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A L G E B R A

2

SIGNED NUMBERS

The two parts of a signed number

IN ARITHMETIC, we cannot subtract a larger number from a smaller:

2 − 3

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

2 − 3 = −1  ("Minus 1" or "Negative 1")

This is, since

2 − 2 = 0,

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?

Do the problem yourself first!

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 + .

When we place a number within vertical lines, |−3|, it means the absolute value of that number.

 |−3| = 3 |3| = 3

Problem 1.   Evaluate each of the following.

 a) |6|  =  6 b) |−6|  =  6 c) |0|  =  0 c) |3 − 1|  =  2 d) |1 − 3|  =  2

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

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.

 a)   3 − 5 = −2 b)  1 − 8 = −7 c)  8 − 14 = −6 d)  20 − 65 = −45

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.

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:

−(−a) = 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

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