Complex Numbers Review by Ron Boutte https://padlet.com/ronjboutte/opteiblcu1z2p6vr Find online resources that support your learning in Pre-Calculus. You may add images and hyperlinks. For each section, summarize the publisher's main points. There are 5 sections: 1. Operations -Complex Numbers 2. Degrees and Radians 3. Polar Form/ Rectangular Form 4. Geometric Effects of Multiplying by Complex Numbers 5. Argument and Modulus of Complex Numbers en-us 2023-10-04 12:30:38 UTC 2023-10-16 14:00:57 UTC hello@padlet.com Mr. Boutte ronjboutte https://padlet.com/ronjboutte/opteiblcu1z2p6vr/wish/2732269794 Taking the square root of negative numbers leads to Complex Number (i). For example, the square root of -1 is equal to i . Complex numbers are in the form a + bi, where (a) is a real number and (bi) is the imaginary number.]]> 2023-10-04 15:12:32 UTC https://padlet.com/ronjboutte/opteiblcu1z2p6vr/wish/2732269794 maria / Tonay/ LaKayla https://padlet.com/ronjboutte/opteiblcu1z2p6vr/wish/2732281365 A example of complex numbers A complex number has the form x + yi where x and y are real numbers and i2 = −1. ]]> 2023-10-04 15:19:41 UTC https://padlet.com/ronjboutte/opteiblcu1z2p6vr/wish/2732281365 Maria,Tonay,and LaKayla https://padlet.com/ronjboutte/opteiblcu1z2p6vr/wish/2732283590 Modulus-take the square root of a^2 and b^2]]> 2023-10-04 15:20:45 UTC https://padlet.com/ronjboutte/opteiblcu1z2p6vr/wish/2732283590 Jesus Trejo, Juan Limon, Matthew Requena https://padlet.com/ronjboutte/opteiblcu1z2p6vr/wish/2732286856 the geometric effect is when a point is changed by rotation,translation,reflection or dilation ]]> 2023-10-04 15:22:41 UTC https://padlet.com/ronjboutte/opteiblcu1z2p6vr/wish/2732286856 tonay, maria, & lakayla https://padlet.com/ronjboutte/opteiblcu1z2p6vr/wish/2732291767 The angle measured from the positive real axis to the line segment; labeled θ ]]> 2023-10-04 15:25:28 UTC https://padlet.com/ronjboutte/opteiblcu1z2p6vr/wish/2732291767 LaKayla,Maria,LaKayla https://padlet.com/ronjboutte/opteiblcu1z2p6vr/wish/2732294354 1st=remains the same
2nd=-180
3rd=+180
4th=-360
https://mathsathome.com/modulus-argument-complex-number/]]> 2023-10-04 15:27:01 UTC https://padlet.com/ronjboutte/opteiblcu1z2p6vr/wish/2732294354 Pablo Sebastian https://padlet.com/ronjboutte/opteiblcu1z2p6vr/wish/2732294470 To convert from degrees to radians, multiply the number of degrees by π/180.]]> 2023-10-04 15:27:05 UTC https://padlet.com/ronjboutte/opteiblcu1z2p6vr/wish/2732294470 Estella , hector , Ignacio https://padlet.com/ronjboutte/opteiblcu1z2p6vr/wish/2732294480 A complex number has two part it has the real number and the imaginary number an example would be z= a+ib a is the real part and ib is the imaginary part .]]> 2023-10-04 15:27:06 UTC https://padlet.com/ronjboutte/opteiblcu1z2p6vr/wish/2732294480 Marielena, Elias https://padlet.com/ronjboutte/opteiblcu1z2p6vr/wish/2732418553 Complex numbers are the numbers that are expressed in the form of a+ib where, a,b are real numbers and  ‘i’ is an imaginary number called “iota”. The value of i = (√-1). For example, 2+3i is a complex number, where 2 is a real number (Re) and 3i is an imaginary number (Im).]]> 2023-10-04 16:37:12 UTC https://padlet.com/ronjboutte/opteiblcu1z2p6vr/wish/2732418553 Mia Portillo and Gizzelle M https://padlet.com/ronjboutte/opteiblcu1z2p6vr/wish/2732421341 the geometric interpretation of multiplication of complex numbers is stretching (or squeezing) and rotation of vectors in the plane. ]]> 2023-10-04 16:39:02 UTC https://padlet.com/ronjboutte/opteiblcu1z2p6vr/wish/2732421341 Nekia, Marcus https://padlet.com/ronjboutte/opteiblcu1z2p6vr/wish/2732429535 Multiplication of the complex numbers multiplies the two magnitudes, resulting in √130, and adds the two angles, 142∘. ]]> 2023-10-04 16:42:55 UTC https://padlet.com/ronjboutte/opteiblcu1z2p6vr/wish/2732429535 Jayla E., Yulissa https://padlet.com/ronjboutte/opteiblcu1z2p6vr/wish/2732431203 The length of the line segment is called the modulus of the complex number and is denoted |z| . The angle measured from the positive real axis to the line segment is called the argument of the complex number, denoted arg(z) a r g ( z ) and often labelled θ.]]> 2023-10-04 16:43:59 UTC https://padlet.com/ronjboutte/opteiblcu1z2p6vr/wish/2732431203 Genesis, Erik https://padlet.com/ronjboutte/opteiblcu1z2p6vr/wish/2732434429 To convert from polar to rectangular, find the real component by multiplying the polar magnitude by the cosine of the angle, and the imaginary component by multiplying the polar magnitude by the sine of the angle.]]> 2023-10-04 16:45:52 UTC https://padlet.com/ronjboutte/opteiblcu1z2p6vr/wish/2732434429 ux natalie. janarius https://padlet.com/ronjboutte/opteiblcu1z2p6vr/wish/2732448594  a different way to represent a complex number apart from rectangular form]]> 2023-10-04 16:54:19 UTC https://padlet.com/ronjboutte/opteiblcu1z2p6vr/wish/2732448594 Leon, Maika, Martin https://padlet.com/ronjboutte/opteiblcu1z2p6vr/wish/2732762991 To change from radians to degrees, you need to multiply the number of radians by 180/π. This number will help you switch between the two units. For example, if you multiply π/2 radians by 180/π, you will get 90 degrees]]> 2023-10-04 20:54:17 UTC https://padlet.com/ronjboutte/opteiblcu1z2p6vr/wish/2732762991 Brennon, Oscar, Jayla https://padlet.com/ronjboutte/opteiblcu1z2p6vr/wish/2732765031 To change from radians to degrees, you need to multiply the number of radians by 180/π. This number will help you switch between the two units. (x)rad × 180/π = Degrees]]> 2023-10-04 20:56:54 UTC https://padlet.com/ronjboutte/opteiblcu1z2p6vr/wish/2732765031 Innocent Mugisha , roxsana salazar,emanuel escobedo https://padlet.com/ronjboutte/opteiblcu1z2p6vr/wish/2732765643  the real component by multiplying the polar magnitude by the cosine of the angle, and the imaginary component by multiplying the polar magnitude by the sine of the angle]]> 2023-10-04 20:57:40 UTC https://padlet.com/ronjboutte/opteiblcu1z2p6vr/wish/2732765643 Leon Maika Martin https://padlet.com/ronjboutte/opteiblcu1z2p6vr/wish/2732768851 2023-10-04 21:01:54 UTC https://padlet.com/ronjboutte/opteiblcu1z2p6vr/wish/2732768851 Kheniyah, Landon, Malik https://padlet.com/ronjboutte/opteiblcu1z2p6vr/wish/2732769355 To convert from polar form to rectangular form use given variables r and θ and use the transformation relations. 
x = r ⋅ cos(θ)
y = r ⋅ sin(θ) 
To convert from rectangular to polar form use:  
r^2 = x^2 + y^2 ---> R
tan^-1 (y/x) ---> θ
]]> 2023-10-04 21:02:35 UTC https://padlet.com/ronjboutte/opteiblcu1z2p6vr/wish/2732769355 Melissa, Hailey, Daniel https://padlet.com/ronjboutte/opteiblcu1z2p6vr/wish/2732772639 When multiplying complex numbers, its useful to remember that the properties we use when performing arithmetic with real numbers work similarly for complex numbers.]]> 2023-10-04 21:06:53 UTC https://padlet.com/ronjboutte/opteiblcu1z2p6vr/wish/2732772639 Nataly, Julian, Abril https://padlet.com/ronjboutte/opteiblcu1z2p6vr/wish/2732774463 Multiplication of  complex numbers multiplies  two quantities, resulting in √130, and adds  two angles, 142∘. In other words, you can look at the second number by scaling and rotating the first (or by scaling the first  and rotating the second).]]> 2023-10-04 21:09:13 UTC https://padlet.com/ronjboutte/opteiblcu1z2p6vr/wish/2732774463 Julian, Nataly, Abril https://padlet.com/ronjboutte/opteiblcu1z2p6vr/wish/2732775501 Multiplication of  complex numbers multiplies  two quantities, resulting in √130, and adds  two angles, 142∘. In other words, you can look at the second number by scaling and rotating the first (or by scaling the first  and rotating the second).]]> 2023-10-04 21:10:34 UTC https://padlet.com/ronjboutte/opteiblcu1z2p6vr/wish/2732775501 unit circle https://padlet.com/ronjboutte/opteiblcu1z2p6vr/wish/2732789078 2023-10-04 21:28:58 UTC https://padlet.com/ronjboutte/opteiblcu1z2p6vr/wish/2732789078 Jesus R, Hector P, Melanie P. https://padlet.com/ronjboutte/opteiblcu1z2p6vr/wish/2734280407 http://pirate.shu.edu/~wachsmut/complex/numbers/proofs/mult_geom.html]]> 2023-10-05 16:46:03 UTC https://padlet.com/ronjboutte/opteiblcu1z2p6vr/wish/2734280407 Alex and Ruben https://padlet.com/ronjboutte/opteiblcu1z2p6vr/wish/2734283218 To convert from degrees to radians, multiply the number of degrees by π/180. To find what it is in radians, you multiply 90 by π/180. This gives you π/2.]]> 2023-10-05 16:47:44 UTC https://padlet.com/ronjboutte/opteiblcu1z2p6vr/wish/2734283218 Derek, Mario, Eleny https://padlet.com/ronjboutte/opteiblcu1z2p6vr/wish/2734284149 Complex numbers are mathematical entities of the form (a + bi). The argument is the angle formed with the positive x-axis.]]> 2023-10-05 16:48:21 UTC https://padlet.com/ronjboutte/opteiblcu1z2p6vr/wish/2734284149 Morgan Campbell & Vxqaimia Lee https://padlet.com/ronjboutte/opteiblcu1z2p6vr/wish/2734289139 The length of the line segment is called the modulus of the complex number and is denoted |z| . The angle measured from the positive real axis to the line segment is called the argument of the complex number, denoted arg(z) a r g ( z ) and often labelled θ .]]> 2023-10-05 16:51:09 UTC https://padlet.com/ronjboutte/opteiblcu1z2p6vr/wish/2734289139 lsrael, Frank https://padlet.com/ronjboutte/opteiblcu1z2p6vr/wish/2734324269 If you have an imaginary number in the denominator of a fraction you have to use something called 'the conjugate'. 
In summary, the conjugate cancels the imaginary part of the denominator.
An example of when to use the conjugate is:

4+2i/3i

the first step is to find the conjugate:

_
3i = -3i

the second step is to multiply the bottom & the top by the conjugate.

(-3i)(4+2i)/(3i)(-3i)

the last step is to multiply & solve

2-4i/3]]> 2023-10-05 17:15:16 UTC https://padlet.com/ronjboutte/opteiblcu1z2p6vr/wish/2734324269 Jose G., Brayan M. https://padlet.com/ronjboutte/opteiblcu1z2p6vr/wish/2734560520 The argument of the complex number Z = a + ib is the angle θ which is the inverse of the tan function of the imaginary part divided by the real part of the complex number.]]> 2023-10-05 20:38:02 UTC https://padlet.com/ronjboutte/opteiblcu1z2p6vr/wish/2734560520 Arlette and Hannah https://padlet.com/ronjboutte/opteiblcu1z2p6vr/wish/2734564886 To convert from degrees to radians, multiply the number by degrees of π/180.]]> 2023-10-05 20:44:13 UTC https://padlet.com/ronjboutte/opteiblcu1z2p6vr/wish/2734564886 Santiago and Yair https://padlet.com/ronjboutte/opteiblcu1z2p6vr/wish/2734576271 To convert from degrees to radians, multiply the number of degrees by pi/180.To find what it is in radians, you multiply 90 by pi/180.This gives you pi/2]]> 2023-10-05 21:02:03 UTC https://padlet.com/ronjboutte/opteiblcu1z2p6vr/wish/2734576271 Ethan Gonzalez, Raul Fuentes https://padlet.com/ronjboutte/opteiblcu1z2p6vr/wish/2734576354 1 radian is the length of the radius wrapped around the circumference of the circle. 
The radius of a circle can wrap around the circle 6 times but with a decimal value remaining ~0.28 so radians are multiplied by pi to make up for it, therefore 2*(~3.14)=(~6.28)=360 degrees]]> 2023-10-05 21:02:10 UTC https://padlet.com/ronjboutte/opteiblcu1z2p6vr/wish/2734576354 complex numbers https://padlet.com/ronjboutte/opteiblcu1z2p6vr/wish/2748845125 this is the thumbnail of a video with good learning source]]> 2023-10-16 13:44:51 UTC https://padlet.com/ronjboutte/opteiblcu1z2p6vr/wish/2748845125 Degrees and Radians https://padlet.com/ronjboutte/opteiblcu1z2p6vr/wish/2748876800 A video that explains how this techniche works]]> 2023-10-16 14:00:57 UTC https://padlet.com/ronjboutte/opteiblcu1z2p6vr/wish/2748876800