Write three examples of complex numbers. Make one example be just an integer, the next example a purely imaginary number, and the last example have both an a and b that is not zero. Then, re-write each of your complex numbers in a + bi form and identify the real and imaginary part.
After you have posted your complex numbers, their a + bi form, and your identification of their real and imaginary part, respond to two of your classmates’ posts. In your response to each post: Re-write their complex numbers in a + bi form. Create rules for how to Identify the real and imaginary part of complex numbers. Explore what effect, if any, negative values and irrational values play in complex numbers.
Example 1: Complex number = 6
a + bi form: 6 + 0i
Real part: 6
Imaginary part: 0
Example 2: Complex number = 4i
a + bi form: 0 + 4i
Real part: 0
Imaginary part: 4
Example 3: Complex number = 2 + 3i
a + bi form: 2 + 3i
Real part: 2
Imaginary part: 3
Response to classmate 1:
Complex number: -8 + 2i
a + bi form: -8 + 2i
Real part: -8
Imaginary part: 2
The real part of a complex number is the coefficient of 'a' in the a + bi form, while the imaginary part is the coefficient of 'b'. Negative values do not change the identification of the real and imaginary part in complex numbers. In this case, the real part is -8 and the imaginary part is 2.
Response to classmate 2:
Complex number: √5 - 3i
a + bi form: √5 - 3i
Real part: √5
Imaginary part: -3
Irrational values, like √5, can be part of the real or imaginary part of a complex number. In this case, √5 is the real part and -3 is the imaginary part. The rules for identifying the real and imaginary part remain the same.
Example 1: Integer complex number
Complex number: 5
a + bi form: 5 + 0i
Real part: 5
Imaginary part: 0
Example 2: Purely imaginary complex number
Complex number: 3i
a + bi form: 0 + 3i
Real part: 0
Imaginary part: 3
Example 3: Complex number with non-zero a and b
Complex number: 2 + 4i
a + bi form: 2 + 4i
Real part: 2
Imaginary part: 4
To rewrite complex numbers in the a + bi form, we simply express the real part as 'a' and the imaginary part as 'bi'. The 'i' represents the imaginary unit, where i^2 = -1.
In general, for any complex number 'z' with a real part 'a' and imaginary part 'b', the a + bi form represents 'z' in the standard complex number notation.
Now let's respond to two classmates' posts.
Response to Classmate 1:
Complex number: -7
a + bi form: -7 + 0i
Real part: -7
Imaginary part: 0
Response to Classmate 2:
Complex number: √2 - i
a + bi form: √2 - i
Real part: √2
Imaginary part: -1
In complex numbers, negative values do not have a significant effect other than indicating the direction of the imaginary part (positive or negative). However, irrational values, like √2 in the second example, can play a role in determining the magnitude and orientation of the complex number on the complex plane.
Example 1: Complex number as an integer
Complex Number: 3
a + bi Form: 3 + 0i
Real Part: 3
Imaginary Part: 0
Example 2: Complex number as a purely imaginary number
Complex Number: 4i
a + bi Form: 0 + 4i
Real Part: 0
Imaginary Part: 4
Example 3: Complex number with non-zero a and b values
Complex Number: 2 + 5i
a + bi Form: 2 + 5i
Real Part: 2
Imaginary Part: 5
Response to Classmate A:
Complex Number: -1/2 - 2i
a + bi Form: -1/2 - 2i
Real Part: -1/2
Imaginary Part: -2
Response to Classmate B:
Complex Number: √2 + √3i
a + bi Form: √2 + √3i
Real Part: √2
Imaginary Part: √3
To identify the real and imaginary part of a complex number in a + bi form, we look at the coefficient of 'a' as the real part and the coefficient of 'b' as the imaginary part. If either 'a' or 'b' is zero, then the complex number is either a purely real or a purely imaginary number, respectively.
Negative values do not have a significant effect on complex numbers as they can still be represented in the a + bi form. In fact, the imaginary unit 'i' itself is defined as the square root of -1. Therefore, negative values can be expressed as a combination of their real and imaginary parts.
Irrational values, like √2 and √3, can still be used as coefficients in the complex numbers. Complex numbers can include irrational values in their a + bi form, where 'a' and 'b' can be irrational numbers. The rules for identifying the real and imaginary parts do not change when dealing with irrational values.