# What is the Difference between imaginary part and imaginary number?

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• SHASHWAT PRATAP SING
In summary: So, basically, it's just a matter of convention and what makes calculations easier. In summary, the imaginary part of a complex number is defined as a real number and this is just a convention to make calculations easier.
SHASHWAT PRATAP SING
I am a high school student and I have a doubt regarding Complex Numbers.
when we define a complex number we say that a number of the form z=a+ib ,where a and b are real numbers is called a complex number.
a is called the real part and b is called the imaginary part, but I have a doubt here we know that ib is an imaginary number then why we say only b is the imaginary part of z why not ib.
for example-
In z=5+3i why we say 3 is the imaginary part of z why not 3i.

This is simply the way things are defined: the imaginary part of a complex number is a real number (the real number multiplying i .)
In fact the imaginary part of a pure imaginary number is a real number. These are just done to make the manipulations easier.

SHASHWAT PRATAP SING said:
I am a high school student and I have a doubt regarding Complex Numbers.
when we define a complex number we say that a number of the form z=a+ib ,where a and b are real numbers is called a complex number.
a is called the real part and b is called the imaginary part, but I have a doubt here we know that ib is an imaginary number then why we say only b is the imaginary part of z why not ib.
for example-
In z=5+3i why we say 3 is the imaginary part of z why not 3i.
It's purely convention. The way things are defined we have: ##z = Re(z) +iIm(z)##, where ##Re(z), Im(z)## are real numbers. You could define ##Re(z) = a## and ##Im(z) = ib##. That wouldn't change anything except a detail about how you name things. For example, normally we have: $$|z|^2 = Re(z)^2 + Im(z)^2$$
Whereas, with your definition we would have: $$|z|^2 = Re(z)^2 + (\frac{Im(z)}{i})^2$$

hutchphd

## 1. What is the difference between the imaginary part and the imaginary number?

The imaginary part is a component of a complex number, while an imaginary number is a number that can be written in the form "bi", where b is a real number and i is the imaginary unit (√-1).

## 2. Can the imaginary part of a complex number be equal to an imaginary number?

Yes, the imaginary part of a complex number can be equal to an imaginary number if the real part of the complex number is equal to zero. In this case, the complex number can be written as "0 + bi", making it an imaginary number.

## 3. How are imaginary parts and imaginary numbers used in mathematics?

Imaginary parts and imaginary numbers are used in mathematics to represent numbers that cannot be expressed using only real numbers. They are particularly useful in fields such as engineering, physics, and computer science.

## 4. What is the significance of the imaginary part in a complex number?

The imaginary part of a complex number is significant because it allows for the representation of numbers that are not on the real number line. It also plays a crucial role in solving equations, particularly in electrical engineering and quantum mechanics.

## 5. Can the imaginary part of a complex number be negative?

Yes, the imaginary part of a complex number can be negative. This means that the number is located below the real number line on the complex plane. For example, the complex number -3 - 2i has a negative imaginary part of -2i.

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