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

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

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.

www.matheass.eu