# Formulas: More examples

# Significant figures

*"The significant figures of a number are digits that carry meaning contributing to its measurement resolution."* (Reference: https://en.wikipedia.org/wiki/Significant_figures).

The Formulas question has no built-in function to display numbers with a given number of significant figures. However, it is easy to work out this function as the following variable assignment which works with any real number:

xr=x==0?x:round(x*pow(10,nsf-1-floor(log10(abs(x)))),0)*pow(10,-nsf+1+floor(log10(abs(x))));

where x is the number to be rounded, nsf the number of significant digits to keep and xr the resulting rounded value.

This example deals with the display of the correct answers with different numbers of significant figures.

The Formulas question should look like this:

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This example is a significant figures drill.

General Question name! Significant figures drill Variables Random variables # a plus or minus sign # b exponent varying from -4 to 4 in steps of 0.01 # rnsf random number of significant figures : 1, 2, 3, or 4 a={-1,1}; b={-4:4.01:0.01}; rnsf={1,2,3,4}; Global variables # nx number x = ± 10^b nx=a*pow(10,b); # Rounding routine: # x number to round # nsf number of significant figures # xr rounded value of x x=nx; nsf=rnsf; xr=x==0?x:round(x*pow(10,nsf-1-floor(log10(abs(x)))),0)*pow(10,-nsf+1+floor(log10(abs(x)))); Main question Question text! Significant figures drill Part 1 Part's mark* 1 Answer type Number Answer* xr Grading criteria* Relative error < 0.0001 Part's text Number to round: {nx} Number of significant digits to keep: {ncs} Rounded number: {_0} Combined feedback For any correct response The correct answer is: {xr} For any incorrect response The correct answer is: {xr}

In the Variables fields (random, global and local), lines starting with # are treated as comments.

In this example, values to be rounded range from -10 000 to +10 000. In order to have the same number of values in each order of magnitude, hence a more interesting exercise, the random variable b is used as a power of 10. By letting:

^{b}

with b ranging from -4 to +4 in steps of 0.01, there are one hundred random values in each order of magnitude (0.000 1 to 0.001, 0.001 to 0.01, 0.01 to 0.1, 0.1 to 1, 1 to 10, 10 to 100, 100 to 1 000 and 1 000 to 10 000), both positive and negative.

**In total, this example generates 6 408 different random questions:**

The Formulas question should look like this:

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# Factorial

"In mathematics, the factorial of a non-negative integer , denoted by , is the product of all positive integers less than or equal to . For example,

The value of is , according to the convention for an empty product." (Reference: https://en.wikipedia.org/wiki/Factorial).

With the built-in function fact(), the Formulas question calculates factorial values up to (which would correspond to the largest number that can be handled by PHP). For larger values of , fact() returns INF for the value of .

However input (with {_0}, {_1}, etc.) is limited to a maximum value corresponding to .^{(Explain why)}

- Factorials can readily be evaluated up to 170! on Google built-in calculator, under 64-bit Windows and Android: 170!

This simple example illustrates the use of the fact() function.

General Question name! Factorial - example 1 Variables Random variables n = {2:8}; m = {2:8}; Global variables x = fact(n)*fact(m); Main question Question text! Factorial - example 1 Part 1 Part's mark* 1 Answer type Number Answer* x Grading criteria* Absolute error == 0 Part's text Calculate: {n}! x {m}! = {_0} (enter value using integer format) Combined feedback For any correct response The correct answer is: {x} For any incorrect response The correct answer is: {x}

The Formulas question should look like this:

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This slightly more elaborate example shows that the fact() function can handle large factorials and also that calculations with factorials can easily be performed using the fact() function.

General Question name! Factorial - example 2 Variables Random variables n = {0:171:1}; nn = {12:171:1}; na = {1:10}; m = {0:103:1}; Global variables x=fact(m); Main question Question text! Factorial - example 2 Part 1 Part's mark* 1 Answer type Number Answer* x Grading criteria* Relative error < 0.01 Part's text 1. Calculate: {m}! = {_0} (use scientific notation for large numbers) 2. It can easily be shown that {nn}! / {=nn-na}! = {=fact(nn)/fact(nn-na)}. Combined feedback For any correct response The correct answer is: {x} For any incorrect response The correct answer is: {x}

The Formulas question should look like this:

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