin

Notes

    1. 5+6i is a complex number
    2. The absolute value of any complex number, a+bi is equal to the square root of (a2+b2)
    1. Remember, the Pythagorean Theorem helps us find the length of the hypotenuse of a right triangle
    2. Later in this tutorial, we'll show you how to use the Pythagorean Theorem to find the absolute value!
    1. In the general form of the solution, 'a' and 'b' stand for any real numbers
    2. We need to find what real numbers in 5+6i are the same as 'a' and 'b'
    3. 'a' is the real number '5'
    4. 'bi' is an imaginary number, but we just want the 'b' in front of the 'i'
    5. If you leave off the 'i', 'b' is just a real number
    6. 'i' means the square root of -1
    7. So 'b' is the real number '6'
    1. Plug in '5' for 'a' and '6' for 'b'
    2. The absolute value of 5+6i is equal to the square root of (52+62)
    1. We can simplify the right side, the part under the square root sign
    2. '5' squared is 25
    3. '6' squared is 36
    4. 25+36 is 61
    1. Since 61 is prime, it doesn't have any perfect square factors we can simplify with the square root
    2. So we could show our answer as a decimal or just leave it as the square root of 61
    1. Graphing this problem in the complex plane can help us understand what the absolute value of 5+6i means
    2. The complex plane is similar to the x-y coordinate plane
    3. The real axis is like the x-axis and the imaginary axis is like the y-axis
    4. The real part of the complex number tells you where to move on the real axis
    5. The imaginary part tells you where to move on the imaginary axis
    6. The distance from the point 5+6i to the point (0,0) is equal to the absolute value of 5+6i
    1. One leg of the triangle is 5 units long on the real axis
    2. The other leg is 6 units long and forms a right angle with the first leg we drew
    3. The Pythagorean Theorem says that for a right triangle, the sum of the legs squared will give us the hypotenuse squared
    4. The hypotenuse, 'c', is the absolute value of 5+6i
    1. The absolute value is the square root of 61