MORE NUMERACY - PRECISION 20040322
Precision, if used correctly, shows that you understand what is a suitable length of number to be accurate. Precision is implied by the use of significant figures. The number 3.14 is implicitly 3 significant figures. Setting the value of g to be 9.8 implies 9.75≤g<9.85 but setting g to be exactly 9.8 is extreme precision. This is one way of arguing that two and two might be five; we tend to assume that integers (whole numbers) are given exactly – if they are not, and we assume one significant figure instead, then two values of say 2.3 add to sufficient to total more than 4.5, and hence round off to five when expressed to only one significant figure.
Useful information for this page:
1 inch = 25.4 mm exactly; £1 = $1.625; I lb (pound) = 454 g (3s.f.)
12 inches = 1 foot; 63360 inches = I mile; C= 2πr A = πr²
1 Convert 16 feet to metres. Make sure you show your working.
2 The equator is 40 million metres long [it’s the original definition of the metre]. Convert this distance to miles. Use 5 significant figures.
3 What is the diameter of the Earth in metres? (4s.f., Std Form)
4 Convert $10,000 to sterling allowing for 0.5% commission taken on the transaction.
5 Convert your answer to the last question back to dollars, still taking commission. Can you see a way to this answer without using the previous result?
6 1000lbs weight lies between 995 and 1005 lbs; what precision has been used? [properly, 995≤”1000”<1005]. What is the equivalent set of numbers when converted to kg? Give your answer to 3 sig fig.
7 Now convert 1000kg to pounds, assuming the same thinking as the last question.
8 A length is 9 metres to 1 sig fig; this length is assumed to be the side of a square. How small could this area be? [Hint: The biggest such area is just less than 90.25 m2 ]
9 I have a cube whose side is measured as 3.60 metres. What is a) the implied precision b) the total edge length? c) the volume of the cube? Think about precision and declare the appropriate number of sig fig.
10 Do this calculation ( 3 π³ x 2.03 x1016 ) / ( 6.02x106 - 3.47x10⁴ )
11 If p = 23,908,750,987 then give p to a) 2, b) 4, c) 8 sig fig in Std Form
and give p² t to d) 3, e) 5 sig fig in Std Form
12 How big could 5 + 5 be? What about 5x5? [extension 5⁵]
Answers:
1. 16x12x2.54/1000=4.8768m say 4.88m. 2. 4x107 /(0.0254 /63360) = 24855 miles to 5 sf.
3. 4x107 /π = 2r = 1.273x107. 4. (10000/1.625)*0.995 = £6123.08
5. £6123.08 *1.625*0.995 = 9900.25 Check .995²=0.990025
6. 2sf used to left and 3sf to right. 0.5% margin, not significant figures. [995,1005] converts to [444, 449]
7. 1000kg±5kg = [2230, 2250]kg. 8. [8.5², 9.5²] = [72.25, 90.25]
9. 3s.f or 2d.p. 3.595≤side<3.605 b) 21.57≤21.6<21.63 c) 46.46≤ 46.656 <46.85 2dp, 4sf 10, 1.88828618 / 5985300 = 315486650599 = 3.15x1011
11. p= 2.4x1010, 2.391x1010, 23908751x1010, p²= 5.72x1020, 25.7163x1020, 5.7162837x1020.
12. 9 = 2*4.5 ≤ “5+5” < 2*5.5 = 11; 20.25 = 4.5² ≤ “5*5” < 5.5² = 30.25 870≤ "5⁵” < 11800
