Calculator Tips and Tricks

Contents:

Appropriate Use
Precision and Accuracy
Memories
Fractions
Order of Operations
Prefix and Post-fix
Nesting Polynomials
Calculator Error

Return to Home Page

Appropriate Use

There will be occasions in this course when a calculator is essential. However, unnecessary calculations are a distracting waste of time and are detrimental to the Mathematics you are learning. Here are some examples of appropriate and inappropriate uses of calculators.

Appropriate Uses:

Inappropriate Uses:

Back to Top

Precision and Accuracy

One of the cardinal virtues of Mathematics is precision. We prefer exact answers so that the accuracy of any decimal approximation can be guaranteed. Mathematical notations like fractions and radicals preserve complete precision by avoiding approximation.

The displayed answer on a decimal calculator is not an exact number, except by accident. The calculator output might be extremely close to the exact answer but you should think of any decimal calculation as an approximation.

When you are calculating an approximation, you should always use the full accuracy of your calculator. Intermediate results should be stored in the calculator memory. This is more convenient than writing out 10 digits on paper and it also preserves the maximum accuracy of the calculation. That's because most calculators use two extra decimal places that are not displayed.

There is usually a FIX key that controls the number of decimal places in the display. Set the maximum number of decimal places (usually 7 to 10). After writing down the best possible approximation you can, round off the final answer to the required accuracy and give any units of measurement.

Back to Top

Memories

Using multiple memories is a lot like juggling. To avoid confusion you should use just one memory. If you're lucky, your calculator has only one memory. One key (usually Min) will store a number in the memory. Otherwise, STO 1 (two keys) will work on most scientific calculators. On other calculators, STO enter stores the number in the A memory.

(To recall a number from memory the keystroke(s) are RM  or  RCL 1  or  memvar enter  respectively.)

Back to Top

Fractions

You should not use a calculator to do fractional arithmetic. For one thing, you need the practice of combining numerical fractions to learn the rules of fractional algebra. For another, the answer is usually displayed as a mixed fraction, which must be converted to an improper fraction.

If you are building a house or measuring flour then mixed fractions can be useful. In Algebra, mixed fractions cause confusion. By convention, when two things are written next to each other the implied operation is multiplication. For example, is 6 1/3 equal to 19/3 or 2? The mixed fraction 6 1/3 really means 6 + 1/3 which simplifies to 19/3 = 18/3 + 1/3. All fractional answers should also be reduced to lowest terms.

Back to Top

Order of Operations

A strict order of operations is programmed into all calculators. If you do not completely understand these rules, the calculator can surprise you. The parentheses keys only make this more confusing. The safest way to deal with this is to use the = key early and often. In other words, force the calculator to do the calculation in pieces. That usually means working from inside to outside rather than from left to right. Use the calculator memory to store partial results.

Generally, you should do the most difficult piece of a calculation first. That's because you may want to use the memory to calculate that piece. If the top of a fraction is calculated first and stored in the memory, you are forced to calculate the bottom divided by the top. Then, the 1/x key will give you the top divided by the bottom. If you have calculated the second minus the first then the +/- key will give you the first minus the second. In both cases, make sure you press = before you use these tricks.

Back to Top

Prefix or Post-fix

Scientific functions on a calculator are programmed in two ways, prefix style or post-fix style. The natural log function ln is a good example.

In prefix style, to calculate ln 2 you would press the ln key first, then type 2, and finally press =.

In post-fix style, you would type 2 and then press the ln key.

Although post-fix style is more efficent for calculation, prefix style looks more like standard mathematical notation. If you happen to have a post-fix style calculator, be careful not to confuse how you write a function on paper (ln 2) with the keys that you press on the calculator (2 ln).

Back to Top

Nesting Polynomials

There is a useful technique for calculating polynomial values on a scientific calculator. Suppose you want to calculate 2x4 - x3 + 3x - 5 when x = 1/3. A little algebra can make this easy to do.

2x4 - 1 x3 + 3x - 5 =
2x4 - 1 x3 + 0x2 + 3x - 5 =
( 2x3 - 1 x2 + 0x + 3 )x - 5 =
(( 2x2 - 1 x + 0 )x + 3 )x - 5 =

((( 2x - 1)x + 0 )x +3 )x - 5

The last line is called the nested form of the polynomial. Note that the coefficients appear in order from left to right.

To do a calculation, first put a number into the memory. For example, enter 1/3 by pressing 1 divided by 3 equals. Store the result in the memory. Then you can calculate the answer using the nested form:

2 X RCL - 1 = X RCL + 0 = X RCL + 3 = X RCL - 5 =

where RCL means recall the memory. (Be certain to press = after each addition or subtraction.) This is the same as synthetic division done on a calculator. The approximate answer is -4.012345679 to 9 decimal places.

Back to Top

Calculator Error

There are some calculations that most calculators and even some computer algebra systems cannot handle. For example, try finding the third root of -8. The answer we expect is -2 but many calculators will show an error. Some calculators may even give a complex number as the answer. The easiest way to get the real number answer is to calculate using positive numbers and determine the sign of the answer yourself.

Back to Top

Last modified January 10, 2009