Scientific Notation
See: Square Roots and Squares
See: Rules of Exponents 

Scientific Notation is a way to work with very large and very small numbers.
It is a way to represent a number with fewer digits, but this also means the possible loss of 'accuracy'. 

A standard level for expression a number in Scientific Notation is:
N.ddd x 10^P

N is a single digit: (1-9).
ddd is the next three digits in the number.
The third 'd' might need to be ROUNDED up based on the next value.
The P is the POWER or the number of times the Decimal was moved.
(The 'x' before the Times 10 is for clarification and readability)
The Letter 'E' or 'e' is sometimes used rather than '10' since without the 'x' times since the N.ddd might run together with the 10 and be confusing;
4.743 x 10^8  could be: 1.783 10^8 without the (x); so 1.783 e^8 is also valid.

Some problems may state the desired number of 'decimals' to have in the answer.

WARNING: Be careful to see if the problem is asking for the answer to be in STANDARD FORM or in SCIENTIFIC NOTATION.  

Two key words often used in problems are:
How much Bigger (or Smaller).  This is a Difference (Subtraction Problem)

How many times Larger (or Smaller) are the values: Multiplication.
       Larger = Smaller x Multiple   or  Multiple = Larger / Smaller

LARGE NUMBERS
1,000 -> 1 x 10^3 (no real benefit here)
1,234 -> 1.234 x 10^3 (no real benefit here)
5,134,128,000 -> 5.134 x 10^9
1,278,231,682 -> 1.278 x 10^9
48,922,176 -> 4.892 x 10^7
158,921,754,180,672 -> 1.589 x 10^14
479,584,207 -> 4.796 x 10^8 (the '5' is rounded up to '6' because of the '8')
50,104,864 -> 5.01 x 10^7
410,085,671 -> 4.1 x 10^8

Do not be confused if a number has decimals:
4,652,129.87 -> 4.652 x 10^6. (you moved the decimal 6 places).

VERY SMALL NUMBERS
Small numbers (Decimal values)
Small numbers can also be expressed in Scientific Notation but the Power is NEGATIVE.
For large numbers, it is relatively easy to find the power because of the Commas (,) in the numbers for place value. Decimal numbers are harder to count the Zeros.
Trailing values are dropped.
The format is still: N.ddd x 10^-P
.58931231 -> 5.893 x 10^-1
.59629781 --> 5.963 x 10^-1 (rounding the 3rd decimal)

There may be a ZERO in front of the decimal. This is usually put in since numbers do not have a leading Zero, it emphasizes that there is a decimal: 0.58931231 

Usually the numbers are really small with many zeros:
.007281021 -> 7.281 x 10^-3
.000000760200021 -> 7.602 x 10^-7
.00087912000 -> 8.791 x 10^-4

 
MATH OPERATIONS WITH SCIENTIFIC NOTATION.

Besides converting numbers to Scientific Notion, there may be problems where you have to add, subtract, multiply and divide numbers.
The 'TRICK' is that the final answer may also be in Scientific Notation: N.ddd x 10^P
(5.4 x 10^3) x (2.6 x 10^5) 
Regroup: 5.4 x 2.6 x 10^3 x 10^5
Rule of Exponents:  Multiplying: Add the exponents:  10^8 
5.4 x 2.6 -> 14.04:  But, This is not in the correct form of: N.ddd 
So: 14.04 becomes: 1.404 x 10^1.
The final answer is 1.404 x 10^1 x 10^8 --> 1.404 x 10^9

While you could expand these numbers, other problems would have too many zeros or be too big!
2.6 x 10^5 -->  260,000
5.4 x 10^3 -->   x 5,400
PRODUCT: 1,404,000,000 --> 1.404 x 10^9

When Adding or Subtracting, The Powers need to be the same.
4.23 x 10^5  - 1.7 x 10^3
423,000 - 1,700  (Standard Form) If the powers are really big or small, you can not expand the values easily)
421,300 --> 4.213 x 10^5
or
423 x 10^3 - 1.7 x 10^3 (Get to have the same powers)
421.3 x 10^3   (421.3 is not in Scientific Notation: N.ddd)
4.213 x 10^2 ^  x 10^3
4.213 x 10^5 
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External Links:

Worksheets:

Notes:
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Revised: April 29, 2024

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