Math 107A
HW #10
Not due
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1. Are there any whole numbers x such that x = − x? Explain. (Your explanation should refer
to the definition of − x.)
The definition of − x says that x +− x = 0. If x =− x, then the equation x + x = 0 must
be true. The only value for which this equation is true is x = 0. Thus 0 is the only whole
number that is its own opposite.
2. Assume that you now know how to add and subtract any two integers.
(a) Compute 5 × − 6
5 × − 6 = − 6 + − 6 + − 6 + − 6 + − 6 = − 30
(b) Compute − 7 × 3
−
7 × 3 = 3 × − 7 = − 7 + − 7 + − 7 = − 21
(c) Explain why you cannot use the definition of multiplication to compute − 8 × − 4.
According to the definition of multiplication I would need to add − 4 to itself negative
8 times, but “negative eight times” doesn’t make sense. Similarly, if we switched the
order we would be adding − 8 to itself negative 4 times, but “negative four times” doesn’t
make sense. Therefore we cannot use the definition of multiplication to compute this.
(d) Assume you now know how to multiply a positive number and a negative number.
Compute − 8 × − 4.
−
−
8×0=0
8 × (4 + − 4) = 0
(− 8 × 4) + (− 8 × − 4) = 0
−
32 + (− 8 × − 4) = 0
Now, since we are trying to figure out what
last equation says,
−
−8
× − 4 equals, let’s call it “ ? ”. So the
32 + ? = 0
Looking at this equation, we realize that ? must equal 32. Thus − 8 × − 4 is equal to 32.
3. Use the definition of division to compute the following. Please show enough work so that I
can tell you are using the definition of division. Also, please circle your final answer.
(a) 40 ÷ 8
8×
= 40
Since 5 fits in the blank, 40 ÷ 8 = 5 .
(b) 40 ÷ − 8
−
8×
= 40
Since − 5 fits in the blank, 40 ÷ − 8 =
(c)
− 40
−
5.
= − 40
Since − 5 fits in the blank, − 40 ÷ 8 =
(d)
5.
÷8
8×
− 40
−
÷
−8
−
8×
= − 40
Since 5 fits in the blank, − 40 ÷ − 8 = 5 .
4. In class we discussed the rule of signs for multiplication?
Rule of Signs for Multiplication
(positive) × (positive) = (positive)
(positive) × (negative) = (negative)
(negative) × (positive) = (negative)
(negative) × (negative) = (positive)
Which rule of signs for multiplication tells us the following.
(a) (positive) ÷ (positive) = (positive)
Using the definition of division to compute (positive) ÷ (positive) we would use the
equation
(positive) ×
= (positive)
So we are multiplying a positive number by something and getting a positive number.
Therefore the first rule of signs says that the blank must be positive.
In other words, the first rule of signs for mulitplication tells us that (positive) ÷ (positive)
= (positive).
(b) (positive) ÷ (negative) = (negative)
Using the definition of division to compute (positive) ÷ (negative) we would use the
equation
(negative) ×
= (positive)
So we are multiplying a negative number by something and getting a positive number.
Therefore the fourth rule of signs says that the blank must be negative.
In other words, the fourth rule of signs for mulitplication tells us that (positive) ÷
(negative) = (negative).
(c) (negative) ÷ (positive) = (negative)
Using the definition of division to compute (negative) ÷ (positive) we would use the
equation
(positive) ×
= (negative)
So we are multiplying a positive number by something and getting a negative number.
Therefore the second rule of signs says that the blank must be negative.
In other words, the second rule of signs for mulitplication tells us that (positive) ÷
(negative) = (negative).
(d) (negative) ÷ (negative) = (positive)
Using the definition of division to compute (negative) ÷ (negative) we would use the
equation
(negative) ×
= (negative)
So we are multiplying a negative number by something and getting a negative number.
Therefore the third rule of signs says that the blank must be positive.
In other words, the third rule of signs for mulitplication tells us that (negative) ÷
(negative) = (positive).