Exponential Notation

• Exponential notation is an alternative method of expressing numbers.
• Exponential numbers take the form aⁿ, where a is multiplied by itself n times.
  A simple example is 8=2³=2×2×2.
• In exponential notation, a is termed the base while n is termed the power or exponent or index.
• Scientific notation is a specific example of exponential numbers, 10 is almost always used as the base number.
  Thus 10³ means 10×10×10, while 10^-3 is the notation for the reciprocal of 10³ namely 1/1000.
• The other name for this mathematical format is standard form ( you may have come across this in GCSE mathematics).
• Expressing numbers which are not whole powers of 10 in scientific notation often requires a further multiplier, termed the coefficient (C), giving the expression in the form C ×10ⁿ.
  For example, 5 ×10³ is the scientific notation for the number 5000, while 3.25×10²is the scientific notation for the number 325.
  Similarly, 3.25×10^-2 would represent
  

  3.25 × 1 100 = 3.25 100 =0.0325.
• Convention
  It is scientific convention to express all such numbers as coefficients between 1 and 10 and followed by an appropriate power of 10.
  Examples
  

  35200 = 3.52 ×104 NOT 352 ×102 NOT 35.2 ×103 1750 = 1.75 ×103 NOT 175 ×10 NOT 17.5 ×102 0.0436 = 4.36 ×10-2 NOT 0.436 ×10-1 NOT 43.6 ×10-3
• You should use such scientific notation whenever you express very large or very small numbers - it is a recognized form of "shorthand", and it avoids spurious accuracy e.g. writing 9 000 000 suggests that the number is exactly 9 million, in contrast to 9.0 ×10⁶ which suggests no such accuracy beyond the first decimal place of the coefficient.