Standard Form

Standard form (or scientific notation) is used to write very large or very small numbers concisely. Every number in standard form must follow the rule: A × 10n, where A is a number between 1 and 10 (1 ≤ A < 10) and n is a whole number power.

a) Converting large ordinary numbers to standard form:

45,000 becomes 4.5 × 104
781,000,000 becomes 7.81 × 108
9,000,000 becomes 9 × 106


b) Converting small ordinary numbers (decimals) to standard form:

0.0034 becomes 3.4 × 10-3
0.00000782 becomes 7.82 × 10-6
0.09 becomes 9 × 10-2


c) Converting standard form back to ordinary numbers:

3.2 × 105 becomes 320,000
6.07 × 10-3 becomes 0.00607
4 × 10-4 becomes 0.0004


d) Multiplying and dividing in standard form:

(2 × 103) × (3 × 104) = 6 × 107     (Multiply front numbers, add the powers)
(8 × 106) ÷ (2 × 102) = 4 × 104     (Divide front numbers, subtract the powers)
(4 × 105) × (3 × 102) = 1.2 × 108     (12 × 107 is adjusted to keep the front number below 10)


e) Adding and subtracting in standard form:

(4.2 × 105) + (3.5 × 105) = 7.7 × 105     (Powers match: simply add the front numbers)
(3 × 104) + (5 × 103)  →  (3 × 104) + (0.5 × 104) = 3.5 × 104     (Powers do not match: adjust one first)



The rules
  1. The Front Number Restriction
    The number out in front (A) must be at least 1, but strictly less than 10.
    For example, 45 × 103 is not in standard form because 45 is larger than 10. It must be written as 4.5 × 104.
    Similarly, 0.3 × 10-2 is incorrect because 0.3 is less than 1. It must be adjusted to 3 × 10-3.

  2. Positive Powers vs Negative Powers
    A positive power tells you how many places the decimal point shifts to the right to make a large number.
    A negative power tells you how many places the decimal point shifts to the left to make a tiny decimal.
    Tip: 10-3 does not mean a negative number, it means a very small fractional number (one thousandth).

  3. Multiplying and Dividing (Index Laws)
    When multiplying expressions, multiply the decimal parts together and add the indices (powers).
    When dividing expressions, divide the decimal parts and subtract the second index from the first.
    Always verify your final answer; if the decimal multiplication results in a number 10 or greater, shift the decimal point left and increase the power by 1.

  4. Adding and Subtracting Pitfall
    You cannot add or subtract the front coefficients unless the powers of 10 are exactly identical.
    If the powers are different, change the expression with the lower power into a standard decimal form that matches the higher power before computing.


Difficulty levels:

This is a table of set questions that do not change, and as such as with all other lists in this system the questions are unaffected by the difficulty level selected.