What are imaginary numbers?
If you are wondering what are imaginary numbers and thinking that there must be a meaning for imaginary number, Then lets get into the article to learn about what exactly are imaginary numbers.
Imaginary numbers :
The imaginary numbers are the numbers which, when squared, give a negative number. Imaginary numbers are the square roots of negative numbers where they do not have any definite value.
The imaginary numbers are represented as the product of a real number and the imaginary value i.
Example β
3i, 5i, 25i are some examples of imaginary numbers.
The value of i2 is given as -1.
So the value of (5i)2 is -25, and that implies is i.
Complex number :
Complex numbers are a combination of real numbers and imaginary numbers.
The complex number is of the standard form β a + bi.
Where a and b are real numbers. i is an imaginary unit.
Now letβs get a quick understanding by using some examples:
- Real numbers β
-1,2, 10, 10000. - Imaginary numbers β
4i, -5i, 2400i. - Complex numbers β
2+3i, -5-4i.
Conjugate pair of an imaginary number :
- a+bi is a complex number and the conjugate pair of a +bi is a-bi.
- When an imaginary number is multiplied by its conjugate pair, then the result will be a real number.
Rules of imaginary numbers :
Arithmetic Operations on Imaginary Numbers :
1. Addition β
- When two imaginary numbers are added, then the real part is added into one , and then the imaginary part is added into one.
Examples β
1. (2 + 2i) + (3 + 4i)
= (2 + 3) + (2 + 4)i
= (5 + 6i)
2. (3 + 4i) + (5 + 3i)
= (8) + (7)i
= 8 + 7i
3. (5 + 3i) + (4 + 2i)
= (5 + 4) + (3 + 2)i
= 9 + 5i.
2. Subtraction β
- When two imaginary numbers are subtracted, then the real part is subtracted, and then the imaginary part is subtracted.
Examples β
1. (2 + 2i) β (3 + 4i)
= (2 β 3) + (2 β 4)i
= (-1 β 2i)
2. (3 + 4i) β (5 + 3i)
= (-2) + (1)i
= -2 + i
3. (5 + 3i) β (4 + 2i)
= (5 β 4) + (3 β 2)i
= 1 + i.
3. Multiplication
- When two imaginary numbers are multiplied, then the result will be as follows β
Examples β
1. (a + bi) (c + di)
= (a + bi)c + (a + bi)di
= ac + bci + adi+
= (ac β bd)+i(bc + ad)
2. (3 + 4i)*(3 β 4i)
= (9 + 12i -12i β 16)
= 25
3. (2 + 2i) * (3 β 4i)
= (6 + 6i -8i β 8)
= (14 β 2i)
4) Division
The numerator and denominator will be multiplied by its conjugate pair of denominators.
Examples β
1. (2 + 2i) / (3 + 4i)
Multiplying the numerator and denominator with the conjugate pair of denominators.
=(2 + 2i) * (3 β 4i) / (3 + 4i)*(3 β 4i)
=(6 +6i -8i β 8) / (9 + 12i -12i β 16)
=(14 β 2i) / 25
2. (3 + 4i) / (2 + 2i)
Multiplying the numerator and denominator with the conjugate pair of denominators.
= ((3 + 4i) * (2 β 2i)) / ((2 + 2i) * (2 β 2i))
=(6 + 2i β 8) / (4-4)
=(14 + 2i) / (8)
=(7 + i) / 4
3. (2 + 2i) / (2 + 2i)
Multiplying the numerator and denominator with the conjugate pair of denominators.
= ((2 + 2i) * (2 β 2i)) / ((2 + 2i) * (2 β 2i))
= (4 β 4) / (4 β 4)
= 8 / 8
= 1
Where do we use imaginary numbers :
- Imaginary numbers are very useful in various mathematical proofs.
- Imaginary numbers are used to represent waves.
- Imaginary numbers show up in equations that donβt touch the x axis.
- Imaginary numbers are very useful in advanced calculus.
- Combining AC currents is very difficult as they may not match properly on the waves.
- Using imaginary currents helps in making the calculations easy.
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