# TI-BASIC:Re Thetai

**Command Summary**

Puts the calculator into re^θ*i* mode.

**Command Syntax**

re^θ*i*

**Menu Location**

Press:

- MODE to access the mode menu.
- Use the arrow keys and ENTER to select re^θ
*i*

TI-83/84/+/SE

1 byte

The re^θ*i* command puts the calculator into polar complex number mode. This means that:

- Taking square roots of negative numbers, and similar operations, no longer returns an error.
- Complex results are displayed in the form re^(θ
*i*) (hence the name of the command)

The mathematical underpinning of this complex number format is due to the fact that if (x,y) is a point in the plane using the normal coordinates, it can also be represented using coordinates (r,θ) where r is the distance from the origin and θ is the angle that the line segment to the point from the origin makes to the positive x-axis (see Polar and PolarGC for more information on polar coordinates and graphing). What does this have to do with complex numbers? Simple: if x+y*i* is a complex number in normal (rectangular) form, and re^(θ*i*) is the same number in polar form, then (x,y) and (r,θ) represent the same point in the plane.

Of course, that has a lot to do with how you define imaginary exponents, which isn't that obvious.

An equivalent form to polar form is the form r[cos(θ)+*i*sin(θ)].

Unfortunately, the calculator seems to have some confusion about the use of degree and radian angle measures for θ in this mode (the answer is: you can only use radians -- degrees make no sense with complex exponents). When calculating a value re^(θ*i*) by using the e^( command and plugging in numbers, the calculator assumes θ is a radian angle, whether it's in Degree or Radian mode. However, when *displaying* a complex number as re^(θ*i*), the calculator will display θ in radian or degree measure, whichever is enabled. This may lead to such pathological output as:

Degree:re^θi Done e^(πi) 1e^(180i) Ans=e^(180i) 0 (false)

It's recommended, then, to use Radian mode whenever you're in re^θ*i* mode.