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Scientific Numbers

Scientists use a standard short form of writing numbers in order that both very small and very large numbers can be expressed clearly in a small number of figures.
If all numbers were written out in full then it would not be obvious at a glance how many figures formed the number and so how large the quantity were. That is especially true of numbers that include figures both before and after a decimal point.

The method or "style" of expressing numerical quantities described here is referred to by several terms, including:

  • Standard Form
  • Scientific Numbers
  • Scientific Notation
  • Exponential Notation.

 

What do Scientific Numbers look like ?

Numbers written in "standard form" or "scientific notation" generally look like:

a x 10b followed by the units of measure, e.g. m

Sometimes they may be even shorter because the "a" (in this example) is often omitted if its value is one.
That makes sense because anything "multiplied by one" is unchanged, so "1 x" is unnecessary.

10b alone is an accurate number only when a=1, when e.g. 103 = 1000 exactly.
However, just the 10b part on its own provides useful information about the scale of the number.

 

What do Scientific Numbers mean ?
Put another way, here's a description of an easy way to understand them ...

Scientific numbers can be thought of as two parts:

The exponent (which is "b" in the example above) provides information about the order of magnitude.
That means it indicates the range of values within which the number falls, e.g. approx. 10-100, 100-1000, 1000-10000 etc..
It actually indicates ranges from the "round" numbers (divisible by ten) up to just less than the next round number, so for example 10-99, or even more accurately 10 to 99.999 etc.. The range 10-99.999 is represented by b=1, the range 100-999.999 is represented by b=2, the range 1000-9999.999 is represented by b=3, and so on. In all cases the limit of the range is "just less than" the next integer that takes the form of a one followed by one zero more than the number that defined the start of the range.

It works like this:

  1. For b=1 ... 101 = 10
  2. For b=2 ... 102 = 10 x 10 = 100
  3. For b=3 ... 103 = 10 x 10 x 10 = 1,000
  4. For b=4 ... 104 = 10 x 10 x 10 x 10 = 10,000
  5. For b=5 ... 105 = 10 x 10 x 10 x 10 x 10 = 100,000
  6. For b=6 ... 106 = 10 x 10 x 10 x 10 x 10 x 10 = 1,000,000 ... and so on.

The pattern is obvious. Some people like to remember this by thinking of the value of the exponent "b" as the number of zeros after the "1" when the full number is written out in its long form. Others prefer to remember it as the numbers of times one writes "10", with multiplication symbols in between each and an "equals" sign at the end. Either way, the result is the same!

So, the exponent indicates the scale of a scientific number by specifying a "round number" consisting of one "1" followed by a specific number of zeros.

The coefficient (which is "a" in the example a x 10b given above) provides information about the actual value of the number.

For example, in the case of the number 6.7 x 103, a=6.7 and b=3.

According to the above, 103 = 10 x 10 x 10 = 1,000.
Therefore it is clear from b that the value is in the thousands, so in the range 1000-9999.999
The value of a indicates that the exact number is 6.7 x 1000 = 6700.

 

Scientific Notation for numbers less than zero

The simple explanation above works well for numbers greater than zero, for which the exponent b is also greater than zero.

Scientific Notation represents numbers smaller than zero by use of negative values of the exponent b.
Negative numbers are represented by use of negative values of the coefficient a.

Here are some examples:

 

Positive Numbers

Negative numbers

Values less than zero

0.0056 = 5.6 x 10-3
So, a=5.6 and b=-3

-0.0008 = -8 x 10-4
So, a=-8 and b=-4

Values greater than zero

9,000,000 = 9 x 106
So, a=9 and b=6

-780 = -7.8 x 102
So, a=-7.8 and b=2

Here's how this can be explained in the case of negative values of b, in a similar way to that for positive values :

Remember that for positive values of the exponent b :

As above ...

The equivalent negative values of b are:

Explanation: The system for negative values of b is very similar to that for positive values of b.
The difference is due to the effect of the negative , or "minus" sign, which means that everything on the right-hand-side of the equation when b is positive forms the denominator (lower section) of a fraction of which the numerator (upper section) has the value "1" when the value of b is negative. Here are some examples:

Positive values of b
Negative values of b
 

For b=1 ... 101 = 10

 
 

For b=2 ... 102 = 10 x 10 = 100

 
 

For b=3 ... 103 = 10 x 10 x 10 = 1000

 

Another way to describe this in words is to say that 10b is the reciprocal of 10-b.

The two columns above suggest an easy way to switch between "scientific numbers" and "decimal numbers":

  1. Think of the number written as a superscript, so the "b" in the example "10b" as indicating the number of zeros to write, and
  2. Remember that the sign (+ or -, indicating "plus" or "minus" on that number) tells you where to put the decimal point.

In the row at the bottom of the table above, b=3 on the left-hand-side and b=-3 on the right-hand-side.
When 103 = 1000 there are 3 zeros and the decimal point goes at the end (because the decimal point goes at the end it may not be written at all but it is still understood to be present).
When 10-3 = 0.001 there are 3 zeros and the decimal point goes after the first zero.

If you check the other examples you will see the same pattern.

 

The importance of specifying the correct Units

The information above is just about numbers.

In science it is usually necessary to state the units (that is, what type of property the numbers refer to) whenever quantites that describe a particular type of amount, e.g. of length, area, volume, temperature, and so on are mentioned e.g. in a report or homework or exam answer.

 

Names (Words!) for particuar sizes / quantities

There is a useful set of standard prefixes used to denote the scale of quantities.

The following uses, as examples, measures of units of length.
The standard scientific unit (SI) of length is the metre*.

Prefix
Unit
Proportion of one metre
Comments
 

femto-

femtometre (fm)

10-15m

pico-

picometre (pm)

10-12m

 

ångström (Å), also "angstrom"

10-10m

 

nano-

nanometre (nm)

10-9m

micro-

micrometre (μm)
also "micron"

10-6m

milli-

millimetre (mm)

10-3m

centi-

centimetre (cm)

10-2m

deci-

decimetre (dm)

10-1m

 

 

metre (m)

100m = 1m

deca-

decametre (km)

101m = 10m

hecto-

hectometre (km)

102m

kilo-

kilometre (km)

103m

mega-

megametre (Mm)

106m

giga-

gigametre (Gm)

109m

tera-

terametre (Tm)

1012m

peta-

petametre (Pm)

1015m

*There are two spellings of this word. "Metre" is the standard spelling in th UK and many Commonwealth countries.
"Meter" is commonly used in "American English". Both spellings have the same scientific meaning and the abbreviation "m".
Beware that the same abbreviation is used for miles, e.g. on both British and American roadsigns.

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