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Hamilton defined a quaternion as the quotient of two directed lines in tridimensional space; [1] or, more generally, as the quotient of two vectors. [2] A quaternion can be represented as the sum of a scalar and a vector. It can also be represented as the product of its tensor and its versor.
For example, density (mass divided by volume, in units of kg/m 3) is said to be a "quotient", whereas mass fraction (mass divided by mass, in kg/kg or in percent) is a "ratio". [8] Specific quantities are intensive quantities resulting from the quotient of a physical quantity by mass, volume, or other measures of the system "size".
For two elements a 1 + b 1 i + c 1 j + d 1 k and a 2 + b 2 i + c 2 j + d 2 k, their product, called the Hamilton product (a 1 + b 1 i + c 1 j + d 1 k) (a 2 + b 2 i + c 2 j + d 2 k), is determined by the products of the basis elements and the distributive law. The distributive law makes it possible to expand the product so that it is a sum of ...
30,561 10 3,G81 20 ÷ ÷ ÷ 61 10 31 20 = = = 501 10 151 20 30,561 10 ÷ 61 10 = 501 10 3,G81 20 ÷ 31 20 = 151 20 ÷ = (black) The divisor goes into the first two digits of the dividend one time, for a one in the quotient. (red) fits into the next two digits once (if rotated), so the next digit in the quotient is a rotated one (that is, a five). (blue) The last two digits are matched once for ...
1 Multiplication. Toggle Multiplication subsection ... So in this example, what remains is a quotient of 485 with a remainder of 16364. ... 5 ⁄ 4: 8 ⁄ 1 18 9: 12321:
The entries are represented by sectors corresponding to their arguments: 1 (green), i (blue), −1 (red), −i (yellow). The two-dimensional irreducible complex representation described above gives the quaternion group Q 8 as a subgroup of the general linear group (,). The quaternion group is a multiplicative subgroup of the quaternion algebra:
Animation showing the use of synthetic division to find the quotient of + + + by . Note that there is no term in x 3 {\displaystyle x^{3}} , so the fourth column from the right contains a zero. In algebra , synthetic division is a method for manually performing Euclidean division of polynomials , with less writing and fewer calculations than ...
Thought of quotitively, a division problem can be solved by repeatedly subtracting groups of the size of the divisor. [1] For instance, suppose each egg carton fits 12 eggs, and the problem is to find how many cartons are needed to fit 36 eggs in total. Groups of 12 eggs at a time can be separated from the main pile until none are left, 3 groups:
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