CHAPTER 1 MATHEMATICS AND UNITS OF MEASUREMENT Mathematics  is  the  Engineering  Aid’s  basic tool.  The  use  of  mathematics  is  found  in  every rating  in  the  Navy,  from  the  simple  arithmetic of   counting   for   inventory   purposes   to   the complicated  equations  encountered  in  computer and engineering designs. In the Occupational Field 13  ratings,  the  Engineering  Aid  is  looked  upon as  superior  in  knowledge  when  it  comes  to  the subject   of   mathematics,   which   generally   is   a correct assumption; however, to be worthy of this calling, you have the responsibility to learn more about  this  subject.  Mathematics  is  a  broad  science that  cannot  be  covered  fully  in  formal  service school  training,  so  it  is  up  to  you  to  devote some  of  your  own  time  to  the  study  of  this subject. The  EA  must  have  the  ability  to  compute easily,  quickly,  systematically,  and  accurately. This  requires  a  knowledge  of  the  fundamental properties of numbers and the ability to estimate the   accuracy   of   computations   based   on   field measurements  or  collected  field  data.  To  compute rapidly,  you  need  constant  practice  and  should be  able  to  use  any  available  device  to  speed up  and  simplify  computations.  In  solving  a mathematical   problem,   you   should   take   a different  approach  than  you  would  if  it  were simply   a   puzzle   you   were   solving   for   fun. Guesswork has no place in its consideration, and the  statement  of  the  problem  itself  should  be devoid  of  anything  that  might  obscure  its  true meaning. Mathematics is not a course in memory but  one  in  reasoning.  Mathematical  problems should be read and so carefully analyzed that all conditions   are   well   fixed   in   mind.   Avoid   all unnecessary   work   and   shorten   the   solution wherever  possible.  Always  apply  some  proof  or check  to  your  work.  Accuracy  is  of  the  greatest importance;  a  wrong  answer  is  valueless. This  chapter  covers  various  principles  of mathematics. The instructions given will aid the EA in making mathematical computations in the field  and  the  office.  This  chapter  also  covers  units of  measurement  and  the  conversion  from  one system to the other; that is, from the English to the metric system. FUNDAMENTALS  OF MATHEMATICS MATHEMATICS  is,  by  broad  definition,  the science that deals with the relationships between quantities  and  operations  and  with  methods  by which  these  relationships  can  be  applied  to determine   unknown   quantities   from   given   or measured  data.  The  fundamentals  of  mathematics remain the same, no matter to what field they are applied.  Various  authors  have  attempted  to classify mathematics according to its use. It has been  subdivided  into  a  number  of  major  branches. Those   with   which   you   will   be   principally concerned are arithmetic, algebra, geometry, and trigonometry. ARITHMETIC  is  the  art  of  computation  by the use of positive real numbers. Starting with the review of arithmetic, you will, by diligent effort, build  up  to  a  study  of  algebra. ALGEBRA is the branch of mathematics that deals with the relations and properties of numbers by means of letters, signs of operation, and other symbols. Algebra includes solution of equations, polynomials, verbal problems, graphs, and so on. GEOMETRY  is  the  branch  of  mathematics that  investigates  the  relations,  properties,  and measurement of solids, surfaces, lines, and angles; it also deals with the theory of space and of figures in space. TRIGONOMETRY  is  the  branch  of  mathe- matics that deals with certain constant relation- ships that exist in triangles and with methods by which   they   are   applied   to   compute   unknown values  from  known  values. STUDY  GUIDES Mathematics is an many   books   on   the exact science, and there are subject.   These   numerous l-l