# Ap1.3.51 are complex numbers, show that

• MHB
• karush
In summary, the problem asks to show that for complex numbers $z$ and $u$, $\bar{z}u=\overline{z\bar{u}}$ and $\overline{\left(\frac{z}{u} \right)}=\frac{\bar{z}}{\bar{u}}$. The bar over a complex number represents its conjugate.
karush
Gold Member
MHB
$\textsf{ If$z$and$u$are complex numbers, show that}$
$$\displaystyle\bar{z}u=\bar{z}\bar{u} \textit{ and } \displaystyle \left(\frac{z}{u} \right)=\frac{\bar{z}}{\bar{u}}$$ok couldn't find good example on what this is
and I'm not good at 2 page proof systemsso much help is mahalo

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I would begin with:

$$\displaystyle u=x_u+y_ui$$

$$\displaystyle z=x_z+y_zi$$

And the see where the algebra leads. :)

MarkFL said:
I would begin with:

$$\displaystyle u=x_u+y_ui$$

$$\displaystyle z=x_z+y_zi$$

And the see where the algebra leads. :)

accually what does the bar over mean

karush said:
accually what does the bar over mean

That means "the conjugate of." And so, using my prior definitions:

$$\displaystyle \overline{u}=x_u-y_ui$$

$$\displaystyle \overline{z}=x_z-y_zi$$

karush said:
$\textsf{ If$z$and$u$are complex numbers, show that}$
$$\displaystyle\bar{z}u=\bar{z}\bar{u} \textit{ and } \displaystyle \left(\frac{z}{u} \right)=\frac{\bar{z}}{\bar{u}}$$

There are some bars missing. I think that the problem should be asking you to show that $$\bar{z}u=\overline{z\bar{u}} \text{ and } \overline{\left(\frac{z}{u} \right)}=\frac{\bar{z}}{\bar{u}}$$

## 1. What are complex numbers?

Complex numbers are numbers that consist of both a real part and an imaginary part. They are typically written in the form a + bi, where a is the real part and bi is the imaginary part, with i being the imaginary unit equal to the square root of -1.

## 2. How do you perform operations with complex numbers?

To add or subtract complex numbers, you simply combine the real and imaginary parts separately. To multiply complex numbers, you use the FOIL method, just like with binomials. To divide complex numbers, you need to rationalize the denominator by multiplying both the numerator and denominator by the complex conjugate of the denominator.

## 3. How can you show that Ap1.3.51 are complex numbers?

To show that a number is a complex number, you need to demonstrate that it has both a real and imaginary part. In the case of Ap1.3.51, you would need to write it in the form a + bi, where a and b are both real numbers.

## 4. What are some common applications of complex numbers?

Complex numbers have many applications in mathematics, physics, and engineering. They are used in electrical engineering to represent AC circuits, in quantum mechanics to describe wave functions, and in signal processing to analyze signals with both real and imaginary components.

## 5. Can complex numbers be plotted on a graph?

Yes, complex numbers can be plotted on a graph known as the complex plane. The real part is plotted on the horizontal axis, and the imaginary part is plotted on the vertical axis. This allows for the visualization of complex numbers and their relationships with each other.

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