But in algebra we can. And to do it, we invent "negative" numbers.

then 2 − 3 is one "less" than 0. We call it −1. −1 is a signed number.

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

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Its *algebraic sign*, + or − , and its *absolute value*. The absolute value is simply the numerical 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? What

5 − 8

1. 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.

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.

4. What is an integer?

Any positive or negative *whole* number, including 0.

0, 1, −1, 2, −2, 3, −3, etc.

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 4. Evaluate the following.

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

d) −(−*x*) = *x*