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The angle between the horizontal line and the shown diagonal is 1 / 2 (a + b). This is a geometric way to prove the particular tangent half-angle formula that says tan 1 / 2 (a + b) = (sin a + sin b) / (cos a + cos b). The formulae sin 1 / 2 (a + b) and cos 1 / 2 (a + b) are the ratios of the actual distances to ...
Simplifying this further gives us the solution x = −3. It is easily checked that none of the zeros of x ( x + 1)( x + 2) – namely x = 0 , x = −1 , and x = −2 – is a solution of the final equation, so no spurious solutions were introduced.
Two fractions a / b and c / d are equal or equivalent if and only if ad = bc.) For example, 1 / 4 , 5 / 6 , and −101 / 100 are all irreducible fractions. On the other hand, 2 / 4 is reducible since it is equal in value to 1 / 2 , and the numerator of 1 / 2 is less than the numerator ...
For example, 6 is highly composite because d(6)=4 and d(n)=1,2,2,3,2 for n=1,2,3,4,5 respectively. A related concept is that of a largely composite number , a positive integer that has at least as many divisors as all smaller positive integers.
Pell's equation for n = 2 and six of its integer solutions Pell's equation , also called the Pell–Fermat equation , is any Diophantine equation of the form x 2 − n y 2 = 1 , {\displaystyle x^{2}-n\,y^{2}=1,} where n is a given positive nonsquare integer , and integer solutions are sought for x and y .
Simplification is the process of replacing a mathematical expression by an equivalent one that is simpler (usually shorter), according to a well-founded ordering. Examples include:
The reciprocal function: y = 1/x.For every x except 0, y represents its multiplicative inverse. The graph forms a rectangular hyperbola.. In mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1/x or x −1, is a number which when multiplied by x yields the multiplicative identity, 1.
49 is the smallest discriminant of a totally real cubic field. [2] 49 and 94 are the only numbers below 100 whose all permutations are composites but they are not multiples of 3, repdigits or numbers which only have digits 0, 2, 4, 5, 6 and 8, even excluding the trivial one digit terms. 49 = 7^2 and 94 = 2 * 47