Pupils  should  see  clearly  that  numbers  are  not  inherently
positive  or  negative,  but  that  the  terms  are  relative,  denoting
Nature  of  ^^^^  classes  of  number  of  opposite  "  sense,"  —  that
WegauVe^*^  both  classes  must  exist  or  neither,  and  that  they
Numbers.  jj^^^y  y^^  interchanged,  the  choice  of  which  shall  be
positive  and  which  negative  being  a  matter  of  convenience.
If,  from  the  wording  of  the  problem  or  otherwise,  one  has  an
expectation  of  the  sense  of  the  result,  there  is  a  tendency  to
take  that  sense  as  positive.  This  is  due  in  part  to  the  name
(a  misnomer)  and  in  part  to  the  preference  given  to  positive
numbers  by  understanding  numbers  to  be  positive  when  no
sign  is  indicated.
A  number    b  is    said    in    mathematics    to    be    less  than  a

number   a   if    there    exists   a    positive     number  c    such    that

b  -\-  c  ^=1  a.     This  definition  includes  the  every-day
Numbers  ^        ^

"Less than       concrete  idea  with  which  the  pupils  are  familiar;

it  also  defines  when  a  negative  number  is  "■  less  "
than  another.     If  the  numbers  are  arranged  on  a  straight  line,

every  number  is  less  than  each
-1*  -z  -2  -i   0   i   %    ^    k   b  number  to  the  right  of  it.     Ac-

cording to  the  definition,  every
negative  number  is  less  than  zero.  This  says  simply  that  if  I
go  in  the  negative  sense  a  certain  distance,  I  must  go  in  the
positive  or  increasing  sense  to  get  to  the  starting-point.
