Working with Numbers & Accuracies
In Math Touch, all numbers can be complex (if the equation can accept a complex value) and significant figures are tracked. For more precision, you can specify an uncertainty for a number and that value will be propagated through the underlying math functions. Using complex numbers and uncertainties will cause approximately a 4-8 x performance decrease.
Format of Numbers
Format of Numbers
You can choose the notation you prefer for numbers by tapping the Info button, then "Preferences" then "Number Notation."
Math Touch can notate numbers as you please:
Standard "Scientific" notation, in which all numbers are printed with a non-zero leading digit, followed by a decimal point and a subsequent exponent on a power of 10 (if this exponent is not 0). You can use the Symbolic Keyboard to type "x10" and superscript-sized exponents. Alternatively, you can use the computer programmer's standard "e" notation in place of "x10". Please note that while non-integer exponents may be understood by Envelop, they may cause the incorrect number of significant figures. At no time, will scientific mode produce a non-integer exponent.
Finally, when a value is infinite to machine precision, the special infinity character will appear to indicate infinity. Negative infinity is also possible. You can type infinity explicitly by using the symbolic keyboard.
Since exponents receive special names in multiples of 3, another kind of standard notation is "Engineering" in which the scientific notation is modified such that the exponent is always a multiple of 3. Engineering notation is the default preference.
No matter the notation you choose for display, Math Touch will understand all three when you type them in. For instance, if you choose "Engineering" notation, but type in "0.0014", Math Touch will re-format your entry as "1.4x10^-3."
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While significant figures are not adequate for high-precision engineering, they are certainly useful, especially when contrasted to the way most computers print numbers, with no regard for the actual precision. Math Touch automatically keeps track of significant figures for all math operations. Note that because Math Touch uses the standard "double" floating-point for its arithmetic, the maximum number of possible significant figures is 16.
As it happens, certain numbers do not have precision associated with them, and are called "pure" numbers. Frequently, these numbers appear as constants in an equation. Some examples include "2" "π" or "e". In Math Touch, while numbers like π and e cannot be represented to infinite precision, they are defined to machine precision, and are marked as "pure" in the included database.
When a decimal is present in a number, the number of significant figures is unambiguous, however, for integers whose trailing digits are all zero, the significant figure is sometimes impossible to distinguish. To the last significant zero, the special Symbolic Keyboard has a key that produces an overbar, a common way to specify the last zero digit. To use this, after typing the last significant zero (or position your cursor after the last significant zero) and press the over bar key on the symbolic keyboard. An overbar will appear over the zero you specify. You can continue typing more zeros. Overbars are not necessary over non-zero characters, and if the last significant zero is in the ones place, it is customary to include the decimal. Note that overbars are never necessary in scientific notation, and seldom necessary in engineering notation.
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For more precision, you can specify the uncertainty of a value explicitly. Both ± and parenthetical notations are accepted, and there is a preference in the Preferences area to choose the one you prefer. Please note that because parnthetical notation replaces the last digits of a number, it is inapprorpiate in some cases and ± will be used instead The uncertainty propagation in Math Touch is linear, so there may be errors caused by the covariance of two numbers. Significant figures are also tracked simultaneously.
± uncertainty is an unambiguous way to indicate the value written is not entirely certain. When using the ± notation, you should use the Symbolic Keyboard to type the ± character after a number, followed by a new number indicating the uncertainty in the result. For instance, "1.234 ± 0.012" has a value of "1.234" and an uncertainty of "0.012." Math Touch knows how to ignore extraneous spaces. The caveat is that you must fully specify the order of the uncertainty using a decimal or exponential notation, or Math Touch will interpret your uncertainty as an integer. For instance, "1.234 ± 12" would represent a value ranging from -10.766 to 12.234.
Parenthetical uncertainty notation is a very compact way to indicate the precision of a value. After the number, parenthesis surround a sequence of numbers indicating the uncertainty in the value. For instance, "1.234(12)" represents the same value and uncertainty as above (1.234 ± 0.012), but in a more compact way. To find the order of the uncertainty, the numbers inside the parenthesis essentially "replace" the last digits in the value to determine their value.
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Math Touch can always accept a complex number. You specify a complex number by adding two numbers in the text area, and placing an "i" next to the imaginary component, which is written second. For example, "1.234 + 0.345 i" would mean a complex number whose real part is 1.234 and whose imaginary part of 0.3345.
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