Find the real and imaginary parts of e-2 + i12
A number refers to a word or symbol which represents a specific quantity. It is with the assistance of numbers only that multiple arithmetic operations are performed and that we are ready to develop such a lot within the field of physics and arithmetic. One cannot live their life without the usage of numbers, even for the foremost basic chores or tasks. Even the cash exchanged for commodities may be a certain value represented by numbers. A bunch of numbers grouped together is employed to assign an individual as their contact number, such is that the prominence of numerals in our lives.
Types of numbers
There are different types of numbers based on their characteristics and the properties they portray. Lets take a look at these different types of numbers in detail,
- Natural Numbers: A group of numbers as is employed to count certain objects are called natural numbers. Such a group of numbers starts with 1 (one) and goes on till infinity. They include only positive integers.
- Whole Numbers: A group of numbers that has all the positive integers and 0 .
- Integers: An integer is defined intrinsically as an entire number which will assume either positive, negative or no value in the least.
- Rational Numbers: They can be expressed in the form of a fraction. These numbers have a terminating decimal expansion.
- Irrational Numbers: Such numbers can’t be expressed as a fraction.
- Real Numbers: The set of real numbers includes both rational numerals and irrational numerals.
Real and Imaginary numbers
The set of real numbers includes both rational numerals and irrational numerals. They are based on the concept of a number line, with zero being the origin and all the numbers to its right, positive and those in the left of the origin, negative.
Often times it so happens while solving quadratic equations, that the discriminant comes out to be a negative value under the square root. This might sound impossible because by following the general rule in mathematics, the square of a negative number is a positive number so it does not make sense for a perfect square or any real number, for that matter, to be negative and under the root altogether. However, numbers can also be depicted as square root of a negative number in mathematics. Example, is an imaginary number, since it depicts the number 100, which is a perfect square as a negative number under a square root. Similarly, , , so on and so forth are all imaginary numbers. Such numbers are not tangible, but still kind of real in the sense that they are used in mathematics. In other words, imaginary numbers are numbers which are the opposite of real numbers. They are not based on the concept of number line, and as a result, cannot be depicted or plotted on one. Another way of defining imaginary number could be: such a number which yields a negative result when multiplied with itself, i.e., squared.
Representing an Imaginary Number Without the Square Root Part
An imaginary number, when written without using the root arithmetic expression can be written as a real number, multiplied by iota, depicted i, which is an imaginary unit and iota (i) = .
Hence, can be written as = 10i.
Powers of i
- i = √-1
- i2 = -1
- i3 = i . i2 = i(-1) = -i
- i4 = i2 . i2 = -1(-1) = 1
- i5 = i . i4 = i
- i6 = i . i5 = i . i = i2 = -1
- i7 = i . i6 = i(-1) = -i
- i8 = (i2)4 = (-1)4 = 1
- i9 = i . i8 = i(1) = i
- i10 = i . i9 = i(i) = i2 = -1
Following this pattern, it can be concluded that i repeats its values after every 4th power.
A complex number can be called a hybrid of real and imaginary numbers, with the real number or constituent being any fraction, rational or irrational integer and its imaginary part being represented as a real number in multiplication with the imaginary unit iota, depicted i. Thus, a complex number shows a real number and an imaginary number combined by either of these two arithmetic operations, addition and subtraction.
Standard Form of a Complex Number
A complex number, in its standard form is expressed as a + ib, where a and b both are real numbers, but b being in multiplication with the imaginary variable i, represents the imaginary part of the whole complex number, which can denoted by ‘z’. Hence, a complex number is usually written in the form z = a + ib, where a depicts the real part and ib or bi would be the imaginary constituent. For that matter 0 + bi would also be regarded as a complex number with the real part being non- existent and bi depicting its imaginary counterpart. Examples are,
- 5 + 2i is a complex number, where 5 is the real part and 2i depicts the imaginary part.
- e2 + 12i is a complex number, where e2 is the real part and 12i is the imaginary part.
- √22 -162i is a complex number, where √22 is the real part and 162i is the imaginary part.
Find the real and imaginary parts of e-2 + i12.
A complex number is usually written in the form z = a + ib, where a depicts the real part and ib or bi would be the imaginary constituent.
Real part = e-2 = 1/ e2 and imaginary part = 12i.
Question 1: Find the real and imaginary parts of ez if z = x + iy.
ez = ex + iy
= ex(cosy + isiny)
= ex cos y + ex isiny
Hence the real part = ex cos y and the imaginary part = ex isiny.
Question 2: Find the real and imaginary parts of 3i20 – i19.
Clearly, i20 = 1 and i19 = i.
So, the expression becomes 3(1) – i = 3 – i
Hence the real part = 3 and the imaginary part = 1.
Question 3: Find the real and imaginary parts of the number q if q ∈ R.
q ∈ R, q is a real number, implying that it does not have any imaginary part. Alternatively one can say that the coefficient of i is zero.
Hence the real part and the imaginary part of q for all q ∈ R are q and zero respectively.
Question 4: Find the real and imaginary parts of 10i100 + 2i99.
Clearly, i100 = 1 and i99 = -i
So, the expression becomes 10(1) + 2(-i)
= 10 – 2i
Hence the real part = 10 and the imaginary part = 2.
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