Overview:
While positive numbers have both positive and negative square roots, negative numbers do not have square roots within the set of real numbers. The set of imaginary numbers can be used in equations with square roots of negative numbers. Complex numbers have real and imaginary components, and can be used in operations similar to real numbers.
What Are Imaginary Numbers?
The imaginary number i is equal to the square root of -1. In other words, i2 equals -1. The square root of a negative number is not a real number and it is not a variable. For example, the square root of -25 is written as 5i because 5i times 5i equals 25 times -1 or -25. The square root of -3 can be written as i√3, because i√3 times i√3 equals -1 times 3, or -3. When the square root of negative numbers is taken, the imaginary number i must be used as a factor.
What Are Complex Numbers?
Complex numbers take the form a +bi, where both a and b are real numbers. If a is equal to 0, then the only part of the number that is left is bi, the form of an imaginary number, where i is equal to the square root of -1. If b is equal to 0, then the only part of the number that is left is a, which is a real number. Every real number is also a complex number.
How Are Complex Numbers Added and Subtracted?
Complex numbers can be added or subtracted in the same way as other expressions. For example, 6 +2i can be added to 3 + 3i by adding 6+3 as 9 and 2i + 3i or 5i. Thus, 6 + 2i + 3 + 3i is 9 + 5i. Similarly, 4 +5i – 2 +3i can be evaluated as 4-2 and 5i-3i or 2 + 2i. The additive inverse of 2 + 3i is -2 -3i.
How Are Complex Numbers Multiplied?
Complex numbers are multiplied similarly to the multiplication of two polynomials. For example, (3 + 2i)(2 +4i) are multiplied as 3(2 +4i) + 2i(2 + 4i) using the distributive property, so that 6 + 12i +4i +8i2. This can be simplified to 6 + 16i -8, because 8i2 is equal to 8 times -1, because i2 equals -1, or -2 +16i.
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