Complex Numbers on the TI-83

The TI-83 has the imaginary number i on its keyboard. We use the key sequence

to generate i. The following screen images demonstrate various operations on complex numbers on the TI-83.

Figure 1 Figure 1 starts with an example of a complex number, namely 4+7i. 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 the calculator computes to be 17 – 4i We follow that by the subtraction problem, (8 + 2i) – (9 – 6i) which the calculator computes to be ^–1 + 8i Note that the parentheses were not needed in the addition problem. However, at least the second set of parentheses are required in the subtraction. If we did not include them then the problem would have been evaluated as 8 + 2i – 9 – 6i 8 + 2i + ^–9 + ^–6i ^–1 – 4i

Figure 2 Now we will examine the multiplication and division of complex numbers. Again, for these problems the parentheses are essential. In Figure 2 we can see that (8 + 2i)(9 – 6i) is evaluated to be 84 – 30i which is the value that we should expect. However, (8 + 2i) / (9 – 6i) 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 + 6i) The result would produce a denominator that is 9²–6i², or 81+36, or 117. The numerator would be 8(9)+8(6i)+(2i)9+2i(6i) which we simplify first to 72+48i+18i–12, and then to 60+66i. Thus, the entire answer is (60+66i)/117, or 60/117 + 66i/117, which we can reduce to 20/39 + 22i/39. And, we note that 20/39 is approximately 0.5128205128 and 22/39 is approximately 0.5641025641. 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.
NOTE, however, that we need to interpret the result carefully. It is given as
but we need to recognize that this is meant as even though the calculator does not supply the extra parentheses. In particular, the result has a fraction times i, which really means, in this case 22i/39.

Figure 5 One of the problems related to the calculator implementation of complex numbers is the fact that we really do need to add the parentheses in these problems. If we forget them, the calculator will produce an answer, but it will be the answer to a different problem. If we repeat out previous problem, but this time remove the parentheses, then the calculator responds as in Figure 5. The calculator is really performing the problem 8 + (2i/9) – 6i which is 8 – (52/9)i.

Figure 6 In Figure 6 we demonstrate squaring a complex number.

The problems presented so far have involved complex numbers of the form a+bi 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-83 does not have the power to simplify the original problem into the answer that we computed. In particular, the TI-83 does not know how to multiply such irrationals and to leave the result in radical form. Instead, the TI-83 will compute the its best approximation to the radicals and then use those approximations to do the problem. However, we can use the TI-83 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 7 In Figure 7 we enter the original problem as which evaluates as . Then we enter our computed answer, namely, , which is then evaluated to the same .

Figure 1	Figure 1 starts with an example of a complex number, namely 4+7i. 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 the calculator computes to be 17 – 4i We follow that by the subtraction problem, (8 + 2i) – (9 – 6i) which the calculator computes to be ^–1 + 8i Note that the parentheses were not needed in the addition problem. However, at least the second set of parentheses are required in the subtraction. If we did not include them then the problem would have been evaluated as 8 + 2i – 9 – 6i 8 + 2i + ^–9 + ^–6i ^–1 – 4i
Figure 2	Now we will examine the multiplication and division of complex numbers. Again, for these problems the parentheses are essential. In Figure 2 we can see that (8 + 2i)(9 – 6i) is evaluated to be 84 – 30i which is the value that we should expect. However, (8 + 2i) / (9 – 6i) 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 + 6i) The result would produce a denominator that is 9²–6i², or 81+36, or 117. The numerator would be 8(9)+8(6i)+(2i)9+2i(6i) which we simplify first to 72+48i+18i–12, and then to 60+66i. Thus, the entire answer is (60+66i)/117, or 60/117 + 66i/117, which we can reduce to 20/39 + 22i/39. And, we note that 20/39 is approximately 0.5128205128 and 22/39 is approximately 0.5641025641. 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. NOTE, however, that we need to interpret the result carefully. It is given as but we need to recognize that this is meant as even though the calculator does not supply the extra parentheses. In particular, the result has a fraction times i, which really means, in this case 22i/39.
Figure 5	One of the problems related to the calculator implementation of complex numbers is the fact that we really do need to add the parentheses in these problems. If we forget them, the calculator will produce an answer, but it will be the answer to a different problem. If we repeat out previous problem, but this time remove the parentheses, then the calculator responds as in Figure 5. The calculator is really performing the problem 8 + (2i/9) – 6i which is 8 – (52/9)i.
Figure 6	In Figure 6 we demonstrate squaring a complex number.