Suppose a and b are real numbers, not both 0. Find real numbers c and
d such that
1=.a C bi/ D c C di:

Given the problem, we want to find real numbers \( c \) and \( d \) such that:

\[
1 = (a + bi)(c + di)
\]

Here, \( a \) and \( b \) are real numbers, and \( c \) and \( d \) are the unknown real numbers we need to determine. Let's expand the expression on the right-hand side:

\[
(a + bi)(c + di) = ac + adi + bci + bdi^2
\]

We know that \( i^2 = -1 \), so this becomes:

\[
ac + adi + bci - bd
\]

This can be rewritten as:

\[
(ac - bd) + (ad + bc)i
\]

For the equation to be equal to 1 (which can be represented as \( 1 + 0i \)), the real part must be 1 and the imaginary part must be 0. Therefore, we set up the following system of equations:

1. \( ac - bd = 1 \)
2. \( ad + bc = 0 \)

Now, we'll solve for \( c \) and \( d \).

### Solving the System of Equations

1. From the second equation \( ad + bc = 0 \), we can express \( d \) in terms of \( c \):

\[
d = -\frac{bc}{a} \quad \text{(assuming \( a \neq 0 \))}
\]

2. Substitute this into the first equation \( ac - bd = 1 \):

\[
ac - b\left(-\frac{bc}{a}\right) = 1
\]

\[
ac + \frac{b^2c}{a} = 1
\]

Multiply through by \( a \) to eliminate the fraction:

\[
a^2c + b^2c = a
\]

Factor out \( c \) on the left-hand side:

\[
c(a^2 + b^2) = a
\]

Thus,

\[
c = \frac{a}{a^2 + b^2}
\]

3. Substitute this value of \( c \) back into the equation for \( d \):

\[
d = -\frac{bc}{a} = -\frac{b \cdot \frac{a}{a^2 + b^2}}{a} = -\frac{b}{a^2 + b^2}
\]

### Final Answer

The real numbers \( c \) and \( d \) that satisfy the given equation are:

\[
c = \frac{a}{a^2 + b^2}, \quad d = -\frac{b}{a^2 + b^2}
\]

Do you have any questions, or would you like more details?

Here are some related questions you might find interesting:

1. How would the solution change if \( a \) and \( b \) were both 0?
2. Can you generalize this method for any complex number \( z \) where \( z = a + bi \)?
3. What are the geometric interpretations of \( c \) and \( d \) in the complex plane?
4. How does the magnitude of the complex number affect the values of \( c \) and \( d \)?
5. What happens if we add a phase shift to \( a + bi \)?

**Tip:** In solving complex number problems, always separate real and imaginary components to set up your system of equations.

Discover how to find real numbers c and d such that the product of (a + bi) and (c + di) equals 1. This problem involves complex number multiplication and solving equations. Suitable for advanced high school students and beyond.

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