1.  What are the two parts of a signed number?

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Its algebraic sign, + or − , and its absolute value, which is simply the arithmetical value, that is, the number without its sign.

The algebraic sign of +3 ("plus 3" or "positive 3") is + , and its absolute value is 3.  

The algebraic sign of −3 ("negative 3" or "minus 3") is − .  The absolute value of  −3 is also 3.

As for the algebraic sign + , normally we do not write it.  The algebraic sign of  2, for example, is understood to be + .

2.  How do we subtract a larger number from a smaller?

5 − 8

1.   What will be the sign of the answer?

It would not be wrong to say that we cannot take 8 from 5.  We can however take 5 from 8 -- and that is what we do -- but we report the answer with a minus sign

5 − 8 = −3.

Even in algebra we can only do ordinary arithmetic.  But then we must choose the correct sign.

We may say that this is the first rule of signed numbers:

To subtract a larger number from a smaller,
subtract the smaller from the larger, but report
the answer as negative.

1 − 5 = −4.

We actually do  5 − 1.

It was in order to subtract a larger number from a smaller that negative numbers were invented.

3.  What is the only difference between  8 − 5  and  5 − 8 ?

The algebraic signs. They have the same absolute value.

Problem 2.   Subtract.

Problem 3.   You have 20 dollars in the bank and you write a check for 25 dollars. Now what is your balance?

20.00 − 25.00 = −5.00

The number line

The number line is a kind of "ruler" centered on 0.  The negative numbers fall to the left of 0; the positive numbers fall to the right.

We imagine every number to be on the number line.  And so the fraction ½ will fall between 0 and 1; the fraction −½ is between 0 and −1; and so on.

4.  What is an integer?

Any positive or negative whole number, including 0.

0, 1, −1, 2, −2, 3, −3, etc.

On the number line, we typically place the integers.

And it is on the number line that we begin to see the practical uses for signed numbers.  In general, they show the "direction" of some quantity.  That quantity might be temperture:  above or below a certain temperature designated as 0.  Or it might be the position or "address" of some object:  left or right of some fixed position chosen as 0.  Or it might be time:  before or after a certain moment that again is chosen as 0.  Or, as we all know, negative numbers can indicate a balance in a checking account

Problem 4.   A rocket is scheduled to launch at precisely 9:16 AM, which is designated t (for time) = 0, and t will be measured in minutes.

a)  What time is it at t = −10?  9:06 AM.

b)  What time is it at t = −1?  9:15 AM.

c)  What time is it at t = +5?  9:21 AM.

d)  What is the value of t at 9:00 AM?  t = −16.

e)  What is the value of t at 9:30 AM?  t = 14.

The negative of each number

Every number will have a negative.  The negative of 3, for example, will be found at the same distance from 0, but on the other side.

It is −3.

Now, what number is the negative of −3?

The negative of −3 will be the same distance from 0 on the other side.  It is 3.

−(−3) = 3.

"The negative of −3 is 3."

This will be true for any number a:

"The negative of −a is a."

What is in the box is called a formal rule .  This means that whenever we see something that looks like this --

−(−a)

-- something that has that form, then we may rewrite it in this form:

a

For example,

−(−12) = 12.

To learn algebra is to learn its formal rules.  For, what are calculations but writing things in a different form?  In arithmetic, we rewrite  1 + 1  as  2.  In algebra, we rewrite  −(−a)  as  a.

Problem 5.   Evaluate the following.

a)  −(−10)  = 10        b)  −(2 − 6)  = 4        c)  −(1 + 4 − 7)  = 2 

d)  −(−x)  = x 

The algebraic definition of the negative of a number

Finally, the way we define a negative number in algebra is as follows.  −5, for example, is that number which when added to 5 itself, results in 0.

5 + (−5) = 0.

That is, to each number  a  there corresponds one and only one number  −a  called its negative, such that when we add it to a, we get 0.

a + (−a) = 0.

Problem 6.   What number is the negative of xyz?  Why?

−xyz , because xyz + (−xyz) = 0.

Problem 7.   What number is the negative of −q?  Why?

−(−q), which is q, because −q + q = 0.

Problem 8.   If

s + t = 0,

then what is the relationship between t and s?

t = −s.

Problem 9.   If you had to prove that

b − a  is the negative of  a − b,

how would you do it?

Show that  a − b  +   b − a  = 0 .

To prove a fact about anything, whether in mathematics, logic, or the law, we have simply to show that it satisfies the definition of that fact.