Is there a difference between imaginary part and imaginary number? Because our teacher says there is.
I mean in $5+3i$ , is it right to say the imaginary part is $3i$ and imaginary number is $3$?
I find it difficult to distinguish between them.
Please tell me the difference and why.
3 Answers
$\begingroup$Let $z = a + bi$ be a complex number. The real part of $z$ is $a$. The imaginary of $z$ is $b$. Thus, the real part of $5 + 3i$ is $5$ and its imaginary part is $3$. Note that the real and imaginary parts of a complex number correspond to its coordinates on the complex plane.
An imaginary number is a complex number of the form $z = 0 + bi = bi$. An example of an imaginary number is $2i$.
Note that an imaginary number is a complex number whose real part is equal to $0$, while a real number is a complex number whose imaginary part is equal to $0$.
$\endgroup$ $\begingroup$In your question, you wrote:
"I mean in $5+3i$ , is it right to say the imaginary part is $3i$ and imaginary number is $3$?"
You have it reversed. $3i$ is a (purely) imaginary number, and $3$ is its imaginary part.
For $3i$, the real part is zero (that's why it's purely imaginary).
For the real number $5$, its real part is $5$ while its imaginary part is zero. Obviously $5$ is a (purely) real number, but remember that the real numbers are a proper subset of the complex numbers.
Finally, consider the complex number $5+3i$ which is obtained by adding a purely real number $5$ to a purely imaginary number $3i$. All complex numbers are constructed this way.
For $5+3i$, the real part is $5$ and the imaginary part is $3$. Both the real and imaginary parts of any complex number are, by definition, real.
$\endgroup$ $\begingroup$Imaginary part is the part of a complex number that is not real, in your example you have $5+3i$, there is no $3i$ on the real number axis as $(3i)^2=-9$ which doesn't work. However a complex number may also contain a real part, the number $5$.
That is you are basically talking about the coordinates in the XY-plane of complex numbers and their respective coordinates with real and imaginary part. However real numbers, when spoken of alone, is only on the "x" axis of this 2 dimensional plane, equally so you can talk about numbers only on our "Y" axis, which are numbers that only have an imaginary component but no real component.
Complex numbers in the most general sense however have both an imaginary component (The y-axis commonly) and a real component (the x-axis component)
This is why we can write complex numbers as, instead of $5+3i$ as $(5,3)$ instead.
To draw the analogy further, the imaginary/real part of a complex number can be said to be the "coordinates" of a point, while when you talk about imaginary number, it's like talking about the y-value alone and real numbers alone are just x-values.
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