Chapter 2. Introduction to Engineering Calculations 6. Handling Numbers i.How do handle extremely large and small numbers? All numbers can be expressed in scientific notation, as N  10 n, where N is a number between 1 and 10, and See Ex. 1 n is a positive/negative integer, named the «exponent». See Ex. 1 ii.How do handle arithmetic operations in scientific notation? A) Addition/Subtraction: Write all the terms with the same exponent. See Ex. 2 B) Addition/Subtraction: Multiply/divide N 1 by N 2. See Ex. 3 Add/substract n 1 by n 2. See Ex. 3 Mass and Energy Balances - Chapter II1

iii.Is it possible to obtain exact values of quantities?  Except integer quantities, generally NO!  For this reason, it is important to indicate the margins of error in a measurement by clearly indicating the number of «significant figures».  Significant figures are the meaningful digits in a measured or calculated quantity. Ex. Ex. A measurement of 1.6 kg indicates that the measured quantitity is between (and including) … kg … kg, in which there are infinite numbers of zeros following both numbers. iv.Guidelines for Significant Figures See Ex. 4 A)Any nonzero digit is significant. See Ex. 4 See Ex. 5 B)Zeros between nonzero digits are significant. See Ex. 5 See Ex. 6 C)Zeros to the left of the first nonzero digit are NOT significant. See Ex. 6 Mass and Energy Balances - Chapter II2

v.Rule of addition/subtraction  Answer can be at most PRECISE as the least precise term. See Ex. 10  Answer cannot have more digits to the right of the decimal point than either of the original numbers. See Ex. 10 vi.Rule of multiplication/division See Ex. 11  The number of significant figures in the final answer is equal to the smallest number of significant figures in the original numbers. See Ex. 11 See Ex. 12,13  Note that exact numbers are considered to have an infinite number of significant figures. See Ex. 12,13 Mass and Energy Balances - Chapter II3 See Ex. 7 D)If a number is greater than 1, all zeros to the right of the decimal point are significant. See Ex. 7 See Ex. 8,9 E)For numbers that do not contain decimal point, the zeros after the last non zero digit MAY OR MAY NOT BE significant. See Ex. 8,9

Mass and Energy Balances - Chapter II4 Note on rounding off: Rounding off is a kind of estimating. The simple rule of rounding off is as follows:  Find the place of the rounding digit and look at the digit just to the right of it.  If that digit is < 5, do not change the rounding digit but drop all digits to the right of it.  If that digit is < 5, add one to the rounding digit and drop all digits to the right of it.  If that digit is = 5, round the number so that rounding digit will be even. Ex. Ex. Round off the following numbers to the following number of sig. fig.’s: a to four s.f.  3 is the rounding digit, and 4 < 5  b to three s.f.  8 is the rounding digit, and 7 > 5  1.69 c to three s.f.  8 is the rounding digit, and 5 = 5  94.8 d to three s.f.  7 is the rounding digit, and 5 = 5  94.8

Mass and Energy Balances - Chapter II5 v.Significant Figures in Chain Calculations  Theoretically, do NOT round off at intermediate steps. too much See Ex. 14  Practically, do NOT round off too much at intermediate steps. See Ex. 14  If the intermediate results are to be reported, i.e. for engineering purposes, then round off these results. 6. Dimensional Homogeneity

Mass and Energy Balances - Chapter II6 All terms on the L.H.S. and R.H.S. have the same dimensions, L/t.  The above eqn is dimensionally homogeneous.