Understanding Algebra

James W. Brennan
 

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Multiplication and Division

Multiplication as Repeated Addition

We think of a multiplication statement like “2 × 3” as meaning “Add two threes together”, or

 

3 + 3

 

and “4 × 9” as “add 4 nines together”, or

 

9 + 9 + 9 + 9.

 

In general, a × b means to add b’s together such that the number of b’s is equal to a:

 

a × b = b + b + b + . . . + b (a times)

Multiplication with Signed Numbers

We can apply this same rule to make sense out of what we mean by a positive number times a negative number. For example,

3 × (–4)

just means to take 3 of the number “negative four” and add them together:

3 × (–4) = (–4) + (–4) + (–4) = –12

Unfortunately, this scheme breaks down when we try to multiply a negative number times a number. It doesn’t make sense to try to write down a number a negative number of times. There are two ways to look at this problem.

One way is to use the fact that multiplication obeys the commutative law, which means that the order of multiplication does not matter:

a × b = × a.

This lets us write a negative times a positive as a positive times a negative and proceed as before:

(–3) × 4 = 4 × (–3) = (–3) + (–3) + (–3) + (–3) = –12

However, we are still in trouble when it comes to multiplying a negative times a negative. A better way to look at this problem is to demand that multiplication obey a consistent pattern. If we look at a multiplication table for positive numbers and then extend it to include negative numbers, the results in the table should continue to change in the same pattern.

For example, consider the following multiplication table:

a

b

a × b

3

2

6

2

2

4

1

2

2

0

2

0

The numbers in the last column are decreasing by 2 each time, so if we let the values for a continue into the negative numbers we should keep decreasing the product by 2:

a

b

a × b

3

2

6

2

2

4

1

2

2

0

2

0

–1

2

–2

–2

2

–4

–3

2

–6

We can make a bigger multiplication table that shows many different possibilities. By keeping the step sizes the same in each row and column, even as we extend into the negative numbers, we see that the following sign rules hold for multiplication:

Sign Rules for Multiplication

(+)(+) = (+)

(–)(–) = (+)

(–)(+) = (–)

(+)(–) = (–)


Multiplication Table

Notice how the step size in each row or column remains consistent, regardless of whether we are multiplying positive or negative numbers.

 

-5

-4

-3

-2

-1

0

1

2

3

4

5

-5

25

20

15

10

5

0

-5

-10

-15

-20

-25

-4

20

16

12

8

4

0

-4

-8

-12

-16

-20

-3

15

12

9

6

3

0

-3

-6

-9

-12

-15

-2

10

8

6

4

2

0

-2

-4

-6

-8

-10

-1

5

4

3

2

1

0

-1

-2

-3

-4

-5

0

0

0

0

0

0

0

0

0

0

0

0

1

-5

-4

-3

-2

-1

0

1

2

3

4

5

2

-10

-8

-6

-4

-2

0

2

4

6

8

10

3

-15

-12

-9

-6

-3

0

3

6

9

12

15

4

-20

-16

-12

-8

-4

0

4

8

12

16

20

5

-25

-20

-15

-10

-5

0

5

10

15

20

25

 

For math purists, here’s the real reason:

The Real Reason

It should be obvious that the presentation of the rules of arithmetic given here is just a collection of motivational arguments, not a formal development. The formal development of the real number system starts with the field axioms. The field axioms are postulated, and then all the other properties follow from them. The field axioms are

    1. The associative and commutative laws for addition and multiplication
    2. The existence of the additive and multiplicative identities (0 and 1)
    3. The existence of the additive inverse (opposites, or negatives) and the multiplicative inverse (the reciprocal)
    4. The distributive law

All of these are essential, but the distributive law is particularly important because it is what distinguishes the behavior of multiplication from addition. Namely, multiplication distributes over addition but not vice-versa.

The rules of arithmetic like “a negative times a negative gives a positive” are what they are because that is the only way the field axioms would still hold. For example, the distributive law requires that

–2(3 – 2) = (–2)(3) + (–2)(–2)

We can evaluate the left side of this equation by following the order of operations, which says to do what is in parentheses first, so

–2(3 – 2) = –2(1) = –2.

Now for the distributive law to be true, the right side must also be equal to -2, so

(–2)(3) + (–2)(–2) = –2

If we use our sign rules for multiplication then it works out the way it should:

(–2)(3) + (–2)(–2) = –6 + 4 = –2

 

Notation for Multiplication

We are used to using the symbol “×” to represent multiplication in arithmetic, but in algebra we prefer to avoid that symbol because we like to use the letter “x” to represent a variable, and the two symbols can be easily confused. So instead, we adopt the following notation for multiplication:

1.      Multiplication is implied if two quantities are written side-by-side with no other symbol between them.

Example:  ab means a × b.

2.      If a symbol is needed to prevent confusion, we use a dot.

 

Example:  If we need to show 3 times 5, we cannot just write them next to each other or it would look like the number thirty-five, so we write 3 · 5.

  • We can also use parentheses to separate factors. 3 times 5 could be written as 3(5) or (3)5 or (3)(5).

Division

There are two ways to think of division: as implying a related multiplication, or as multiplying by the reciprocal.

Division as Related Multiplication

The statement “12 ÷ 3 = 4” is true only because 3 × 4 = 12. A division problem is really asking the question “What number can I multiply the divisor by to get the dividend?”; and so every division equation implies an equivalent multiplication equation. In general:

a ÷ b = c if and only if a = b × c

This also shows why you cannot divide by zero. If we asked “What is six divided by zero?” we would mean “What number times zero is equal to six?”, but any number times zero gives zero, so there is no answer to this question.

Multiplicative Inverse (The Reciprocal)

For every real number a (except zero) there exists a real number denoted by 1/a such that

a(1/a) = 1

  • The number 1/a is called the reciprocal or multiplicative inverse of a.
  • Note that the reciprocal of 1/a is a. The reciprocal of the reciprocal gives you back what you started with.

This allows us to define division as multiplication by the reciprocal:

a ÷ b = a × (1/b)

This is usually the most convenient way to think of division when you are doing algebra.

Notation for Division

Instead of using the symbol “ Έ ” to represent division, we prefer to write it using the fraction notation:

Sign Rules for Division

Because division can always be written as a multiplication by the reciprocal, it obeys the same sign rules as multiplication.

If a positive is divided by a negative, or a negative divided by a positive, the result is negative:

but if both numbers are the same sign, the result is positive:

 
      

 


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James W. Brennan