# 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} = a^{2} + 2ab + b^{2} 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·x^{7}−1344·x^{6}· y +6048·x^{5}· y^{2}−15120·x^{4}· y^{3}+22680·x^{3}· y^{4}−20412·x^{2}· y^{5}+10206·x · y^{6}−2187·y^{7}

## 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*).

x^{4}+ 2·x^{3}- 3·x^{2}+ 5·x - 5 = 0 <=> (x - 1)·(x^{3}+ 3·x^{2}+ 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 x^{2} + y^{2} = z^{2}, 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