Read about . . . |
Standard Form |
e-Format |
Formatting and Accuracy in JavaScript |
Significant Figures |
Spurious Accuracy |
Significant Figures v. Decimal Places |
Standard form Consider the number 47350000000000 Yes, it is a big number, but it is not possible to appreciate just how big it is at a glance. Dividing it up in the usual way with commas 47,350,000,000,000 is some help but not a lot, especially if it is required to compare it with another number which is (very probably) not on the same page. (Remember that the SI preferred way is to use spaces and NOT commas)
The mathematicians/scientists who have to work with large numbers devised a system known variously as Standard Form or Scientific Notation. This requires that the first number is always given a value between 1 and 10 (it can also be 1); followed by the number of 10's required to multiply it so as to make it the correct size. The above number then becomes 4.735 x 10^{13} The '13' is known as an 'index number' and most people finish up remembering that the index number shows how many places the decimal point has to be moved to the right putting in the 0's as necessary. And it works. |
The great merit of this system for users is that the associated arithmetic becomes very easy. For readers its benefit is that comparative sizes can be seen at a glance. For example (a) 9.987 x 10^{15} . . . (b) 4.103 x 10^{21} . . . (c) 5.7332 x 10^{18} can be ranked in size order, largest to smallest as (b, c, a) merely by looking at the index values for each.
What about very small numbers? Something like
0.000 000 000 000 000 037 92 In Standard Form that becomes 3.792 x 10^{-17} The minus sign is the important thing to notice here. It indicates that we actually have to divide by 10 (17 times) An explanation of why this is so will not be attempted here. (For the curious: Find out about the Laws of Indices.) The simple thing to remember is: minus means the decimal point must be moved to the left putting in 0's as necessary. Again - it works. |
So, what about the e-Format? The IEEE [= Institute of Electrical and Electronic Engineers (US) ] used the above idea but changed the way it was written. Essentially they said that the index numbers being written 'up in the air' like that were an inconvenience. Also, 'everyone' knows about the 'x10' bit so, we will replace the 'x10' with an 'e' and put the index number down on the line with the rest. Like this 6.1053 x 10^{17} becomes 6.1053e17 (In IEEE numbers the 'e' signals that the number which follows is an exponent; and 'exponent' is another word for an 'index number') So, with those 10's out of the way, we just remember that we have to move the decimal point the number of places given by the number after the 'e'. Take notice of the sign of that number following the 'e' If it has a positive sign (or none) - move right (make it bigger) (The positive sign is optional) If it has a negative sign - move left (make it smaller) Examples 4.11765e19 represents 41 176 500 000 000 000 000 4.11765e-19 would be 0.000 000 000 000 000 000 411 765 |
Formatting and Accuracy in JavaScript The Conversion Calculator is written in two languages - HTML and JavaScript. HTML is the 'normal' language of the Web and (usually) presents no problems. JavaScript is another matter. It is still developing and, at the moment, it is a little 'uncertain' when it handles numbers. This uncertainty applies to both the accuracy of its arithmetic and the presentation of numbers. (More precisely, the problem arises from how the Browser handles JavaScript.) Within the range of numbers encountered in 'ordinary everyday' life there should be no problem. The formatting should have all spacing properly placed, and the accuracy should be given to the number of significant figures stated at the top. But, when dealing with very large and very small numbers, discrepancies will be noticed. So, if you really wish to know how many centimetres there are in 1000 light years - do not expect complete accuracy or a neat format! For the curious - or those who want to try it - it should be 94605 followed by 16 zeros. |
Significant Figures (sig.figs or s.f.) |
The idea of significant figures concerns all those who have to work with numbers, especially with regard to seeing that they are not invested with an importance or implied accuracy that is not warranted. |
The MOST significant figure in any number is the first digit on the left which is NOT a zero. Look at these numbers. Identify which is the first significant figure in each. (a) 239 . . . (b) 58124 . . . (c) 702100 . . . (d) 0.3605 (e) 0.0000439 . . . (f) 810.427 . . . (g) 0.007400 (Answers at the bottom)
Then, starting with the most significant figure and moving right, ALL digits (including zeros) are now counted until, either the end of the number is reached, or the only digits remaining are zeros - these zeros are not counted (they may be sometimes, see later) Count how many sig.figs there are in each of the above numbers. That is a broad outline, which works well most times. However cases can be complicated by zeros. It depends where they come in the number. As a rough rule, if they appear WHEN NOT NECESSARY then they should be considered as significant. So, in 0.04600 the last two zeros do nothing for the value of the number, but they do signal that (whatever it is the number is about) has been measured to an accuracy of 4 sig.figs. It was just that the last two happened to be zero. Put another way, 0.04600 implies a greater accuracy than 0.046 (Answers, in order, are: 2, 5, 7, 3, 4, 8, 7) (No of sig.figs are: 3, 5, 4, 4, 3, 6, 2) |
Care must be taken. It is not merely a matter of merely adding zeros on the end of a number. they should be put there only if it is known the value is that. Measuring a line as 23.5cm and calling it 23.5000cm has not made the measurement any more accurate.
Zeros which appear BETWEEN non-zeros (as in 62.07) must be significant since we can see that measurement has gone beyond that place of accuracy. |
Spurious Accuracy This is a form of innumeracy which all users of calculators and computers need to be aware of. It consists of claiming a greater accuracy than is justified, and usually arises when numbers are being combined (arithmetically) to produce an answer to a problem. As an example, consider the arithmetic used in working out the circumference of a circle. A measurement has been taken of the diameter of the circle and done with great care to produce the value of 17.32 cm. (Note that this is saying the measurement is accurate to one-tenth of a millimetre.) It is only necessary to multiply this by 'pi' to get the circumference. Using the pi-value given by the calculator (3.1415927) gives 54.412385
If stated as that, the implied accuracy is to one-millionth of a millimetre! This is clearly ridiculous and the number needs to be truncated to some suitable degree of accuracy. By how much? A good rule of thumb is that no final answer should have any more accuracy attached to it than the LOWEST accuracy to found amongst the numbers being worked with. In the above case, the diameter was given to 4 sig.figs. and pi to 8. So the best answer that should be offered is 54.41 (to 4 s.f.) In most practical work it is rare that anything beyond 3 sig.figs is needed, or justified. But, this truncation should only be carried out on the final answer after all the arithmetic has been done. |
Significant Figures v. Decimal Places Why use significant figures? Why not decimal places - which are so much easier to understand? The key to that lies in the passage above about spurious accuracy. Using sig.figs. allows us to specify a 'blanket' accuracy, regardless of how big or how small the numbers turn out to be. That way the accuracy is always the relative to the size of the stated number. For instance, suppose it is decided that 4 sig.figs is accurate enough. Then we can look at (say) the mean radius of the Earth's orbit about the Sun as 149 600 000 000 metres and the radius of the calcium atom as 0.000 000 000 195 8 metres and know that, in spite of their vast difference in size, a variation of 1 in the last significant figure of either (6 in the first, 8 in the second) is of the same degree of importance in both cases. There is one important exception to all of this. That is when it comes to working with money. It is accepted then that EVERY penny or cent must be accounted for and so, no matter how big the amount is, there are nearly always two decimal places (sometimes four) on the end. In the European Union, the rules demand that six decimal places are used for accounting purposes but, as a concession to clarity and ease of reading, that only two need to be displayed. But in general use (conversation, newspaper headlines and similar writings) it is customary to use only 1 significant figure, though words like 'nearly' or 'over' are often attached as an acknowledgement that it is a rounded number. In connection with all of this, it is worth noting that spreadsheets use as many as 16 significant figures for all their working regardless of what is actually being displayed. |
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