Positive   and   Negative   Integers
The   first   step   to   understanding   algebra   is   to   have   a   solid   grasp   of   what   numbers   are   and   how   they   can   be   manipulated.
Before   we   can   do   anything   algebraic,   we   must   look   back   at   what   we   learned   in
elementary   school.
The   first   thing   that   we   learn   to   do   in   math   is   to   count.
From   there   we   can   make   number   lines,   combine   numbers   together,   make   charts   and   graphs,   and   start   understanding   patterns   and   how   to   write
expressions.
It   makes   sense   then,   to   start   with   the   counting   numbers   1,   2,   3,   4,   ....
and   so   on.
We   have   had   plenty   of   experience   adding,   subtracting,   looking   at   fractions   and   patterns   that   involve   whole   numbers   such   as   1,   2,   3   and   so   on.
We   hit   a   point,   however,   where   positive   numbers   don't   make
sense   for   a   situation.
What   happens   if   you   go   below   zero?
Consider   the   following   example.
The   temperature   outside   is   3   degrees   Celsius.
This   evening   it   will   drop   5   degrees.
How   do   we   deal   with   this?
How   would   we   come   up   with   an   answer   for   3   –   5   ?
It   can’t   be   2   degrees   because   it   has   to   drop   5   and   that   would   only   be   a   drop   of   one   degree.
This   is   the   point   where   we   have   to   create   a   new   kind   of   number...a
negative.
Negatives   are   basically   the   same   thing   as   the   positives   except   that   they   are   below   zero.
If   you   think   of   a   number   line,   treat   zero   like   the   center   that   cuts   the   line   into   two   parts,   the   positives   on   the   right   and   the   negatives   on   the   left.
If   you   have   ever   played   a   piano   you   can   compare   this   to   thinking   about   how   the   middle   C   note   splits   the   keys   into   an   upper   half   (right   hand)   and   a   bottom   half   (left   hand).
Or   if   you   have   seen   an   American   football   field   where   the   50   yard   line   splits   the   field   into   two   halves   and   two   directions,   a   positive   one   going   toward   your   endzone   and   a   negative   one   going   toward   your
opponent's.

However   you   choose   to   think   about   it,   the   important   things   to   realize   are:
1.
The   number   line   is   symmetrical.
It   looks   the   same   in   the   positive   half   as   it   does   in   the   negative   half.
2.
Zero   is   in   the   middle   and   is   neither   positive   nor   negative .
We   call   it   "non ‐ negative"   and
"non ‐ positive".
3.
Every   positive   number   has   a   negative   that   matches.
These   values   are   called   opposites .
(2   and  ‐ 2   for   example).
Opposite   numbers   are   always   the   same   distance   away   from   0.
You   have   heard   the   term   "whole   number"   before   when   referring   to   the   positive   counting   numbers.
When   we   take   all   those   whole   numbers   and   combine   them   with   zero   and   all   the   negative   counting   numbers,   we   get   what   is   called   an   integer .
In   other   words,   an   integer   is   basically   any   number   (positive   or
negative   or   zero)   that   does   not   have   any   decimal   values   after   it.
Some   examples   are   below.
Definition:    An   integer   is   any   of   the   counting   numbers   0,   1,   2,   3,   ....
and   their   opposites  ‐ 1,  ‐ 2,  ‐ 3,   ...
Integers
7            ‐ 29                  427
200%           ‐  28395             0
12
1
9.00
4
30
5
‐ 15
3.9
5/8              29.39
π
‐ 873.001
−
42
1
2
Non ‐ Integers
0.0002
97%
‐ 30.2
−
1
1
100
When   you   are   working   with   positives   and   negatives   together,   the   values   of   the   numbers   do   not   change,   it   is   merely   the   direction   that   changes.
So   if   you   see   the   number   13,   you   should   now   start   to   wonder:
Is   this   an   increase   of   13   (moving   right   on   the   number   line)?
Or   is   this   a   decrease   of   13   (moving   left   on   the   number   line)?
And   guess   what….you
have   actually   been   doing   this   already   and   not   even   knowing   it!
Take   a   look   at   what   happens   when   you   subtract   two   numbers:     10 7
Is   the   10   going   in   the   positive   direction   or   negative?
What   about   the   7?
Positive   or   negative?

Think   about   them   as   two   different   values:     +10     and    ‐  7.
If   you   move   10   in   the   positive   direction   followed   by   7   in   the   negative   direction,   look   where   you   end   up.
Starting   at   0   and   moving   +10   units   and  ‐ 7   units   puts   us   at   +3   units.
So   our   answer   10   –   7   =   3.
Another   way   to   write   this   would   be   10   +   ( ‐ 7)   =   3.
This   brings   up   an   important   rule:   subtracting   a   number   is   the   same   thing   as   adding   it’s   opposite .
10   –   7   =   10   +   ( ‐ 7)       2   –   8   =   2   +   ( ‐ 8)         ( ‐ 5)   –   6   =   ( ‐ 5)   +   ( ‐ 6)      2   –   14   =   2   +   ( ‐ 14)
It’s   all   just   about   the   direction   you   go!
Take   the   following   example:    8   +   ( ‐ 3)
Moving   8   in   the   positive   direction   and   3   in   the   negative   direction   puts   us   at   positive   5.
So   any   of   the   following   could   be   true   statements:
8   +   ( ‐ 3)   =   5     8   –   3   =   5
But   what   if   we   add   two   negatives   together?
What   about   ( ‐ 3)   +   ( ‐ 4)   ?
( ‐ 3)   +   8   =   5
This   means   we   go   3   units   in   the   negative   direction   and   another   4   units   in   the   negative   direction:

This   should   make   sense   since   we   are   starting   in   the   negatives   and   adding   ( ‐ 4)   which   goes   even   further   down   in   the   negatives.
Based   on   the   picture   on   the   number   line,   any   of   the   following   equations   could
represent   this.
See   if   you   can   use   the   number   line   to   explain   why   each   of   the   following   works:
( ‐ 3)   +   ( ‐ 4)   =  ‐ 7      ( ‐ 4)   +   ( ‐ 3)   =  ‐ 7     (  ‐ 3)   –   4   =  ‐ 7     ( ‐ 4)   –   3   =  ‐ 7
Now   let’s   try   subtraction   with   some   negatives.
Subtracting   positive   and   negative   numbers   can   be   done   the   same   way,   you   just   have   to   be   more   careful   of   the   direction   you   are   travelling   on   the   number   line.
Take   the   following   example:    6   –   9.
First   of   all,   can   you   see   another   way   to   write   this?
What   numbers   are   we   dealing   with?
Is   the   6   positive   or   negative?
Is   the   9   positive   or   negative?
This   statement   is   really   telling   us   to   combine   +6   and  ‐ 9   so   let’s   look   at   that   on   the   number   line.
We   can   see   by   counting   it   out   on   the   number   line   that   the   answer   will   be  ‐ 3.
We   also   could   have   noticed   this   by   just   looking   at   the   numbers   +6   and  ‐ 9.
Or   we   could   have   noticed   that   this   is   really   just    6   +   ( ‐ 9).
We   also   know   that   the   9   is   larger   which   means   the   answer   is   going   to   be   a   negative   number   since   there
are   more   negatives   than   positives.
How   many   more   negatives   are   there?
Well   6   and   9   are   3   apart   so   there   are   going   to   be   3   negatives   left.
(See   how   you   can   get   that   off   the   number   line?)
Let’s   do   one   more   number   line   example   and   then   look   at   some   general   rules   and   shortcuts.
What   if   we
had   to   do:    ( ‐ 3)   –   ( ‐ 7).
What   numbers   are   involved   here?
It’s   a   little   tougher   with   all   the   minus   signs   involved.
This   expression   says   to   start   at  ‐ 3   then   subtract  ‐ 7.
Hmmm.
How   do   you   subtract   a   negative?
Remember   the   examples   above   where   we   said   that   subtraction   is   the   same   thing   as   just   adding   a
number’s   opposite?
So    ( ‐ 3)   –   ( ‐ 7)   =   ( ‐ 3)   +   7
(7   and  ‐ 7   are   opposites)

Let’s   think   about   this   in   terms   of   the   number   line.
Remember   that   addition   means   we   travel   to   the   right   and   subtraction   means   we   travel   to   the   left.
But   wait,   subtracting   7   would   go   to   the   left,   we   want   to   subtract  ‐ 7   so   doesn’t   it   make   sense   that   this
would   go   to   the   right   instead?
The   two   negatives   cancel   each   other   out.
It’s   like   you’ve   turned   around   to   walk   backwards   and   then   turned   around   again   which   puts   you   back   in   a
positive   direction.
So   our   solution   would   be   ( ‐ 3)   –   ( ‐ 7)   =   4.
Remember:   Subtracting   a   value   is   the   same   as   adding   its   opposite.
Take   a   look   at   a   few   more   examples   below   where   we   have   rewritten   some   expressions   with   addition   instead   of   subtraction.
4   –   ( ‐ 5)   =   4   +   5     1   –   ( ‐ 2)   =   1   +   2     (  ‐ 13)   –   ( ‐ 6)   =   ( ‐ 13)   +   6     (  ‐ 4)   –   ( ‐ 9)   =   ( ‐ 4)   +   9
This   can   be   a   handy   little   trick   if   you   get   into   situations   where   it   looks   too   confusing   with   all   the   subtraction   and   negatives.
A   lot   of   times   it’s   easier   to   understand   addition   with   negatives   so   you   can
always   fall   back   on   this   method   if   you   need   to.
Let’s   now   take   a   look   at   some   shortcuts   so   you   don’t   have   to   always   rely   on   the   number   line   to   add   and   subtract   numbers.
There   are   many   shortcuts   and   patterns   to   combining   positive   and   negative   numbers   and   it   is   really   up   to   you   to   find   the   one   that   makes   the   most   sense.
Some   people   like   to   memorize   rules   and   others   just   like   to   use   the   number   line   to   count.
It’s   up   to   you,   but   the   key   is   to   PRACTICE,   PRACTICE,
PRACTICE   so   you   can   eventually   do   them   without   needing   to   rely   on   a   rule   or   a   number   line.
Here   is   a   common   shortcut   for   adding   and   subtracting   two   integers.
You   need   to   remember   the   following   rules:
1.
Before   you   start,   turn   any   double   negatives   into   positives   (ie.
2   –   ( ‐ 6)    Æ    2   +   6   )
2.
If   the   signs   are   the   same,   you   add   the   values.
If   the   signs   are   different,   you   subtract   the   values.
(big   –   small)
3.
Always   keep   the   sign   of   the   biggest   number.
For   this   to   work,   you   just   ignore   the   operation   and   focus   on   the   numbers   you   are   combining.
BE   CAREFUL….This
will   always   work   (even   with   non ‐ integers),   but   this   ONLY   APPLIES   TO   2   NUMBERS   AT
A   TIME   so   be   careful   with   larger   problems.

Here   are   some   examples:
‐ 15   –   ( ‐ 5)
1.
Turn   any   double   negatives   into   positives    Æ
‐ 15   –   ( ‐ 5)    =   ‐ 15   +   5
2.
We   have  ‐ 15   and   +5   so   the   signs   are   different.
We   will   subtract.
15   –   5   =   10
3.
Keep   the   sign   from   the   larger   number.
The   15   was   bigger   and   it   was   negative,   so   our   answer   is   negative.
‐ 10
7   +   ( ‐ 19)
1.
Turn   any   double   negatives   into   positives    Æ          none,   skip   this   step
2.
We   have   +7   and  ‐ 19   so   the   signs   are   different.
We   will   subtract.
19   –   7   =   12
3.
Keep   the   sign   from   the   larger   number.
The   19   was   bigger   and   it   was   negative,   so   our   answer   is   negative.
‐ 12
‐ 3  ‐  8
1.
Turn   any   double   negatives   into   positives    Æ          none,   skip   this   step
2.
We   have  ‐ 3   and  ‐ 8   so   the   signs   are   the   same.
We   will   add.
3   +   8   =   11
3.
Keep   the   sign   from   the   larger   number.
The   8   was   bigger   and   it   was   negative,   so   our   answer   is   negative.
‐ 11

‐ 22   +   ( ‐ 5)
1.
Turn   any   double   negatives   into   positives    Æ          none,   skip   this   step
2.
We   have  ‐ 22   and  ‐ 5   so   the   signs   are   the   same.
We   will   add.
22+   5   =   27
3.
Keep   the   sign   from   the   larger   number.
The   22   was   bigger   and   it   was   negative,   so   our   answer   is   negative.
‐ 27
17 ‐  ( ‐ 9)
1.
Turn   any   double   negatives   into   positives    Æ
17   –   ( ‐ 9)    =    17   +   9
2.
We   have   +17   and   +9   so   the   signs   are   the   same.
We   will   add.
17   +   9   =   26
3.
Keep   the   sign   from   the   larger   number.
The   17   was   bigger   and   it   was   positive,   so   our   answer   is   positive.
26
13 +   ( ‐ 7)
1.
Turn   any   double   negatives   into   positives    Æ          none,   skip   this   step
2.
We   have   +13   and  ‐ 7   so   the   signs   are   different.
We   will   subtract.
13   –   7   =   6
3.
Keep   the   sign   from   the   larger   number.
The   13   was   bigger   and   it   was   positive,   so   our   answer   is   positive.
6

This   is   the   only   shortcut   that   we   will   address   here   since   the   more   we   do   here,   the   easier   it   is   to   get   confused.
There   are   others   out   there   but,   again,   it’s   a   matter   of   just   practicing   as   much   as   you   can   with
whichever   method   makes   the   most   sense   to   you.
If   you   like   the   number   line,   use   it.
If   you   like   the   shortcut,   use   it.
If   you   know   of   other   ways   or   if   you   come   up   with   your   own   method,   fantastic….use
it.
Whatever   you   do,   just   make   sure   you   practice,   practice,   practice.
This   is   a   huge   skill   that   will   come   up   for   the   rest   of   your   math   career   so   put   in   the   time   now   to   really   master   it   and   feel   comfortable   with   it   so   you   don’t   have   to   waste   your   energy   in   algebra   deciding   whether   your   answers   should   be   positive   or   negative.
That’s   all   for   part   1.
We   will   address   multiplication   and   division   with   negatives   in   part   2.
The   rules   for   multiplication   and   division   are   very   simple   but   it   is   really   easy   to   confuse   them   with   the   rules   for   addition   and   subtraction   so   we   suggest   practicing   this   skill   a   great   deal   before   you   jump   into   part   2   to   avoid   getting   your   rules   mixed   up.
Remember   to   slow   down,   check   that   your   answers   make   sense,   and   don’t   be   afraid   to   draw   a   number
line   on   your   paper   if   you   ever   get   confused   about   whether   a   value   should   be   positive   or   negative.
There’s   nothing   more   frustrating   than   going   through   all   the   correct   steps   on   a   test   or   assignment   and
messing   up   on   the   positive/negative   sign.
Now   is   the   time   to   really   pay   attention   to   detail   in   your   work.
Good   luck!
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