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We'll now show you how easy and fun it is to use Xzfracal. If you are using Xzfracal for handheld devices, then the CALC key is at the lower-right corner of the home keyboard. It is the key with the equal “=” sign on it.
D1. Calculate \( 3 + 4.1\overline{3} - \dfrac{2}{5} + 6\dfrac{5}{21} \) .
Xzfracal uses the special character “&” to mark the beginning of the repeating decimal group. The repeating decimal \( 4.1\overline{3} = 4.13333\ldots \) is represented by 4.1&3 . Xzfracal uses the special character “:” to delimit the whole (integer) part, numerator and the denominator of a fraction. The fractions \( \dfrac{2}{5} \) and \( 6\dfrac{5}{21} \) are represented by 2:5 and 6:5:21 respectively. The given expression is represented by the following expression in Xzfracal:
3 + 4.1&3 - 2:5 + 6:5:21The spaces are optional, they enhance readability. We'll include them whenever it is appropriate. If you don't feel it worth the extra effort to type extra spaces on a handheld device; please do feel free to skip the spaces in your input.
Go ahead, type the above expression without the spaces. Xzfracal constantly monitor all the keystrokes. When the input contains syntax error, the background of the input area is red, signaling that something is wrong. When the input is free of syntax error, the background of the input area is green, signaling that Xzfracal is ready to evaluate the expression. Please fix any typo in the input expression. If there is no mistake, the background of the input area should be green. Press CALC now.
Xzfracal, by default, prints the answer in multiple formats. You can choose the formats in the preference menu. Here are the answer in various formats (Approx n means rounding to n decimal places):
Fraction | 454/35 | |
Mixed Fraction | 12:34:35 | |
Exact Decimal | 12.9&741285 | |
Approx 2 | 12.97 | |
Approx 5 | 12.97413 | |
Approx 15 | 12.974128574128574 |
The first three answers are exact. The rest are approximations. Xzfracal uses the common "/" notation in fraction format to produce a "friendly" output. It uses the special “:”notation in the mixed fraction format.
If you did't see all these formats at the same time, please explore the preference menu on your own. Here are the exact answers in familiar formats: \( \dfrac{454}{35} \), \( 12\dfrac{34}{35} \), \( 12.9\overline{741285} \) . Here are the approximate answers in familiar formats: \( 12.97 \), \( 12.97413 \), \( 12.974128574128574 \) .
You may notice the underscore character “_” in Xzfracal's answer. The special character “_”is a number spacer. It can be inserted inside a number to enhance the readability of a number.
Depending on the preference configuration, Xzfracal may, for example, display decimals in groups of 5 digits. Thus the last answer may appear as 12.97412_85741_28574 . Number spacer is very flexible. You can, for example, write 12.97412857412857474128574 = 12.9_741285_74128574_74128574 to emphasis the repeating decimals.
Upon evaluation, the result may be valid or invalid. Inavlid result happens whenever some error (such as division by zero) happens somewhere during the calculation. Xzfracal saves all successful (valid) calculations onto a stack. If this is a new session, then the result will be labelled as $$1 . The result of next successful calculations will be labelled as $$2, $$3, ... and so. Theses results can be referenced again in later calculation such as: 5 * $$4 + 3.7 .
D2. Calculate \( 3 + 4.5 \div \left( 6\dfrac{2}{5} \right) + \left( 4\dfrac{2}{3} \right)^3 - 23.7\overline{43} \times 3 - {4.\overline{7}}^2 \) .
Enter the following and press CALC (all spaces are optional):
3 + 4.5 / 6:2:5 + 4:2:3 ^ 3 - 23.7&43 * 3 - 4.&7 ^ 2Here are the answer in various formats (Approx n means rounding to n decimal places):
Fraction | 3214811/285120 | |
Mixed fraction | 11:78491:285120 | |
Decimal | 11.275291&105499438832772166 | |
Approx 4 | 11.2753 | |
Approx 10 | 11.2752911055 |
Please explore the preference menu to see the various formats. The exact decimal has a repeating group of 18 digits. The same answer in friendlier formats are: \( \dfrac{3214811}{285120} \) \( = \) \( 11\dfrac{78491}{285120} \) \( = \) \( 11.275291\overline{105499438832772166} \) .
IMPORTANT: It is not necessary to protect the numbers 6:2:5, 4:2:3, 23.7&43 and 4.&7 with pairs of matching parentheses in the input expression! Xzfracal understands the meaning of the characters “&”and ":"; it parses the numbers accordingly. Xzfracal treats 6:2:5 , for example, as a single unbreakable entity.
The power in the exponentiation operator "^" must be an integer. Fractional powers are illegal, the result will be an invalid number: INVALID. Any computation involving INVALID will also become INVALID.
D3. Calculate \( \left( \left( 3\dfrac{2}{3} - 4\dfrac{3}{5} \right)^3 + 3.2\overline{3} \right) \times \left( 7 \div 1\dfrac{3}{5} + 2\dfrac{1}{4} - 2\dfrac{3}{8} \right) \) .
Enter the following and press CALC :
( (3:2:3 - 4:3:5)^3 + 3.2&3 ) * ( 7 / 1:3:5 + 2:1:4 - 2:3:8 )The answer is 277729/27000 = 10:7729:27000 = 10.286&259 , or: \( \dfrac{277729}{27000} \) \( = \) \( 10\dfrac{7729}{27000} \) \( = \) \( 10.286\overline{259} \)
D4. Reduce the fraction \( \dfrac{36}{45} \) to the lowest term.
Xzfracal automatically reduces all fractions to their lowest terms. Enter 36:45 and press CALC to obtain the answer 4/5 .
D5. Convert the fraction \( \dfrac{12}{21} \) to a decimal number.
Enter 12:21 and press CALC to obtain the result: 4/7 = 0.&571428 . Hence the exact decimal is \( 0.\overline{571428} \) \( = \) \( 0.571428571428571428\ldots \), which allows us to write down the following sequence of increasingly more accurate approximated answers:
0.60.571
0.5714
0.57143
0.571429
0.5714286
0.57142857
0.571428571
0.5714285714
...
D6. Convert the decimal number \( 12.45\overline{7824} \) to a fraction.
Enter 12.45&7824 and press CALC to obtain the result: 4152193/333300 = 12:152593:333300 . Hence the answer is: \( 12.45\overline{7824} \) \( = \) \( \dfrac{4152193}{333300} \) \( = \) \( 12\dfrac{152593}{333300} \). .
D7. Convert the improper fraction \( \dfrac{280}{78} \) to a mixed fraction.
Enter 280:78 and press CALC to obtain the result: 3:23:39 . Hence the answer is \( 3\dfrac{23}{39} \) .
D8. Convert the mixed fraction \( 5\dfrac{7}{12} \) to an improper fraction.
Enter 5:7:12 and press CALC to obtain the result: 67:12 . Hence the answer is \( \dfrac{67}{12} \) .
D9. Find the larger of the two numbers: \( \dfrac{328}{77} \) and \( 4\dfrac{2}{37} \) .
Xzfracal has many built-in functions. Please see the Function Reference in Xzfracal's Home page for more details. Enter max(328:77, 4:2:37) and press CALC to obtain the answer: 328:77 . The function max takes two or more arguments.
D10. Find the smaller of the numbers: \( \dfrac{123}{44} \) , \( 2\dfrac{43}{51} \) and \( 2.7\overline{634} \) .
Enter min(123:44, 2:43:51, 2.7&634) and press CALC to obtain the answer: 2.7&634 . The function min can take more than two arguments, as illustrated by this example.
D11. What is the least common denominator of \( \dfrac{7}{30} \) , \( \dfrac{5}{12} \) .
Enter lcd(7:30, 5:12) and press CALC to obtain the answer: 60 . The function lcd takes two or more arguments.
D12. Compare \( \dfrac{738}{83} \) , \( \dfrac{491}{55} \) . Are they the same? Which one is larger, if not.
The function cmp(a,b) compares the numerical values of two fractions a and b . It returns -1 , 0 or 1 according to if the first number is less than, equals to, or greater than the second number respectively.
Enter cmp(738:83, 491:55) and press CALC . You'll see the answer -1 , which means that the first number is less than the second number.
Note: The functions max and min return the actual larger or smaller number respectively; they do not tell you explicitly whether it is the first or the second number that they return. On the other hand, cmp returns a "three-way code" which encodes the numerical relationship of the two numbers.
D13. Compare \( \dfrac{413}{52} \) , \( 7.92\overline{37} \) . Are they the same? Which one is larger, if not.
Enter cmp(413:52, 7.92&37) and press CALC . You'll see the answer 1 , which means that the first number is larger than the second number.
D14. What is the greatest common divisor of 120 and 105?
Enter gcd(120,105) to obtain the answer 15 . The function gcd can take more than two arguments. Note that the arguments of gcd must be integers. The result is zero if either argument is zero; the result is positive otherwise.
D15. What is the least common multiple of 40 and 32?
Enter lcm(40,32) to obtain the answer 160 . The function lcm can take more than two arguments. Note that the arguments of lcm must be integers. The result is zero if either argument is zero; the result is positive otherwise.
D16. What is the factorial of 39?
Enter 39! to obtain the 47-digit answer of \( 39! = \) \(1 \times 2 \times 3 \times \ldots \times 38 \times 39 \) :
20_39788_20811_97443_35864_02817_39902_89735_68000_00000The underscore character “_”is used to enhance the reability.
D17. You have 23 trillion pieces of candies, and you need to package them into 36-piece bags. How many bags will you have? How many candies will be left alone?
One trillion is one followed by twelve zeros. If you write all the zero’s out; it is impossible, at a glance, to see how many zeros are there. The standard way to write such big numbers is to use the scientific notation: \( 1{,}000{,}000{,}000{,}000 \) \( = 10^{12} \) . Xzfracal uses the special character “@” to write scientific notation. Hence one trillion is 1@12 in Xzfracal. Likewise, the prefix nano, as in nanosecond, is \( \dfrac{1}{1{,}000{,}000{,}000} \) \( = 10^{-9} \) in the scientific notation, and 1@-9 in Xzfracal's notation.
Note that exponents, just like the radixes, themselves are always in base 10. For example, consider the number 101.1_@4_#2 : the exponent 4 is in decimal, the radix 2 is in decimal, the digits in 101.1 are all in binary; hence the number is: 101.1_@4_#2 \( = \) \( (1\times4+1+\tfrac{1}{2} )\times 2^4 \) \( = \) \( 5.5 \times 16 \) \( = \) \( 88 \) in decimal. Likewisely: 12_@3_#4 \( = \) \( (1\times4+2)\times 4^3 = 384 \) in decimal.
This notation is quite flexible, the following representations are all equivalent:
1782000 = 1.782@6 = 178.2@4 = 0.01782@8 = 1782000000@-3Xzfracal provides another way to enhance the readability of long numbers. You can insert the spacer character “_” any number of times anywhere inside a number. For example, the following representations are all equivalent:
100_000 = 10_00_00 = _100_000 = _1_0_000_0 = __1_0_000_0_.0_000_ .As another example:
873.45&679@12 = 873.45_&_679_@_12 = __8_73_._4_5_&_679_@__1__2_ .As long as the components are integers, you can also use the scientific notations to simplify the writing of fractions. For example:
700000:45:6720 = 700_000:_4_5_:67_20 = 7@5:0.45@2:6720_00@-2 .The above examples are meant to demonstrate the flexibility of the “@” and “_” . You should, however, use them in a more sensible manner. Remember your goal is to use them to enhance readability!
Let’s now return to the original question. This is a two-part question. The first part requires integer division. Enter the expression 23@12 \ 36 and obtain the answer: 638_888_888_888 , which is close to 639 billion. Note, you can configure the behavior of the spacers in the output so that spacers are inserted every three digits (please explore the preference menu non your own).
The second part requires modulo division. Enter the expression 23@12 % 36 and obtain the answer: 32.
Please look up the two operatos "\" and "%" in the Operator Reference posted in the Xzfracal's Home page for details.
D18. Evaluate the polynomial: \( p(x) = 2 + \dfrac{5}{4} x - 6.\overline{34} x^2 + \dfrac{9}{8} x^5 \) at \( x = \dfrac{2}{3} \) .
The polynomial is represented by the list
[ 2, 5/4, -6.&34, 0, 0, 9/8 ] ,which consists of the coefficients in the ascending order. Note the two zero terms, corresponding to the missing coefficients of the third and fourth degrees. The function p#eval can be used to evaluate a polynomial. Enter
p#eval( [2, 5/4, -6.& 34, 0, 0, 9/8], 2/3)and press CALC to obtain the answer: 289/1782 .
D19. Multiply the following two matrices: \[ \left( \begin{array}{cc} 7 & 1 \\ 1 & 5 \\ 0 & 3 \end{array} \right) \left( \begin{array}{ccc} 4 & -6 & -2 \\ 1 & 2 & -2 \end{array} \right) \]
A matrix is represented by listing its dimension (number of rows and columns), and its entries row by row. The dimensions of the two matrices are \( 3\times2 \) and \( 2\times3 \) respectively. Hence they are represented by:
[ 3, 2, 7, 1, 1, 5, 0, 3 ][ 2, 3, 4, -6, -2, 1, 2, -2 ]
The extra spaces between the numbers are optional. The function m#mul can be used to multiply two matrices as follows:
m#mul( [3,2, 7,1, 1,5, 0,3] ,[2,3, 4,-6,-2, 1,2,-2] )
= [3,3, 29,-40,-16, 9,4,-12, 3,6,-6]
Hence the result is the following 3x3 matrix: \[ \left( \begin{array}{ccc} 29 & -40 & -16 \\ 9 & 4 & -12 \\ 3 & 6 & -6 \end{array} \right) \] .
D20. Find the determinant of the matrix: \[ \left( \begin{array}{ccc} 1 & 2 & 3 \\ 5 & 6 & 7 \\ 8 & 9 & 4 \end{array} \right) \] .
The functions m#det and s#det can be used to find the determinant. Functions involving matrices usually start with the prefixes: m# or s# . Functions with the prefix s# explicitly expect square matrices only. The matrix representations, in such cases, do NOT include dimension information. Here are the two ways of finding the determinant of the above matrix:
m#det( [3,3, 1,2,3, 5,6,7, 8,9,4] ) = 24s#det( [ 1,2,3, 5,6,7, 8,9,4] ) = 24
D21. Solve the system of linear equations: \[ \left\{ \begin{array}{rcrcrcr} \frac{1}{2} x& +& \frac{3}{2} y& -& 2 z& \, &=\,& \frac{3}{4} \\ 3 x& +& 2 y& +& \frac{5}{4} z& \, &=\,& 3 \\ 5 x& +& 6 y& +& \frac{7}{2} z& \, &=\,& -2 \end{array} \right. \]
The function x#solve can be used to solve the system:
x#solve([ 1:2, 3:2, -2, 3:4, 3, 2, 5:4, 3, 5, 6, 7:2, -2 ] ) =[ 511/181, -725/362, -212/181]
Hence the solution is: \[ x=\dfrac{511}{181} \qquad y=-\dfrac{725}{362} \qquad z=-\dfrac{212}{181} \]
D22. Find the quotient of the following quadratic surd expressions: \[ \dfrac{ 1-4\sqrt{3}+\sqrt{75}-2\sqrt{8} }{ -2+4\sqrt{3}-\sqrt{27}-\sqrt{2} } \]
A quadratic surd expression is an expression of the following form: \[ a_0 + a_1\sqrt{b_1} + a_2\sqrt{b_2} + \cdots + a_n\sqrt{b_n} \] where \( a_0 \), \( a_1 \),\( a_2 \), \( \ldots \), \( a_n \) are any fractions; \( b_1 \),\( b_2 \), \( \ldots \), \( b_n \) are any non-negative fractions. The expression is represented by a list of the fractions. \[ [ a_0, a_1, b_1, a_2, b_2, \ldots, a_n, b_n ] \] NOTE: The list must start with the term \( a_0 \) even if \( a_0 \) is zero, followed by successive pairs of the fractions \( a_1 \),\( b_1 \), \( \ldots \), \( a_n \), \( b_n \). Here are two examples:
\( 1 + 4\sqrt{3} + \tfrac{5}{6}\sqrt{10} - 14\sqrt{3} \) is [1, 4,3, 5/6,10, -14,3] .
\( -5\sqrt{\tfrac{12}{17}} + 5.2\overline{74}\sqrt{11} \) is [0, -5,12/17, 5.2&74,11] . Note: the first term is zero.
We use q#div to divide quadratic surd expressions. Hence the solution to the original question is:
q#div( [1, -4,3, 1,75, -2,8], [-2, 4,3, -1,27, -1,2] ) = [47/23, -9/23,2, 33/23,3, -21/23,6]
Thus: \[ \dfrac{ 1-4\sqrt{3}+\sqrt{75}-2\sqrt{8} }{ -2+4\sqrt{3}-\sqrt{27}-\sqrt{2} } = \dfrac{47}{23} - \dfrac{9}{23}\sqrt{2} + \dfrac{33}{23}\sqrt{3} - \dfrac{21}{23}\sqrt{6} \]
D23. Evaluate the following expression, where the subscripts are the radixes: \[ \left( \dfrac{2}{3} \right)_{4} \times \left( 5\dfrac{A}{B} \right)_{12} + h.k\overline{n}_{30} \]
Xzfracal has a native support of expressing numbers in various number basis (from 2 to 36). It uses the special character “#” to indicate the base. NOTE: The base itself must be in decimal. For example, the binary number 10110.01#2 is the decimal number \( 2^4 + 2^2 + 2^1 + \dfrac{1}{2^2} \) \( = \) \( 16+4+2+0.25 \) \( = 22.25 \) . Letters are used to represent digits larger than 10: 'A'='a'=10 , 'B'='b'=11 , 'C'='c'=12 , ... . Numbers must start with digits in the range 0, 1, 2, ..., 9 . The digit 0 can be used for this purpose. For example, the hexadecimal number 0B3#16 is the decimal number \( 11\times 16 + 3 \) \( = 179 \) . You can mix numbers with different radixes in any expressions involving polynomials, matrices, ... etc.
Hence the above expression is:
2#4 / 3#4 * ( 5#12 + 0A#12 / 0B#12 ) + 0h.k&n#30
Or much simpler, cleaner, clearer, more intuitive and more compact if you use the : notation for fractions and mixed fractions (the underscore _ is optional):
2:3_#4 * 5:A:B_#12 + 0h.k&n_#30
You can configure the answer to display in different formats and radixes. For example: The answer in selected formats and radixes are:
\( \dfrac{207023}{9570} \), \( 21 \dfrac{6053}{9570} \),
\( 21.6\overline{3249738766980146290491118077} \) ,
\(\left( \dfrac{302202233}{2111202} \right)_{4}\) , \(\left( 111 \dfrac{1132211}{2111202} \right)_{4}\) ,
\({ 111.2\overline{2013223112110223120321002001100131233300303210301022223330311332312120}}_{4} \),
\(\left( \dfrac{9B97B}{5656} \right)_{12} \), \(\left( 19 \dfrac{3605}{5656} \right)_{12} \), \({ 19.7\overline{70B5} }_{12} \),
\(\left( \dfrac{7K0N}{AJ0} \right)_{30} \), \(\left( L \dfrac{61N}{AJ0} \right)_{30} \), \({ L.I\overline{T7CQFIA4L1} }_{30} \) .