Choose the imaginary unit i to satisfy i2=-1. The two square roots of -1 are i and -i. A complex num

Question

Choose the imaginary unit i to satisfy i2=-1. The two square roots of -1 are i and -i. A complex number in Cartesian form is z=x+yi, where x,y in R. Re(z)=x, Im(z)=y. The powers repeat every four steps: i0=1, i1=i, i2=-1, i3=-i, and in+4=in. For example, i58=i2=-1. Algebra **Operation** | **Result for z₁=x₁+y₁i and z₂=x₂+y₂i** Equality | z_1=z_2 x_1=x_2 and y_1=y_2 Addition | z_1+z_2=(x_1+x_2)+(y_1+y_2)i Subtraction | z_1-z_2=(x_1-x_2)+(y_1-y_2)i Multiplication | z_1z_2=(x_1x_2-y_1y_2)+(x_1y_2+x_2y_1)i Real scalar | r(x+yi)=rx+ryi, r in R Addition and multiplication obey the commutative, associative and distributive laws, as they do for real numbers. A number is real when its imaginary part is zero; it is purely imaginary when its real part is zero.

Accepted Answer

$\mathbb N\subset\mathbb Z\subset\mathbb Q\subset\mathbb R\subset\mathbb C,\quad z=a+b\mathbf i,\ a,b\in\mathbb R,\ \mathbf i^2=-1$

Operation	Result for z₁=x₁+y₁i and z₂=x₂+y₂i
Equality	\(z_1=z_2\iff x_1=x_2\) and \(y_1=y_2\)
Addition	\(z_1+z_2=(x_1+x_2)+(y_1+y_2)i\)
Subtraction	\(z_1-z_2=(x_1-x_2)+(y_1-y_2)i\)
Multiplication	\(z_1z_2=(x_1x_2-y_1y_2)+(x_1y_2+x_2y_1)i\)
Real scalar	\(r(x+yi)=rx+ryi\), \(r\in\mathbb R\)

Cartesian Complex Numbers and Algebra

Cartesian Complex Numbers and Algebra

Video Solution:

Solution:

Solution locked