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Let's look at our objectives for today. We'll start by reviewing imaginary and complex numbers. We'll then look at how to add and subtract imaginary numbers. And finally, we'll look at how we add and subtract complex numbers.
Let's start by reviewing imaginary and complex numbers. The square root of a negative number is non-real, or an imaginary number. We define the imaginary unit, i, as the square root of negative 1.
A complex number is a value in the form a plus bi, where a and b are real numbers, and i is the imaginary unit. In a complex number, a, is the real part, and b times i is the imaginary part. We use complex numbers in fields such as engineering and physics.
Now let's look at how we add and subtract imaginary numbers. First, recall that the product property for square root says that the square root of a times b is equal to the square root of a times the square root of b. For example, if we have the square root of negative 25, we can write that as the square root of 25 times negative 1.
We then apply the property, which gives us square root of 25 times the square root of negative 1. The square root of 25 is 5. And the square root of negative 1 is i. So we have 5i.
Now let's do an example adding and subtracting imaginary numbers. We have the square root of negative 4 plus the square root of negative 49 minus the square root of negative 9. Applying the product property for square roots, the square root of negative 4 is 2 times i.
The square root of negative 49 is 7i. And the square root of negative 9 is 3i. 2i, 7i, and 3i are all like terms. Therefore, we can combine them together by adding or subtracting their coefficients. 2 plus 7 gives us 9. And 9 minus 3 is 6. So our final answer is 6i.
Now let's do some examples adding and subtracting complex numbers. Adding and subtracting complex numbers is also similar to combining like terms. We add or subtract the real parts together and add or subtract the coefficients of the imaginary parts together. We can add or subtract complex numbers in this way because of the commutative property of addition.
Here's our first example. We want to add the complex numbers 4 plus 8i and 2 plus 3i. We start by adding our real parts together, 4 plus 2. And then we add our imaginary parts together, 8i plus 3i.
Combining our real parts together, 4 plus 2, gives us 6. And combining our imaginary parts together, 8i plus 3i, gives us 11i. So our final answer is 6 plus 11i.
Here's a second example. We want to subtract the complex numbers 11 minus 6i minus 7 plus 9i. We start by subtracting our real parts, 11 minus 7. And then we subtract our imaginary parts, negative 6i minus 9i.
Combining our real parts together, 11 minus 7, gives us 4. And combining our imaginary parts together, negative 6i minus 9i, gives us negative 15i. So our final answer is 4 minus 15i.
Let's go over our important points from today. Make sure you get these in your notes so you can refer to them later. The square root of a negative number is non-real, or an imaginary number. We define the imaginary unit i as i is equal to the square root of negative 1.
A complex number is a value in the form a plus bi, where a is the real part, and b times i is the imaginary part of the complex number. When we add or subtract complex numbers, we add or subtract the real parts together and add or subtract the coefficients of the imaginary parts together.
So I hope that these important points and examples helped you understand a little bit more about adding and subtracting complex numbers. Keep using your notes, and keep on practicing, and soon you'll be a pro. Thanks for watching.
A value of the form a + bi, where a and b are real numbers and 'i' is the imaginary unit.