Module 2 - Computer Algebra

Introduction | Lesson 1 | Lesson 2 | Lesson 3 | Self-Test

Lesson 2.1: Numerical Calculations

In this lesson you will use the TI-89 computer algebra system to perform exact and approximate calculations.

• Turn your calculator on and go to the Home screen by pressing
• Clear the History Area and Edit Line by pressing

Mode Settings

Modes and their settings control how numbers, commands, and graphs are displayed and interpreted. The Mode dialog box is the place where you can change the mode settings.

• Display the Mode dialog box by pressing

You can go to different pages within the dialog box by pressing , , and , and you can scroll up and down the pages with the cursor movement keys, and .

The Exact/Approx Mode Setting

The Exact/Approx setting determines the way calculations are carried out and how the results of those calculations are displayed. By default, the Exact/Approx setting is AUTO, but you can select two other settings: EXACT or APPROXIMATE. All three settings are discussed in this lesson.

The EXACT Mode

In EXACT mode, all values entered as part of an expression are considered to be exact, all computations are done using exact methods, and all results are displayed in exact form (using fractions, square roots, etc.).

To select the EXACT mode,

• Go to page 2 of the Mode dialog box by pressing
• Highlight Exact/Approx by using the down cursor movement key,
• Display the Exact/Approx menu by pressing
• Highlight EXACT by pressing

• Select EXACT and exit this menu by pressing
• Save the mode settings by pressing again
 You must press to save the Mode settings and to exit the Mode dialog box. If you use some other key, such as or , the changes you made in the Mode settings will not be saved.

You should be on the Home screen. Notice one of the current Mode settings shown on the Status Line under the Edit Line is EXACT.

Now you are ready to perform exact calculations. If you make mistakes while entering the following calculations, you can erase with the key or clear the entire Edit Line and start over with .

Exact Calculations of Fractions

• Find the exact value of by pressing

After you press , the expression being evaluated appears on the left side of the History Area and the exact result appears on the right side. Any fractions in the result are written in reduced form.

Treatment of Decimal Numbers in Exact Mode

• Find the value of 0.123 + 0.5 in Exact Mode.

Notice that the TI-89 treats the decimal fractions 0.123 and 0.5 as exact.

Exact Calculations Involving Square Roots

The next calculation involves the square root function, which is a second function above the multiplication key in the right column of keys.

• Find the exact value of by pressing

The TI-89 returns by rationalizing the denominator. The calculator will also reduce a fraction that is the result of rationalizing the denominator.

In the next calculation, you are going to find the value of a trig function. Your calculator should say "RAD" in the Status Line, indicating that you are in radian mode. Radian mode is one of the factory default settings selected when you reset to the factory setting in Lesson 1.1. If your calculator is not in radian mode, you can change the Angle setting to Radian on page 1 of the Mode dialog box.

Exact Calculations with Trig Functions

• Find the exact value of by pressing .

The exact value of is .

The APPROXIMATE Mode

To see the difference between EXACT and APPROXIMATE modes, set your calculator to APPROXIMATE mode and then repeat the above calculations.

To change to APPROXIMATE mode,

• Display the Exact/Approx menu by pressing then
• Select APPROXIMATE by pressing
• Save the new settings by pressing
• Your calculator should say APPROX in the Status Line.

Repeat the three calculations you found in Exact mode: , , and . The results are now decimal approximations.

 Display Digits Factory Setting The default setting for "Display Digits" is FLOAT 6, not FLOAT. If you reset your calculator to its default settings in Lesson 1.1, your calculator will display a maximum of six significant digits for any number in the result.

The AUTO Mode

AUTO is the other setting available in the Exact/Approx menu of the Mode dialog box. AUTO uses APPROXIMATE mode whenever the expression to be evaluated contains a decimal point. Otherwise, AUTO uses EXACT mode whenever possible. Certain higher order functions (e.g. "integrate" and "solve") may return APPROXIMATE results even if your entry does not contain a decimal point, but only when an EXACT answer cannot be found.

Calculating an Exact Value in AUTO Mode

To find in AUTO mode,

• Clear the Home screen and Edit Line by pressing
• Change to AUTO mode by pressing
• Calculate by pressing

The TI-89 returns , an exact result with a simplified radical, and AUTO is shown in the Status Line.

Calculating an Approximate Value in AUTO Mode

To calculate an approximate value of ,

• Enter by pressing

The result is approximate because the entered expression contained a decimal point.

Viewing Large Results

Compute 2100 by pressing .

The TI-89 computer algebra system computes the exact answer, but the result is too big to fit on the screen, as indicated by the small arrow at the right of the result.

To see the part of the result that is hidden,

• Move up into the history area by pressing
• Scroll to the right by pressing

Now the first part of result is hidden, as indicated by the small arrow shown at the left of the result.

• Move back down to the Edit Line by pressing

Approximate Results Using

Another way to obtain an approximate result is to press at the end of a calculation instead of . Notice the approximate symbol in green above the key.

You should still have the expression 2^100 in the Edit Line.

• Find an approximate value for 2100 by pressing .

The approximate result is given in scientific notation.

also works in both AUTO and EXACT mode.

Calculating log x

The last calculation in this lesson involves logarithms. The natural logarithm (ln) is a second function above the x-variable key, which is in the left column of keys. There is no dedicated key for the common logarithm (log), but it can be found in the catalog, which is an alphabetical listing of all the commands and functions available. Pressing accesses the list. You could also type in log using the keyboard.

To find the value of log(25),

• Go to the catalog entries that begin with the letter "L" by pressing . Notice that there is an "L" above the key but that you don't need to press before pressing .
• Move the cursor until it points to log( by repeatedly pressing

• Paste the log( function to the Edit Line by pressing
• Finish entering the expression by pressing

The result, which is exact, is expressed in terms of natural logarithms.

• Press to see a decimal approximation