STANDARD FORM 20040320
Standard Form is used to represent large and small numbers in a consistent way. Standard Form shows exactly one non-zero digit before the decimal point and is then multiplied by a power of ten to indicate the size. Thus the absolute number and the general size have been separated. Rounding is carried out as for significant figures (indicated by sig.fig., SF, s.f., Std Fm), and usually Std Fm requires you to think about both the rounding and the power of ten.
For example, the speed of light is 299,792,458 m/sec to the nearest whole unit. This is 2.99792 x 10⁸ m/s to 6 sig.fig. in SF. The index on the ten indicates how many places the decimal point has been moved. With large numbers it helps to space digits in threes and remember a few landmarks; with small numbers there is an easy way to do this if you keep the leading zero (before the decimal point). So 0.00417 (look, three zeroes) becomes 4.17x10⁻³. That number squared is 1.739x10⁻⁵ to 4 sig.fig., 0.000 017 39, five written zeroes.
Do these for necessary practice; use standard form throughout.
The reciprocal button may prove useful on your calculator [labelled x-1 or 1/x].
1 Write the speed of light to a) 7 sig.fig. b) 4 sig.fig c) 3 sig.fig.
2 a) A picometre is a millionth of a millionth of a metre. Write this to 3 sig.fig.
b) How many micrometres in a kilometre?
c) How many millimetres in a mile? [to 7 sig fig and still in Std Form].
3 A second is 9 192 631 770 oscillations of a particular type of caesium crystal. Write this number to a) 3 and b) 4 sig fig (in Standard Form).
How long is a single oscillation cycle to c) 1, d) 3, e) 4 sig fig?
4 I say a year is 3.6524x10² days to 5 sig.fig.
a) Write this number to 3 sig fig
b) How long is this version of a year in seconds?
The smallest my year could be – and still correct – is 365.235 and the biggest is just less than 365.245.
c) How long are these two lengths of time in seconds? So now can you answer why is it still correct to only use 5 sig fig?
d) How many years per day?
e) years per hour?
f)f) years per second?
So just why is a baby nettle like 10-12?
Answers:
1. 2.99,792,500m/s, 3.000x108 m/s, 3.00x108 m/s,
2. 1 picometre = 1x10-12metres. 1km = 109 micrometers, 1609344 mm/mile, 1.609344x106 mm/mile.
3. 9.19x109 , 9.193x109 , 1.08782775708x10-10, which is 1x10-10, 1.09x10-10, 1.088x10-10 seconds.
4. 3.6524x10² days is 3.65x10² days to 3 sig.fig.
3.6524x10² x24x60x60 = 31556736 seconds (8s.f.)
3.65235x10² x24x60x60 = 31556304 seconds (8s.f.)
3.65245x10² x24x60x60 = 31557168 seconds (8s.f.) all three are 3.1556x107.
1/365.24 = 2.7379x10-3 yrs/day = 1.1408x10-4 yrs/hr = 3.1690x10-8 yrs/sec
they’re both small numbers!!
