# Finding Real Part of $z$ for Complex Numbers

• MHB
• anemone
In summary, the real part of a complex number is the numerical value that is not affected by the imaginary unit, i. To find the real part, you simply take the numerical value in front of i. It can be negative and is used in various fields such as engineering, physics, and mathematics. The difference between the real part and the imaginary part is that the imaginary part is multiplied by i.
anemone
Gold Member
MHB
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Let $z_1=18+83i,\,z_2=18+39i$ and $z_3=78+99i$, where $i=\sqrt{-1}$. Let $z$ be the unique complex number with the properties that

$\dfrac{z_3-z_1}{z_2-z_1}\cdot \dfrac{z-z_2}{z-z_3}$ is a real number and the imaginary part of $z$ is the greatest possible. Find the real part of $z$.

[TIKZ]
\begin{scope}
\draw (0,0) circle(3);
\end{scope}
\node (1) at (0,0) {c};
\draw (0,0) node[anchor=south] {.};
\coordinate[label=left: $z_2$] (E) at (-2,-2.236);
\coordinate[label=left: $z_1$] (A) at (-2,2.236);
\coordinate[label=above: $z$] (B) at (-1,2.828);
\coordinate[label=above: $z_3$] (C) at (1.2,2.75);
\coordinate[label=below: $z$] (D) at (2,-2.236);
\draw (E) -- (A);
\draw (E) -- (B);
\draw (E) -- (C);
\draw (E) -- (D);
\draw (A) -- (B);
\draw (B) -- (C);
\draw (C) -- (D);
\node (1) at (-1.8,2) {$\theta_1$};
\node (2) at (-0.8,2.6) {$\theta_2$};
\node (3) at (1.8,-2.0) {$\theta_2$};
[/TIKZ]

Let $\dfrac{z_3-z_1}{z_2-z_1}=r_1\cis(\theta_1)$, where $0<\theta_1<180^{\circ}$.

If $z$ is on or below the line through $z_2$ and $z_3$, then $\dfrac{z-z_2}{z-z_3}=r_2\cis(\theta_2)$, where $0<\theta_2<180^{\circ}$. Because $r_1 \cis(\theta_2)\cdot r_2 \cis(\theta_2)=r_1\cdot r_2\cdot \cis(\theta_1+\theta_2)$ is real, it follows that $\theta_1+\theta_2=180^{\circ}$, meaning that $z_1,\,z_2,\,z_3$ and $z$ lie on a circle.

On the other hand, if $z$ is above the line through $z_2$ and $z_3$, then $\dfrac{z-z_2}{z-z_3}=r_2\cis(-\theta_2)$, where $0<\theta_2<180^{\circ}$. Because $r_1 \cis(\theta_1)\cdot r_2 \cis(\theta_2)=r_1\cdot r_2\cdot \cis(\theta_1-\theta_2)$ is real, it follows that $\theta_1=\theta_2$, meaning that $z_1,\,z_2,\,z_3$ and $z$ lie on a circle.

In either case, $z$ must lie on the circumcircle of $\triangle z_1 z_2 z_3$ whose center is the intersection of the perpendicular bisectors of $\overline{z_1z_2}$ and $\overline{z_1z_3}$, namely, the lines $y=\dfrac{39+83}{2}=61$ and $16(y-91)=-60(x-48)$.

Thus the center of the circle is $c=56+61i$. The imaginary part of $z$ is maximal when $z$ is at the top of the circle, and the real part of $z$ is 56.

## What is the real part of a complex number?

The real part of a complex number is the portion of the number that does not contain the imaginary unit, i.e. the part without the "i" term. It is represented by the symbol Re(z) or simply by the letter "a".

## How do you find the real part of a complex number?

To find the real part of a complex number, simply take the coefficient of the "real" term, which is usually denoted by "a". For example, in the complex number z = 3 + 4i, the real part is 3.

## Why is it important to find the real part of a complex number?

The real part of a complex number helps us understand the location of the number on the complex plane. It also allows us to perform operations such as addition, subtraction, and multiplication with other complex numbers.

## Can the real part of a complex number be negative?

Yes, the real part of a complex number can be negative. This simply means that the number is located in the left half of the complex plane. For example, in the complex number z = -2 + 3i, the real part is -2.

## Is the real part of a complex number always a real number?

Yes, by definition, the real part of a complex number is always a real number. This means that it can be represented on the number line and it does not contain the imaginary unit "i".

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