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Adults.
Algebra for Adults:
Part 3, Review, Multiplication and Division:
December 2, 2002:
The terms used in multiplication are the factors, the product,
the multiplicand, and the multiplier:
Factors are the numbers being multiplied.
The result of the multiplication is the product.
The normal sign for multiplication is x (times).
When we deal with a 2-factor multiplication problem, by convention
the first factor is called the multiplicand and the second factor
is the multiplier such that,
multiplicand X multiplier = product.
We notice that multiplication can be broken down into a series
of additions of the same number.
For example:
4 x 7 = 4 sevens or 7 + 7 + 7 + 7 = 28
The Commutative Law of Multiplication, states that interchanging
factors does not change their product.
For example:
7 x 4 = 28 is equivalent to 4 x 7 = 28
6 x 7 = 42 is equivalent to 7 x 6 = 42
The Multiplicative Identity Property, is give as,
The product of any number multiplied by 1 is the identical number.
The Multiplicative Property of Zero is given as,
The product of any number multiplied by zero is equal to zero.
We can multiply more than two factors in a problem.
For example:
3 x 5 x 7 = 105
4 x 2 x 6 = 48
3 x 7 x 5 = 105
We find that the order in which we multiply numbers does not change their product. This is called the Associative Law of Multiplication.
Once we knew how to multiply two simple numbers, we are able to construct the Multiplication Table. It turns out that constructing the table is just counting by numbers other than 1.
We can use the Multiplication Table to find the products of two numbers.
We learned that the product of a number multiplied by itself
is called the Square.
For example:
The square of 7 = 7 x 7 = 49
We also discovered how to multiply numbers by aligning them
vertically. This allows us to multiply numbers of more than one
digit. We apply the properties of the base-10 numbering system
to do our problem in parts. Thus we are able to find products
of numbers which are not part of the multiplication table, numbers
larger than 9.
For example:
We also learned to deal with the carry in multiplication. We
found that it was nothing more than a matter of adding the carry
to the next, left most, product in the problem.
For example:
We also confirmed that, because of the structure of our base-10 numbering system, we will not have to deal with factors larger than 9.
We also learned how to deal with multiplication by place values
and numbers that are multiples of place values.
For example:
We learned to find products of multi-digit factors by solving
the problem in parts and adding the separate products.
For example:
The Distributive Law of multiplication is given as,
To multiply a number by the sum of two addends, multiply the number by each addend and add the resulting products together.
Review of Division:
The terms we use in division are dividend, divisor, and quotient:
The basic operation is, dividend / divisor = quotient.
Dividend is the number we divide.
Divisor is the number we divide it by.
The quotient is the result of the division.
Division is directly related to multiplication so that the divisor is equivalent to one factor and the quotient is equivalent to the other factor. The dividend is then equivalent to the product.
Thus, we can still use the multiplication table in reverse to do simple divisions.
We understand that dividing any number by 1 results in the identical number.
The rule for zero is the reverse of the Multiplicative Property of Zero. That is: If zero is divided by any nonzero number, the quotient is zero.
Division by zero is undefined or impossible.
The 4 procedural steps for long division are divide, multiply, subtract, and bring down.
Here is an example of long division:
Here is the same problem using short division wherein we keep some of the notation in our head.
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