# MatheAss 9.0 − 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 all prime number twins (p, p + 2), prime number cousins (p, p + 4), sexy primes (p, p + 6) and prime number triplets in an interval [a, b].

```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 breaks down natural numbers into their prime powers.

```  99999999999901 = 19001 5262880901
99999999999001 = 107 401 1327 1756309
99999999990001 = prime number
3938980639167 = 3 14 7 7
999330136292431 = 99971 2 99991```

## GCD and LCM

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

```a = 24
b = 256

greatest common factor gcd = 8
lowest common multiple lcm = 768

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

## Calculating percentages   (new in version 9.0)

The base value G, the percentage value W, the percentage p or p%, the growth factor q and the final value E are calculated if two independent values 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  ```

## Decimal numbers into fractions

The program converts periodic and terminating decimal fractions into fractions.

```Non-periodic part: 1.20
Period: 045
___
1.20045 = 120/100 + 1/2220 = 533/444```

## Fractions into decimals

The program converts fractions into periodic decimal fractions and determines the period and its length.

```numerator   : 533
denumerator : 444
___
533/444 = 1.20045

The recurring decimal starts with the
3rd digit following the decimal point
and is 3 digits long.```

## Binomials of n-th Degree

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

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-valued solutions of an equation of 4th or smaller degree. For equations of a higher degree there is no algebraic solution method apart from approximate calculations (zeros in the program Calculus of Arbitrary Functions).

```x4 + 2·x3 - 3·x2 + 5·x - 5 = 0   <=>   (x - 1)·(x3 + 3·x2 + 5) = 0
L = {-3,42599;  1}```

## Diophantine Equations

The program computes the integer solutions of the equation a·x - b = m·y with m>0.
This for example permits the determination of the integer points in a straight line.

```7·x − 3·y − 5 = 0 ;   x,y integer
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 in right triangles.

For x, y, z between 100 and 400 we get:

```( 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

• The calculator for fractions can do the four basic arithmetic operations and can raise fractions to the power.
• The calculator for place value systems works with every base between 2 and 16.
• In addition to the usual functions, the calculator for complex numbers also calculates the conjugate complexes of a number.   ## Calculating with large numbers (new in version 9.0 from April 2021)

The calculation is based on whole numbers with a maximum of 10,000 digits. ```1 267 650 600 228 229 401 496 703 205 376 div 1 125 899 906 842 624
=  1 125 899 906 842 624  Remainder  0
=  1,13 · 10^15 Remainder 0

nCr(100,50)  =  100 891 344 545 564 193 334 812 497 256 = 1,01 · 10^29                        ```