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MULTIPLICATION

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The concepts for this exercise are summarized below; for a complete discussion, read the text.

With the trend towards more and earlier calculator usage, some people have lost a comfort with basic arithmetic operations like  5·7 = 35  and  8+6 = 14 .
It is a waste of valuable time to use your calculator for problems such as these.
In this section, your basic multiplication skills are brought "up to speed" so you won't be wasting mental energy on arithmetic and will be able to concentrate on higher-level ideas.

In mathematics, a vertically centered dot " · " is often used to denote multiplication.
Thus,  3 · 5  denotes " 3 times 5 ".

Consider the multiplication problem  3 · 5 .
One interpretation is  3  groups of  5 , so  3 · 5 = 5 + 5 + 5 = 15 .
Another interpretation is  5  groups of  3 , so  3 · 5 = 3 + 3 + 3 + 3 + 3 = 15 .
Thus, multiplication is a shorthand for repeated addition.

Algebra uses letters to represent numbers. (LOTS more on this later on!)
The expression  2x  means " 2 times x " which is a shorthand for the addition problem  x + x .
Similarly, the expression  3y  means " 3 times y " which is a shorthand for the addition problem  y + y + y , and so on.

Thus, for example,  5 · 3x  means  3x+3x+3x+3x+3x  which is  15x .
Of course, it's much easier to think in terms of multiplication rather than repeated addition, so you immediately get  5 · 3x = 15x .
Similarly,  6 · 7t = 42t .

To commute means to change places.
The Commutative Property of Multiplication states that for all numbers  x  and  y ,   x · y = y · x .
That is, you can change the places of the numbers in a multiplication problem, and this does not affect the result.

The expression  x · y  can more simply be written as  xy .
That is,  juxtaposition (placing two things side-by-side) can be used to denote multiplication.
Of course, this notation can't be used with numbers like 2 and 3, because "23" means the number "twenty-three," not 2 times 3!
So, we can restate the Commutative Property of Multiplication:   For all numbers  x  and  y ,  xy = yx .

Multiplying a number by  1  does not change it.
In other words, multiplying by  1  preserves the identity of the original number.
That is, for all numbers  x ,   x · 1 = 1 · x = x .
For this reason, the number  1  is called the multiplicative identity.

Multiplying a number by  0  always gives  0 .
That is, for all numbers  x ,   x · 0 = 0 · x = 0 .
This is sometimes called the Multiplication Property of Zero.

In this exercise, you will practice multiplication problems of the form "x · y"
where  x  and  y  can be any of these numbers: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 .

Follow the instructions below, and time yourself.
If you average less than 3 seconds per problem, then you'll get an online reward!

Click on "new problem" to get started!


Multiply:


Put your answer here:


Click here or press "tab" to check your answer:


When you're ready to time yourself, use these buttons.
When you "end timing," you'll get a summary sheet of your results. Good luck!
     


   (press the "BACK" key to return to this page after printing)

CONCEPT QUESTIONS EXERCISE:
On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.

Click on "new problem" to get started!
             Want to practice a particular problem type?





Table of Contents for "One Mathematical Cat, Please!"

One Mathematical Cat, Please! A First Course in Algebra
© 2004  Carol J.V. Fisher 		Creative Commons License
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