and there are two of those oblongs; so that in the whole the number of counters is twice 28, or 28 x 2, or 7 first multiplied by 4, and that product multiplied by 2. In figure II. it is shewn that 7 multiplied by 8 is also 7 first multiplied by 2, and that product multiplied by 4. The same method may be applied to other numbers. Thus, since 80 is 8 times 10, 256 times 80 is 256 multiplied by 8, and that product multiplied by 10. If we use the signs, the foregoing assertions are made thus:
7 × 8 = 7 × 4 × 2 = 7 × 2 × 4.
256 × 80 = 256 × 8 × 10 = 256 × 10 × 8.
EXERCISES.
Shew that 2 × 3 × 4 × 5 = 2 × 4 × 3 × 5 = 5 × 4 × 2 × 3, &c.
Shew that 18 × 100 = 18 × 57 + 18 × 43.
- Articles (51) and (55) may be expressed in the following way, where by ab we mean a taken b times; by abc, a taken b times, and the result taken c times.
ab = ba.
abc = acb = bca = bac, &c.
abc = a × (bc) = b × (ca) = c × (ab).
If we would say that the same results are produced by multiplying by b, c, and d, one after the other, and by the product bcd at once, we write the following:
a × b × c × d = a × bcd.
The fact is, that if any numbers are to be multiplied together, the product of any two or more may be formed, and substituted instead of those two or more; thus, the product abcdef may be formed by multiplying
| ab | cde | f |
|---|---|---|
| abf | de | c |
| abc | def | &c. |
- In order to multiply by 10, annex a cipher to the right hand of the multiplicand. Thus, 10 times 2356 is 23560. To shew this, write 2356 at length which is
2 thousands, 3 hundreds, 5 tens, and 6 units.
Take