Complex Numbers on the TI-85 and TI-86

Neither the TI-85 nor the TI-86 have the imaginary number i on its keyboard. Rather, these two calculators use a different notation to represent a complex number. In particular, the TI-85 and the TI-86 represent the complex number a + bi as (a,b). Thus, 8 + 2i would be represented as (8,2). Recognizing this convention, the following screen images demonstrate various operations on complex numbers on the TI-85 or TI-86.

Figure 1	Figure 1 starts with an example of a complex number, namely 4+7i, but which we need to represent as (4,7). We generate that value via the keys . Then, when we press the calculator accepts our value and redisplays it on the right side of the screen. Figure 1 continues with the problem (8 + 2i) + (9 – 6i) which we represnt via (8,2) + (9,^–6) The calculator computes this to be (17, ^–4) which we could rewite as 17 – 4i. We follow that by the subtraction problem, (8 + 2i) – (9 – 6i) which we represnt via (8,2)–(9,^–6) The calculator computes this answer to be (^–1,8) which we could rewite as ^–1 + 8i.
Figure 2	Now we will examine the multiplication and division of complex numbers. In Figure 2 we can see that (8,2)(9,^–6) is evaluated to be (84,^–30) which is the value that we should expect. However, (8,2)/(9,^–6) is evaluated as where the three dots at the right end of the screen indicate that there is more to the answer. We use the key to shift the display to the right.
Figure 3	Now we can see the rest of the answer, namely, If we could put the two pieces together we would have This still looks pretty strange. To do the problem on paper we would multiply numerator and denominator by the complex conjugate of the denominator, in this case by (9,6) The result would produce a denominator that is 117. The numerator would be (60,66). Thus, the entire answer is (60,66)/117, or (60/117 , 66/117), which we can reduce to (20/39 , 22/39). And, we note that 20/39 is approximately 0.512820512821 and 22/39 is approximately 0.564102564103. It seems that we have the correct answer, but we would prefer that the calculator provide it in fractional form.
Figure 4	Figure 4 restates the division problem but with the symbol appended to the end of the command. The result is exactly as we had determined above.

The problems presented so far have involved complex numbers of the form (a,b) where a and b have been rational numbers. However, the definition of complex numbers only requires that a and b be real numbers. Thus, we could have a problem such as

We can do this problem by hand and obtain the complex number

The TI-85 and TI-86 do not have the power to simplify the original problem into the answer that we computed. In particular, the TI-85 and TI-86 do not know how to multiply such irrationals and to leave the result in radical form. Instead, the TI-85 and TI-86 will compute the its best approximation to the radicals and then use those approximations to do the problem. However, we can use these calculators to verify our algebraic work. We can enter the original problem and allow the calculator to compute its approximate answer. Then we can enter our algebraic answer and let the calculator compute its approximate value. The two results should be essentially identical.

Figure 5 In Figure 5 we start by demonstrating the squaring a complex number. This has nothing to do with the discussion above, but it does provide a small example.
Then, Figure 5 shows how we enter the original problem and follow that by our computed answer. As we can see, in both cases the calculator evaluates the expressions to the same approximations.