Rational Numbers on the TI-86 (TI-85)

Note that the TI-86 and the TI-85 have slightly different keys. This page uses the keys associated with the TI-86. The differences are in the "2^nd" functions on some of the keys used here. The TI-85 keys will have the same key-face symbol unless otherwise noted.

We need to pay attention to the use of rational numbers on the TI-86. For one thing, unlike many less expensive calculators, the TI-86 does not have an key to deal with fractions. This page presents the TI-86 approach to using rational numbers.

We start by assuming that you have turned on the calculator. (Press the key. If there is text on the screen, you can clear the screen by pressing the key.)

Figure 1 To generate the first screen, press the and keys. The next thing that we need to generate is the entity. This does not appear on the keyboard. We will need to open a menu to find it. Press the key to open the math menu.

Figure 2 Figure 2 shows the math menu at the bottom of the screen. As it turns out, the item that we want is not here. In fact, it is in the MISC sub-menu. We can select that sub-menu by pressing the key.
Figure 2a Figure 2a shows the MISC math sub-menu at the bottom of the screen. The choice that we want, , is still not on the screen so we will press the key to see more options. Note that the small arrow at the right end of the sub-menu indicates that there are more items to see.
Figure 2b Figure 2b shows that we want in the first position, above the F1 key. We can select that item by pressing the key.
Figure 3 This figure shows the result of our efforts. We are asking the calculator to change 0.5 into a fraction. We press the to signal that we are done and that the calculator should carry out the instruction.
Figure 4 Figure 4 shows the calculator's response.

Figure 5 We change Figure 4 into Figure 5 by using the keystrokes to produce the 0.3 and then the resulting answer 3/10. And, the keys to produce the 1/8 result. This is similar to the screen that appears in the text.

Figure 6 The text goes on to point out that the operation does not always work. For example, if we enter .333 the calculator responds with 333/1000, as seen in the middle of Figure 6. However, if we add one more digit so that we enter .3333 then the calculator refuses to convert to a fraction and instead displays the result as a decimal, as shown in the bottom of Figure 6.

Figure 7 Figure 7 demonstrates that we can fool the calculator. At the top of Figure 7 we have tried to convert 0.33333333333 to a fraction. The calculator refuses to do this and, instead, displays the value as a decimal. However, if we add one more digit, so that we have 12 three's, then the calculator incorrectly converts this to 1/3. Essentially, the calculator determines that it can not tell the difference between 0.333333333333 and 1/3. Therefore, the calculator has no problem making the conversion, even though it is wrong. The two values are close, in fact, very close, but they are not the same.
In preparation for Figure 8, we want to close the sub-menu and the menu at the bottom of the screen. We press the key to close the sub-menu, and we press the key again to close the menu.

Figure 8 Figure 8 explores some more of the concepts of rational numbers. In this figure we look at the decimal expansion of 5/11, 5/7, and 8/13. It is important to note a few things here. First, these are all repeating decimals. 5/11 is not equal to 0.454545454545 as is shown on the calculator. In order for us to have equality, we need to indicate that the decimal version continues the pattern of digits. Thus we can say 5/11=0.454545... That is a true equlaity. The calculator has a limited number of digits in its answers. It actually has more digits than it will display. It uses the extra digits to help keep calculations accurate and to be able to try to make great guesses at more answers.
The problem that we saw with 5/11 shows up again in 5/7 and in 8/13. The true equality for the former is
5/7=0.71428571428571428571428... but the calculator only displays the answer with 12 digits. Therefore, it rounds the answer off to 12 digits and shows us 0.714285714286, which is not at all the value of 5/7.
8/13 produces a similar, incorrect, rounded answer.

Figure 9 Figure 9 demonstrates another problem caused by the limited number of digits that the calculator uses. The change from Figure 8 to Figure 9 is the addition of a new problem, namely, 2/17. Back when we did 8/13 the calculator gave an approximation, but it was enough for us to correctly guess that the true answer is 8/13=0.61538461384615384... However, for the new problem, 2/17, the calculator produces an answer but that answer does not even start to show the complete set of repeating digits. It turns out that the decimal version of 2/17 repeats a pattern of 16 digits. Since the calculator shows only 12 digits, we can not see the entire pattern.

Figure 10 Figure 10 starts a sequence of 5 frames that demonstrate how we can "see" the extra digits that the calculator keeps. We start with 5/37, which the calculator displays as the decimal .135135135135. Then we multiply this by 100. It is importnat to note that we are not multiplying 0.135135135135 by 100, but rather that we are multiplying the previous answer by 100. We can do this in two ways. For the first method, we can recall the previous answer by using the keys . Then we use the keys . Notice that the result has exactly the same number of digits showing.
We can use the second method to do the next step, subtract 13 from the result. For this we press the key and the calculator immediately displays "Ans-". We follow that by , to produce the rest of Figure 10. Again, there are the same number of digits in the answer. Notice that at the right end of the number, the new digits 1 and 4 have appeared. They came from the extra digits that the calculator keeps internally.

Figure 11 In Figure 11 we will continue the process that we used in Figure 10, namely, multiply the old answer buy 100 to shift the decimal point two places to the right, and then subract the whole number part of the answer. Again, new values appear at the right end of the final answer in Figure 11, and those values are still part of the extra internal digits in the calculator.

Figure 12 In Figure 12, we apply the process yet another time, but this time we get different results. Multiplying by 100 accomplishes shifting the decimal point. However, when we subtract the 35 no new values are added to the right of our answer. We are now down to an 8 digit answer.

Figure 13 This new situation continues in Figure 13 where we drop down to a mere 6 digits in the answer.

Figure 14 And, for a last look, Figure 14 applies the technique one more time and we are down to a mere 4 digits.

We are going to make a small diversion here to take into account one of the differences between the TI-86 and the TI-83. The text is based on the TI-83. The diagrams in the text were taken from an 83. One difference that you may have noted in the Figures above is that the TI-86 displays 12 digits for its answers where the TI-83 only shows 10 digits. A second difference is the use of menus and even the appearance of menus. If you look at the corresponding pages for the TI-83 you will see that the menus show up as full screen menus. On the TI-86 the menus appear at the bottom of the screen. We have seen these menus and sub-menus as we located the symbol in the MISC sub-menu of the MATH menu. Any time that we want to use the operation we will need to go to that sub-menu. As long as the menu stays visible, the is immediately available. However, it is often the case that we need to be moving around through the menu system to do our work. The symbol is buried in the sub-menu. The following screens will allow us to "install" the symbol on our CUSTOM menu so that it is more available to us.

Figure 14a The first step in installing the symbol on our CUSTOM menu is to find that symbol in the CATALOG of the calculator. We open the CATALOG by pressing first the key, and then the key. This willopen the CATALOG menu as is shown in Figure 14a.

Figure 14b Now we want to see the items in the CATALOG. To do this we select the first selection in the menu, namely CATLG, by pressing the key. This will start a display of the items in the CATALOG of the calculator. Figure 14b shows such a list, along with the new menu for looking through the CATALOG. Note that the display need not start with the abs item. Your display may start somewhere else in the CATALOG.
The arrow to the left of "abs" indicates that "abs" is the currently selected item. We could use the cursor keys to move down, or up the list. We can also use the F1 key to move down a page at a time. Press the key to move to the display in Figure 14c.

Figure 14c Moving a page at a time is going to require many keystrokes to get to the end of the list, which is where we want to go. There should be a better way to move through the CATALOG and there is. We can press one of the keys with the alphabetic characters above and to the right of the key to move to that part of the list. To change Figure 14c to the display in Figure 14d, press the key to move to the "Z" portion of the CATALOG.

Figure 14d Figure 14d shows us the start of the "Z" portion of the CATALOG. We need to move further down the list to find the symbol. Therefore, we will need to press the key about 7 times to move to the display in Figure 14e.

Figure 14e In Figure 14e we have moved down the CATALOG to the point where we can see the symbol. However, that is not the symbol that is selected in Figure 14e. We will need to press the key 5 times to move the selection arrow on the screen so that it points to the symbol, as is shown in Figure 14f.

Figure 14f Now that we have found the symbol and have selected it, our next step is to put it into our CUSTOM menu. To do this we press the key to choose the CUSTM option.

Figure 14g In Figure 14g the CUSTOM menu has been opened as a sub-menu. We can place the symbol into any of the cells that we want, simply by pressing the function key below the desired cell. Let us press the key to insert the symbol into the first position. The result is shown in Figure 14h.

Figure 14h At this point we have installed the symbol into the first cell. We could move through the CATALOG and select other items to put into other cells, but we will not do that now. Rather, let us close the menus by pressing the key to close the CUSTOM menu and key to close the CATALOG. W

Figure 14i Now we can try out our new CUSTOM menu. We start by entering the value .0625 onto the screen.

Figure 14j Now we press the key to open the CUSTOM menu. Figure 14j shows that menu at the bottom of the screen. Now we can to select the symbol from our CUSTOM menu. The result is shown in Figure 14k.

Figure 14k Figure 14k shows the symbol having been inserted, and after pressing the calculator performs the operation as expected.

The previous Figures demonstrated the process for installing items from the CATALOG to the CUSTOM menu. We use this process to move the things that we want to use most often to our CUSTOM menu so that they are easy to use.

Figure 15 Entering a mixed number into the calculator is a pain. In Figure 15 we duplicate the steps from the text. Specifically, to enter 4¹/₂ we use the keys . Then we can select the option by pressing the key. That results in the input line which we terminate via to produce the result 9/2.
Notice that we run into a problem if we try to do the same thing for -4¹/₂. The calculator does exactly what we asked it to do, namely, add 1/2 to -4. The result is -3.5 or -7/2.
Be careful when entering -4! The appropriate key for the negative sign is the key not the key.

Figure 16 There are two methods for entering -4¹/₂. First, we can recognize that we are really talking about -4 minus 1/2, and we enter the value in that fashion. Second, we can express the value as the negative of the quantity 4 and 1/2. Figure 16 shows both methods.
The remaining Figures on this page assume that you have the "Quotient" program loaded onto your calculator. This can be done in a number of ways. First, you can transfer the program from a calculator that has it to yours. Second, you can transfer it from a computer that has

the TI Graph-Link program
cable for the TI-86
the source code for the program as a file on the computer. (quotient.85p is a link to the TI-85 version of the program; quotient.86p is a link to the TI-86 version of the program; quotient.89p is a link to the TI-89 version of the program; quotient.92p is a link to the TI-92 version of the program; quotient.9xp is a link to the TI-92+ version of the program; quotient.83p is a link to the TI-83 version of the program)

And third, you can enter the program yourself, assuming that you have a listing of the program. Here is a listing of the TI-86 version of the program.

Clearly the easiest way is to transfer the program between calculators.

The calculator that was used to prepare these figures has other programs in it. They are not needed and they do not affect the Quotient program.

The Quotient program was written to overcome the problem that we experienced in Figure 9. At that point we learned that the fraction 2/17 can not be displayed as a decimal value using the normal calculator functions. With 17 in the denominator, this fraction has a repeating sequence of digits that is 16 digits long. With the Quotient program we will be able to look at the decimal representation of fractions that have long sequences of repeating digits.

Figure 17 To run a program on the TI-86 we enter the PROGRAM menu via the key. This opens a menu that has two choices, NAMES and EDIT. We want the NAMES sub-menu. Therefore we press the key. The resulting display depends upon the programs that have been loaded into the calculator. Figure 17 identifies that there are many programs loaded into this particular calculator. The one we want is Quotient. It is not in Figure 17. The arrow at the right of the last cell at the bottom of Figure 17 indicates that there are more programs.

Figure 18 We can use the key as many times as we need to use it to bring the Quotient program into view. Notice, in Figure 18 that the Quotient program is displayed in the menu in a truncated form, "Quotie". The calculator does not have enough room for the entire name, so it truncates the name in the menu.

Figure 19 To obtain Figure 19 we select the Quotient program from the sub-menu by pressing the key, because "Quotie" is above the F3 key. This places the name of the program, Quotient, onto the screen. We can start the program by pressing the key.

Figure 20 At the start of the program, the screen clears and the user is prompted for a value for the numerator.

Figure 21 For this example, we will choose the numerator to be 53. We then press the key and the program asks for a value for the denominator.

Figure 22 We have chosen the denominator to be 37 so we are looking for the decimal representation of 53/37. Had we done this without the program the calculator would respond with 1.43243243243, which we recognize from the earlier Figures to be the calculator's approximation to the true value.

Figure 23 We move to Figure 23 by pressing the key. The program reproduces the problem on the top line and then computes each digit of the answer below. Since this is a repeating decimal, there is no nice place to stop. When the calculator runs out of space, as it has done on the bottom of Figure 23, the program asks if we want to get more of the answer, or do we want to stop? We can see that the pattern just keeps going, so we will press the key to stop working on this problem.

Figure 24 The program stops and returns to accept a new numerator and denominator. In Figure 24 those have been entered as 5 and 43, respectively. We might add that 5/43 on the calculator produces the approximate answer 0.116279069767.

Figure 25 Press to leave Figure 24 and generate Figure 25. Here we can see a more extended decimal representation of the value 5/43. We note that it repeats a pattern of digits, "116279069767441860465". Since we do not need to see more of the value, we again select to stop working on this problem.

Figure 26 Figure 26 starts yet another example. This time we have a denominator of 289.

Figure 27 The program gives the start of the decimal equivalent to 5895674527/289. In examining the decimal version we can not pick out a repeating pattern. Therefore, we can choose the key at the bottom to let the process continue.

Figure 28 Here we have move of the decimal expansion of our fraction. We still do not find the pattern repeating. Therefore, we can go on to look at more of the expansion.

Figure 29 More and more numbers. Can we find a repeating pattern? Yes. Starting on the second line of numbers, immediately below the "S" of CONTINUES in the top line, we find the sequence 87889273356... This is exactly the same as the digits immediately after the decimal point back in Figure 27. It took 272 digits before the pattern repeats.

Figure 30 So far we have seen how to start the program, Figures 17-19, how to enter values for the numerator and denominator, how to read the program output and to ask for additional screens of output, and how to stop the current problem and start over with a new numerator and denominator. At some point, however, we want to end the program.
We break out of the program by pressing the key. This will bring up a screen such as is shown in Figure 30. On this screen we have two choices, Quit and Goto. Quit will terminate the program, and for most of us, most of the time, this is the choice that we want. To perform the Quit command, press either the key.
DANGER: if you choose the Goto command, then calculator will place you into the edit mode for changing the program Quotient. This is fine if you know what you are doing. However, it is also a danger because any change that you make is automatically saved and there is no "undo" capability. In short, it is extremely easy to destroy a program once you are in edit mode. If you do destroy it, then the easiest fix is to reload the program from some other calculator.

Figure 1	To generate the first screen, press the and keys. The next thing that we need to generate is the entity. This does not appear on the keyboard. We will need to open a menu to find it. Press the key to open the math menu.
Figure 2	Figure 2 shows the math menu at the bottom of the screen. As it turns out, the item that we want is not here. In fact, it is in the MISC sub-menu. We can select that sub-menu by pressing the key.
Figure 2a	Figure 2a shows the MISC math sub-menu at the bottom of the screen. The choice that we want, , is still not on the screen so we will press the key to see more options. Note that the small arrow at the right end of the sub-menu indicates that there are more items to see.
Figure 2b	Figure 2b shows that we want in the first position, above the F1 key. We can select that item by pressing the key.
Figure 3	This figure shows the result of our efforts. We are asking the calculator to change 0.5 into a fraction. We press the to signal that we are done and that the calculator should carry out the instruction.
Figure 4	Figure 4 shows the calculator's response.
Figure 5	We change Figure 4 into Figure 5 by using the keystrokes to produce the 0.3 and then the resulting answer 3/10. And, the keys to produce the 1/8 result. This is similar to the screen that appears in the text.
Figure 6	The text goes on to point out that the operation does not always work. For example, if we enter .333 the calculator responds with 333/1000, as seen in the middle of Figure 6. However, if we add one more digit so that we enter .3333 then the calculator refuses to convert to a fraction and instead displays the result as a decimal, as shown in the bottom of Figure 6.
Figure 7	Figure 7 demonstrates that we can fool the calculator. At the top of Figure 7 we have tried to convert 0.33333333333 to a fraction. The calculator refuses to do this and, instead, displays the value as a decimal. However, if we add one more digit, so that we have 12 three's, then the calculator incorrectly converts this to 1/3. Essentially, the calculator determines that it can not tell the difference between 0.333333333333 and 1/3. Therefore, the calculator has no problem making the conversion, even though it is wrong. The two values are close, in fact, very close, but they are not the same. In preparation for Figure 8, we want to close the sub-menu and the menu at the bottom of the screen. We press the key to close the sub-menu, and we press the key again to close the menu.
Figure 8	Figure 8 explores some more of the concepts of rational numbers. In this figure we look at the decimal expansion of 5/11, 5/7, and 8/13. It is important to note a few things here. First, these are all repeating decimals. 5/11 is not equal to 0.454545454545 as is shown on the calculator. In order for us to have equality, we need to indicate that the decimal version continues the pattern of digits. Thus we can say 5/11=0.454545... That is a true equlaity. The calculator has a limited number of digits in its answers. It actually has more digits than it will display. It uses the extra digits to help keep calculations accurate and to be able to try to make great guesses at more answers. The problem that we saw with 5/11 shows up again in 5/7 and in 8/13. The true equality for the former is 5/7=0.71428571428571428571428... but the calculator only displays the answer with 12 digits. Therefore, it rounds the answer off to 12 digits and shows us 0.714285714286, which is not at all the value of 5/7. 8/13 produces a similar, incorrect, rounded answer.
Figure 9	Figure 9 demonstrates another problem caused by the limited number of digits that the calculator uses. The change from Figure 8 to Figure 9 is the addition of a new problem, namely, 2/17. Back when we did 8/13 the calculator gave an approximation, but it was enough for us to correctly guess that the true answer is 8/13=0.61538461384615384... However, for the new problem, 2/17, the calculator produces an answer but that answer does not even start to show the complete set of repeating digits. It turns out that the decimal version of 2/17 repeats a pattern of 16 digits. Since the calculator shows only 12 digits, we can not see the entire pattern.
Figure 10	Figure 10 starts a sequence of 5 frames that demonstrate how we can "see" the extra digits that the calculator keeps. We start with 5/37, which the calculator displays as the decimal .135135135135. Then we multiply this by 100. It is importnat to note that we are not multiplying 0.135135135135 by 100, but rather that we are multiplying the previous answer by 100. We can do this in two ways. For the first method, we can recall the previous answer by using the keys . Then we use the keys . Notice that the result has exactly the same number of digits showing. We can use the second method to do the next step, subtract 13 from the result. For this we press the key and the calculator immediately displays "Ans-". We follow that by , to produce the rest of Figure 10. Again, there are the same number of digits in the answer. Notice that at the right end of the number, the new digits 1 and 4 have appeared. They came from the extra digits that the calculator keeps internally.
Figure 11	In Figure 11 we will continue the process that we used in Figure 10, namely, multiply the old answer buy 100 to shift the decimal point two places to the right, and then subract the whole number part of the answer. Again, new values appear at the right end of the final answer in Figure 11, and those values are still part of the extra internal digits in the calculator.
Figure 12	In Figure 12, we apply the process yet another time, but this time we get different results. Multiplying by 100 accomplishes shifting the decimal point. However, when we subtract the 35 no new values are added to the right of our answer. We are now down to an 8 digit answer.
Figure 13	This new situation continues in Figure 13 where we drop down to a mere 6 digits in the answer.
Figure 14	And, for a last look, Figure 14 applies the technique one more time and we are down to a mere 4 digits.

Figure 14a	The first step in installing the symbol on our CUSTOM menu is to find that symbol in the CATALOG of the calculator. We open the CATALOG by pressing first the key, and then the key. This willopen the CATALOG menu as is shown in Figure 14a.
Figure 14b	Now we want to see the items in the CATALOG. To do this we select the first selection in the menu, namely CATLG, by pressing the key. This will start a display of the items in the CATALOG of the calculator. Figure 14b shows such a list, along with the new menu for looking through the CATALOG. Note that the display need not start with the abs item. Your display may start somewhere else in the CATALOG. The arrow to the left of "abs" indicates that "abs" is the currently selected item. We could use the cursor keys to move down, or up the list. We can also use the F1 key to move down a page at a time. Press the key to move to the display in Figure 14c.
Figure 14c	Moving a page at a time is going to require many keystrokes to get to the end of the list, which is where we want to go. There should be a better way to move through the CATALOG and there is. We can press one of the keys with the alphabetic characters above and to the right of the key to move to that part of the list. To change Figure 14c to the display in Figure 14d, press the key to move to the "Z" portion of the CATALOG.
Figure 14d	Figure 14d shows us the start of the "Z" portion of the CATALOG. We need to move further down the list to find the symbol. Therefore, we will need to press the key about 7 times to move to the display in Figure 14e.
Figure 14e	In Figure 14e we have moved down the CATALOG to the point where we can see the symbol. However, that is not the symbol that is selected in Figure 14e. We will need to press the key 5 times to move the selection arrow on the screen so that it points to the symbol, as is shown in Figure 14f.
Figure 14f	Now that we have found the symbol and have selected it, our next step is to put it into our CUSTOM menu. To do this we press the key to choose the CUSTM option.
Figure 14g	In Figure 14g the CUSTOM menu has been opened as a sub-menu. We can place the symbol into any of the cells that we want, simply by pressing the function key below the desired cell. Let us press the key to insert the symbol into the first position. The result is shown in Figure 14h.
Figure 14h	At this point we have installed the symbol into the first cell. We could move through the CATALOG and select other items to put into other cells, but we will not do that now. Rather, let us close the menus by pressing the key to close the CUSTOM menu and key to close the CATALOG. W
Figure 14i	Now we can try out our new CUSTOM menu. We start by entering the value .0625 onto the screen.
Figure 14j	Now we press the key to open the CUSTOM menu. Figure 14j shows that menu at the bottom of the screen. Now we can to select the symbol from our CUSTOM menu. The result is shown in Figure 14k.
Figure 14k	Figure 14k shows the symbol having been inserted, and after pressing the calculator performs the operation as expected.

Figure 17	To run a program on the TI-86 we enter the PROGRAM menu via the key. This opens a menu that has two choices, NAMES and EDIT. We want the NAMES sub-menu. Therefore we press the key. The resulting display depends upon the programs that have been loaded into the calculator. Figure 17 identifies that there are many programs loaded into this particular calculator. The one we want is Quotient. It is not in Figure 17. The arrow at the right of the last cell at the bottom of Figure 17 indicates that there are more programs.
Figure 18	We can use the key as many times as we need to use it to bring the Quotient program into view. Notice, in Figure 18 that the Quotient program is displayed in the menu in a truncated form, "Quotie". The calculator does not have enough room for the entire name, so it truncates the name in the menu.
Figure 19	To obtain Figure 19 we select the Quotient program from the sub-menu by pressing the key, because "Quotie" is above the F3 key. This places the name of the program, Quotient, onto the screen. We can start the program by pressing the key.
Figure 20	At the start of the program, the screen clears and the user is prompted for a value for the numerator.
Figure 21	For this example, we will choose the numerator to be 53. We then press the key and the program asks for a value for the denominator.
Figure 22	We have chosen the denominator to be 37 so we are looking for the decimal representation of 53/37. Had we done this without the program the calculator would respond with 1.43243243243, which we recognize from the earlier Figures to be the calculator's approximation to the true value.
Figure 23	We move to Figure 23 by pressing the key. The program reproduces the problem on the top line and then computes each digit of the answer below. Since this is a repeating decimal, there is no nice place to stop. When the calculator runs out of space, as it has done on the bottom of Figure 23, the program asks if we want to get more of the answer, or do we want to stop? We can see that the pattern just keeps going, so we will press the key to stop working on this problem.
Figure 24	The program stops and returns to accept a new numerator and denominator. In Figure 24 those have been entered as 5 and 43, respectively. We might add that 5/43 on the calculator produces the approximate answer 0.116279069767.
Figure 25	Press to leave Figure 24 and generate Figure 25. Here we can see a more extended decimal representation of the value 5/43. We note that it repeats a pattern of digits, "116279069767441860465". Since we do not need to see more of the value, we again select to stop working on this problem.
Figure 26	Figure 26 starts yet another example. This time we have a denominator of 289.
Figure 27	The program gives the start of the decimal equivalent to 5895674527/289. In examining the decimal version we can not pick out a repeating pattern. Therefore, we can choose the key at the bottom to let the process continue.
Figure 28	Here we have move of the decimal expansion of our fraction. We still do not find the pattern repeating. Therefore, we can go on to look at more of the expansion.
Figure 29	More and more numbers. Can we find a repeating pattern? Yes. Starting on the second line of numbers, immediately below the "S" of CONTINUES in the top line, we find the sequence 87889273356... This is exactly the same as the digits immediately after the decimal point back in Figure 27. It took 272 digits before the pattern repeats.
Figure 30	So far we have seen how to start the program, Figures 17-19, how to enter values for the numerator and denominator, how to read the program output and to ask for additional screens of output, and how to stop the current problem and start over with a new numerator and denominator. At some point, however, we want to end the program. We break out of the program by pressing the key. This will bring up a screen such as is shown in Figure 30. On this screen we have two choices, Quit and Goto. Quit will terminate the program, and for most of us, most of the time, this is the choice that we want. To perform the Quit command, press either the key. DANGER: if you choose the Goto command, then calculator will place you into the edit mode for changing the program Quotient. This is fine if you know what you are doing. However, it is also a danger because any change that you make is automatically saved and there is no "undo" capability. In short, it is extremely easy to destroy a program once you are in edit mode. If you do destroy it, then the easiest fix is to reload the program from some other calculator.