MatheAss 10.0 − Algebra

Prime Numbers

The program calculates all prime numbers between two numbers.

Prime numbers between 1000000000 and 1000000300:

1000000007 1000000009 1000000021 1000000033 1000000087 1000000093
1000000097 1000000103 1000000123 1000000181 1000000207 1000000223
1000000241 1000000271 1000000289 1000000297

16 prime numbers

Prime Tuples   (New in Version 9.0)

The program determines in an interval [a,b] all twin primes (p,p+2), cousin primes (p,p+4), sexy primes (p,p+6) and prime triplets.

Prime triplets between 1 and 200

(3|5|7) (5|7|11) [7|11|13] (11|13|17) [13|17|19] (17|19|23) [37|41|43] 
(41|43|47) [67|71|73] [97|101|103] (101|103|107) [103|107|109] (107|109|113)
(191|193|197) [193|197|199] 

15 prime triplets
7 of the form (p|p+2|p+6) and 7 of the form [p|p+4|p+6]

Prime Factorization

The program decomposes natural numbers into their prime power factors.

  99999999999901 = 19001 · 5262880901
  99999999999001 = 107 · 401 · 1327 · 1756309
  99999999990001 = prime number 
    3938980639167 = 314 · 77
999330136292431 = 999712 · 99991

GCD and LCM

For two numbers a and b, the greatest common divisor, the least common multiple and their sets of divisors are determined.

a = 24
b = 256

greatest common divisor               GCD = 8
least common multiple                 LCM = 768  

Sets of divisors:
T(a) = { 1 2 3 4 6 8 12 24}
T(b) = { 1 2 4 8 16 32 64 128 256}

Percentage Calculation   (New in Version 9.0)

The program calculates the base value G, the percentage value W, the percentage rate p or p%, the growth factor q and the final value E, when any two of them are entered.

Given:
¯¯¯¯¯¯¯¯
Percentage value W = −120
   Growth factor q = 95% = 0.95 = 19/20

Results:
¯¯¯¯¯¯¯¯¯¯
        Basic value G = 2400
     Percentage p% = −5% = −0.05 = −1/20      
          End value E = 2280  

Decimals to Fractions

The program converts repeating and terminating decimals into fractions.

Non-repeating part : 1.20
Repeating part : 045
    ___
1.20045 = 120/100 + 1/2220 = 533/444

Fractions to Decimals

The program converts fractions into repeating decimals and determines the repeating part and its length.

Numerator : 533
Denominator : 444
              ___
533/444 = 1.20045
repeating from the 3rd digit after the decimal point
the repeating part is 3 digits long

Binomials

One of the best-known formulas in school mathematics is certainly the binomial formula (a + b)² = a² + 2ab + b² .

The program calculates the more general case (a·x + b·y)ⁿ.

(2·x  − 3·y)7 =       +128 · x7
                           −1344 · x6 · y
                            +6048 · x5 · y2
                          −15120 · x4 · y3   
                          +22680 · x3 · y4
                          −20412 · x2 · y5
                          +10206 · x · y6
                            −2187 · y7  

4th Degree Equations

The program determines the real solutions of an equation of 4th degree or lower. For higher-degree equations, apart from numerical approximations (zeros in the curve analysis module), there is no algebraic solution method.

x4 + 2·x3 - 3·x2 + 5·x - 5 = 0   <=>   (x - 1)·(x3 + 3·x2 + 5) = 0
L = {-3.42599;  1}

Diophantine Equations

Named after Diophantus of Alexandria (around 250), who in his book *Arithmetica* studied the solution of linear and quadratic equations, especially their integer solutions.
The program computes the integer solutions of the equation  a·x - b·y - c = 0. This allows determining the integer points on a straight line.

7·x − 3·y − 5 = 0 ;   x,y integers
L = { ( 2 + 3t | 3 + 7t ) }

Pythagorean Triples

Pythagorean triples are the integer solutions (x,y,z) of the equation x² + y² = z² , which applies to the sides of right triangles.

For x, y, z between 100 and 400 one obtains:

( 119, 120, 169 )    ( 104, 153, 185 )    ( 133, 156, 205 )    ( 105, 208, 233 )    
( 140, 171, 221 )    ( 115, 252, 277 )    ( 120, 209, 241 )    ( 161, 240, 289 )    
( 160, 231, 281 )    ( 207, 224, 305 )    ( 175, 288, 337 )    ( 135, 352, 377 )    
( 136, 273, 305 )    ( 204, 253, 325 )    ( 225, 272, 353 )    ( 189, 340, 389 )    
( 180, 299, 349 )    ( 252, 275, 373 )    ( 152, 345, 377 )    ( 228, 325, 397 )  

Calculators

There are four calculators:

• The fraction calculator handles the four basic operations and exponentiation.
• The numeral system calculator works with any base between 2 and 16.
• The complex number calculator computes, in addition to the usual functions, the complex conjugate of a number.
• The 4. calculator handles the basic operations and combinatorial formulas with integers of up to 10 000 digits.

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