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In mathematics, summation is the addition of a sequence of numbers, called addends or summands; the result is their sum or total.Beside numbers, other types of values can be summed as well: functions, vectors, matrices, polynomials and, in general, elements of any type of mathematical objects on which an operation denoted "+" is defined.
Now we insert the number 6 at the end of the sequence to get (5, 8, 3, 2, 6). What is the total of that sequence? Answer: 5 + 8 + 3 + 2 + 6 = 24. This is arrived at by simple summation of the sequence. But if we regarded 18 as the running total, we need only add 6 to 18 to get 24. So, 18 was, and 24 now is, the running total.
The arithmometer (French: arithmomètre) was the first digital mechanical calculator strong and reliable enough to be used daily in an office environment. This calculator could add and subtract two numbers directly and perform long multiplications and divisions effectively by using a movable accumulator for the result.
To add a new list of numbers and arrive at a total, the user was first required to "ZERO" the machine. Then, to add sets of numbers, the user was required to press numbered keys on a keyboard, which would remain depressed (rather than immediately rebound like the keys of a computer keyboard or typewriter or the buttons of a typical modern machine).
The calculator was sold in a hinged mahogany case 46 by 12 by 11 centimetres (18.1 in × 4.7 in × 4.3 in) which, if required, holds the instrument when in use by means a brass support that can be latched to the outer end of the case. [6] [7] Out of its case the calculator weighs about 900 grams (32 oz). [8]
Comptometer type calculators were often retained for much longer to be used for adding and listing duties, especially in accounting, since a trained and skilled operator could enter all the digits of a number in one movement of the hands on a comptometer quicker than was possible serially with a 10-key electronic calculator. In fact, it was ...
The Comptometer was the first commercially successful key-driven mechanical calculator, patented in the United States by Dorr Felt in 1887.. A key-driven calculator is extremely fast because each key adds or subtracts its value to the accumulator as soon as it is pressed and a skilled operator can enter all of the digits of a number simultaneously, using as many fingers as required, making ...
The total amount to be subtracted is 4 + 8 + 12 + 16 + ⋯, which is 4 times the original series. These relationships can be expressed using algebra. These relationships can be expressed using algebra.