Standard form 2

Standard Form is a (sometimes difficult and often misunderstood) way of representing numbers which are 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, and Standard Form usually uses both skills (rounding correctly, identifying the order of magnitude)

For example, an astronomic unit, the mean diameter of the earth’s orbit, is 149 597 870 000 metres, or 149 597 870 kilometres, or, in words, just less than 150 million kilometers. In Standard Form this is 1.4959787 x 1011 m/s to 8 sig.fig. So the 1.49… indicates the significant figures and the x 1011 gives the size of the number – where the decimal point belongs. You can see that it helps if I space the digits in threes. With long numbers it does help readability to space digits in threes and I recommend this in your writing. Notice also that Standard Form is often rounded as well, so this large number is 1.496 x 1011  to 4 sig.fig, 1.50 x 1011 to three sig.fig. and 1.5 x 1011 to 2 sig fig.


Here are some exmples for you to consider. Where I have given answers, make sure you can make your cakulator produce the number in that form. Some calculators will not convert numbers until you make them answers; put them in and multiply by one (go x 1 =) before you start fiddling.

1.  A nautical mile is 1 852 metres. Write this in (i) metres in Standard Form (ii) kilometres,

2.  An imperial mile is 1.609 344 x100 km. Write this (i) in metres (ii) in metres to 3 sig.fig.

3.  An atmosphere, atm, is 101 325 kg/sq.m. Write this in (i)  Standard Form (ii) tonnes/sq.m.

4.  A fathom is 7 200 / 3 937 metres. Convert this (i) by division to 7 sig.fig.  and then (ii) to 3 sig.fig.

If you can, see how to do the conversion without the machine. Best practice is to look at what the machine is doing and recognise when you have a wrong answer. That is our long-term objective.

Small numbers are in some way easier, because written in full they have the right number of zeros before the first non-zero figure: for exmple a tablespoonful (tbsp) is 0.000 014 79 of a cubic metre to 4 sig.fig. The first five zeros are shown by the -5 in 1.479 x 10-5 .


5.  A gallon is 4.546 x 10-3 cubic metres [m³]. Write this as (i) a decimal (ii) in litres (you might know this number).

6.  A  teaspoon (tsp) is 4.929 x 10-6 m³. Write this as a decimal (i) in m³ (ii) in litres and (ii) ml.

7.  4.470 x 10-1 m/s = 1 mph. Write this (i) as a decimal (ii) as km per second and (iii) kmph 

8.  A fluid ounce is 0.000 028 413 07 cubic metres. Convert this (i) to Standard Form, (ii) to 3 sig.fig in Standard Form and (iii) try to convert it to teaspoonfuls using question 6.

kmph = kilometres per hour;  mph = miles per hour;  m/s = metres per second. m³ = cubic metres. 
SF & Std Fm mean Standard Form;  sig.fig and s.f. mean significant figures. I have avoided using s.f. and S.F..

Did you read all the words? If you did, what two words were spelled incorrectly? Where did I use a plural verb (‘are’) when it should have been (‘is’)  singular?
                                                                                                         Ring these on your copy.

Question: How many teaspoonsful make a tablespoonful? It must be a whole number !            Related sheets: 0320, 1115, 1116,



Answers:

1.    1.852 x 10³ m   1.852 km

2.   1609.344 m,  1610m  (1.61x10²)

3.   1.01325x 10⁵, 101.325 t/sqm, (1.01325 x 10²)

4.   1.82880365761 > 1.828 804 (7s.f), 1.83 (3s.f.)   

5.   0.004546,  4.546 x 100  or   4.546

6.   0.000 004 929m³,   0.004 929 litres,  4.929 ml

7.   0.4470 m/s = 4.47x10-4km/s   1.609kmph

8.   2.841 307 x 10-5 ,  2.84 x 10-5,     5.764 495 tsp  /fl.oz.

cakulator, exmples  - spelling / typos

top line, standard form …is a way which IS used…


Units and Measures page explains tablespoons ands teaspoons – the answer is three or four depending whether you use US or UK units. Find the page by googling "site:scoins.net tablespoon"

© David Scoins 2017