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 10^b 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.

10^b alone is an accurate number only when a=1, when e.g. 10³ = 1000 exactly.
However, just the 10^b 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 10^b 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 :

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 10^b 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 "10^b" 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.

For example: In everyday conversation one may ask a guest how many teaspoons of sugar he or she would like in his or her cup of tea. The guest may reply "two, please" or even "three" and it will be understood from the context that he or she would like two (or three!) teaspoons of sugar, not two or three grains of sugar - or two or three handfuls of sugar, etc.. In science it is not acceptable to make such assumptions, one must always specify the units of measure being used, e.g. the distance is 2 mm (millimetres), 2 " (inches), 2 km (kilometres) or 2 miles, etc.. "Standard Form" takes this rule even further by stipulating use of particular types of units of measure for particular types of quantities, e.g. length, volume, speed, pressure, temperature, etc.. However it is useful to know about other units as well because preference can vary from place to place and between different uses, e.g. inches are now rarely used in science in Europe but are still used in some workshops and dots-per-square-inch (dpi) is widely used as the standard for print resolution, possibly thanks to American manufacturers!

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
	(for measurement of length)	(as a scientific number)	(as a decimal number)	(expressed in words)
femto-	femtometre (fm)	10-15m	0.000000000000001m	one quadrillionth	Rarely used.
pico-	picometre (pm)	10-12m	0.000000000001m	one trillionth	Used in particle and quantum physics, e.g. sizes of atoms approx. 60 pm to 520 pm diameter.
	ångström (Å), also "angstrom"	10-10m	0.0000000001m		Used to state sizes of atoms, lengths of chemical bonds, wavelengths of radiation, and in biology.
nano-	nanometre (nm)	10-9m	0.000000001m	one billionth	Commonly used to specify wavelengths in optics, esp in the "visible" region of the electromagnetic spectrum..
micro-	micrometre (μm) also "micron"	10-6m	0.000001m	one millionth	Sometimes used to specify infrared wavelengths and for some cells in biology, e.g. sizes of blood cells.
milli-	millimetre (mm)	10-3m	0.001m	one thousandth	Widely used, sometimes even when cm seem the more obvious choice, so always specify and check units.
centi-	centimetre (cm)	10-2m	0.01m	one hundredth	In common use, especially in non-technical situations e.g. paper sizes..
deci-	decimetre (dm)	10-1m		one tenth	Rarely used. Approx 4" (4 inches), which is one "hand", the units in which the height of horses and ponies is measured.
	metre (m)	100m = 1m	1m	one	The standard unit for length according to the International System of Units (SI).
deca-	decametre (km)	101m = 10m	10m	ten times	Very rarely used.
hecto-	hectometre (km)	102m	100m	one hundred times	Very rarely used.
kilo-	kilometre (km)	103m	1,000m	one thousand times	Widely used to indicate distances along highways in continental Europe (not in UK, where miles are used).
mega-	megametre (Mm)	106m	1,000,000m	one million times	Rarely used outside of science fiction. Beware similaity in symbol between mm (millimetre) and Mm (megametre).
giga-	gigametre (Gm)	109m	1,000,000,000m	one billion times	Rarely used outside of astronomy! Beware similaity in symbol between gm (gram) and Gm (gigametre).
tera-	terametre (Tm)	1012m	1,000,000,000,000m	one trillion times	Very rarely used.
peta-	petametre (Pm)	1015m	1,000,000,000,000,000m.	one quadrillion times	Very rarely used.

*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.