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Multiplication can also be thought of as scaling. In the above animation, we see 3 being multiplied by 2, giving 6 as a result. One theory of learning multiplication derives from the work of the Russian mathematics educators in the Vygotsky Circle which was active in the Soviet Union between the world wars. Their contribution is known as the ...
Four bags with three marbles per bag gives twelve marbles (4 × 3 = 12). Multiplication can also be thought of as scaling. Here, 2 is being multiplied by 3 using scaling, giving 6 as a result. Animation for the multiplication 2 × 3 = 6 4 × 5 = 20. The large rectangle is made up of 20 squares, each 1 unit by 1 unit.
The fractional portion is discarded (5.5 becomes 5). 5 is halved (2.5) and 6 is doubled (12). The fractional portion is discarded (2.5 becomes 2). The figure in the left column (2) is even, so the figure in the right column (12) is discarded. 2 is halved (1) and 12 is doubled (24). All not-scratched-out values are summed: 3 + 6 + 24 = 33.
The products of small numbers may be calculated by using the squares of integers; for example, to calculate 13 × 17, one can remark 15 is the mean of the two factors, and think of it as (15 − 2) × (15 + 2), i.e. 15 2 − 2 2.
Unit fractions can also be expressed using negative exponents, as in 2 −1, which represents 1/2, and 2 −2, which represents 1/(2 2) or 1/4. A dyadic fraction is a common fraction in which the denominator is a power of two , e.g. 1 / 8 = 1 / 2 3 .
The method for general multiplication is a method to achieve multiplications with low space complexity, i.e. as few temporary results as possible to be kept in memory. . This is achieved by noting that the final digit is completely determined by multiplying the last digit of the multiplic
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