phy1020.DVI

Appendix G

Mathematical Subtleties

• When taking the square root of both sides of an equation, a ̇sign must always be introduced. For
  example:

x^2 Da ) xD ̇

p

a

Both roots may be valid, or, depending on the problem, it may be that one root or the other may be

rejected on mathematical or physical grounds.

• Dividing an equation through by a variable may result in losing roots. For example, suppose we have

x^2 axD 0

Dividing through by the variablexwill result in one solution,xDa; the solutionxD 0 has been lost.

Instead of dividing through by the variablex, the proper procedure is tofactor outanx:

x.xa/D 0

Since the product on the left-hand side is zero, it follows that eitherxD 0 orxaD 0 , and we retain

both roots.

• The relation
  p
  x
  p
  yD
  p
  xy (G.1)

is valid only forx;y    0.

• Some mathematical conventions:

?1isnotconsidered a prime number.

?0ŠD 1

?0^0 D 1

?Towers of exponents are evaluated from the top down:ab

c

Da.b

c/