Scientific Notebook returns an exact, or symbolic, answer whenever appropriate and otherwise returns a numerical approximation. You can force a numerical result to any evaluation either by choosing Evaluate Numerically from the Maple menu or by starting with numbers in decimal notation. You obtain numerical approximations in response to any operation when you enter numbers with decimals because such a number is interpreted as a floating-point real number and not as a rational number.
Numerical analysis may be described as the study of errors introduced by using floating point arithmetic (round-off errors) and by using a finite number of terms when an infinite number of terms are required for exactness (truncation errors). Floating point is a data type that is machine dependent. It is important to understand that a floating point number is neither rational nor irrational; indeed, each floating-point number represents an infinite number of possible rational numbers and an infinite number of possible irrational numbers.
Computer algebra systems use what is called infinite precision or
extended precision to represent integers and rationals exactly.
Numbers such as sin 1,
, and π are examples of numbers that are represented exactly.
Numerical evaluation leads to the following approximations, with Digits Used in Computations and Digits Used in Display both set to
25 in Maple SettingsMaple settings.
sin 1 | = | . 841 470 984 807 896 506 652 502 3 | |
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= | 1. 414 213 562 373 095 048 801 689 | |
π | = | 1. 414 213 562 373 095 048 801 689 |
Notice that the approximations are broken into blocks of length 3 decimal digits in order to make them more readable. The numbers on the left are exact, while the numbers on the right are merely approximations. In particular,
Contrast the results of evaluating the following expressions with Evaluate and Evaluate Numerically.
Evaluate | Evaluate Numerically |
---|---|
82÷37 = ![]() |
82÷37 = 2.216216216 6pt |
936/14 = ![]() |
936/14 = 66.85714286 6pt |
936/14.0 = 66.85714286 | 936/14.0 = 66.85714286 6pt |
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You can change the number of digits displayed in these approximations by changing the setting in a dialog box. You can also change the number of digits used in computations and the threshold for scientific notation in that dialog box. These options are discussed in greater detail in the following section.