Lecture on Significant Digits Rounding and Chopping

Numerical analysis is the study of algorithms (computational technique) that use numerical approximation to solve for the problems related to mathematical relation •Computing the trajectory of a spacecraft requires the accurate numerical solution of a system of ordinary differential equations. •Car companies can improve the crash safety of their vehicles by using computer simulations of car crashes. Such simulations essentially consist of solving partial differential equations numerically. •Hedge funds (private investment funds) use tools from all fields of numerical analysis to attempt to calculate the value of stocks and derivatives more precisely than other market participants. •Airlines use sophisticated optimization algorithms to decide ticket prices, airplane and crew assignments and fuel needs. Historically, such algorithms were developed within the overlapping field of operation research. Integers include positive whole numbers, negative whole numbers, and zero. Integers = {… -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, …} A decimal is any number in our base-ten number system. Specifically, we will be using numbers that have one or more digits to the right of the decimal Below point inwe this have unitexpressed of lessons.the The number decimalinpoint expanded is usedform to separate and in decimal the ones form. place from the tenths place in decimals. (It is also used to separate dollars from cents in money.) As we move to the right of the decimal point, each number place is divided by 10. Mixed Number ------------- E x p a n d e d F o r m -----Decimal Form -----= (5 x 10) + ( 7 x 1) + (4 x ) + (9 x ) = 57.49 What is a "significant figure“ or significant digit ? The number of significant figures in a result is simply the number of figures that are known with some degree of reliability. The number 13.2 is said to have 3 significant figures. The number 13.20 is said to have 4 significant figures. Rules for deciding the number of significant figures in a measured quantity: (1) All nonzero digits are significant: 1.234 g has 4 significant figures, 1.2 g has 2 significant figures. (2) Zeroes between nonzero digits are significant: 1002 kg has 4 significant figures, 3.07 mL has 3 significant figures. (3) Leading zeros to the left of the first nonzero digits are not significant; such zeroes merely indicate the position of the decimal point: 0.001 oC has only 1 significant figure, 0.012 g has 2 significant figures. (4) Trailing zeroes that are also to the right of a decimal point in a number are significant: 0.0230 mL has 3 significant figures, 0.20 g has 2 significant figures. (5) When a number ends in zeroes that are not to the right of a decimal point, the zeroes are not necessarily significant: 190 miles may be 2 or 3 significant figures, 50,600 calories may be 3, 4, or 5 significant figures. The potential ambiguity in the last rule can be avoided by the use of standard exponential, or "scientific," notation. For example, depending on whether the number of significant figures is 3, 4, or 5, we would write 50,600 calories as: 4 5.06 × 10 calories (3 significant figures) 4 5.060 × 10 calories (4 significant figures), or 4 5.0600 × 10 calories (5 significant figures). By writing a number in scientific notation, the number of significant figures is clearly indicated by the number of numerical figures in the 'digit' term as shown by these examples. This approach is a reasonable convention to follow. • Here are the basic rules for significant digits: • All non-zero numbers ARE significant. ... • Zeros between two nonzero digits ARE significant. ... • Leading zeros are NOT significant. ... • Trailing zeros to the right of the decimal ARE significant. ... • Trailing zeros in a whole number with the decimal shown ARE significant. What is an "exact number"? Some numbers are exact because they are known with complete certainty. Most exact numbers are integers: exactly 12 inches are in a foot, there might be exactly 23 students in a class. Exact numbers are often found as conversion factors or as counts of objects. Exact numbers can be considered to have an infinite number of significant figures. Rules for rounding off numbers (1) If the digit to be dropped is greater than 5, the last retained digit is increased by one. For example, 12.6 is rounded to 13. (2) If the digit to be dropped is less than 5, the last remaining digit is left as it is. For example, 12.4 is rounded to 12. (3) If the digit to be dropped is 5, and if any digit following it is not zero, the last remaining digit is increased by one. For example, 12.51 is rounded to 13. (4) If the digit to be dropped is 5 and is followed only by zeroes, the last remaining digit is increased by one if it is odd, but left as it is if even. For example, 11.5 is rounded to 12, 12.5 is rounded to 12. This rule means that if the digit to be dropped is 5 followed only by zeroes, the result is always rounded to the even digit. The rationale for this rule is to avoid bias in rounding: half of the time we round up, half the time we round down. What is truncation? In simplest terms, truncation means to chop off the decimal portion of a number. This means: Truncating 3.3 returns 3 Truncating 3.8 returns 3 Significant Digits in Multiplication, Division, Trig. functions, etc.In a calculation involving multiplication, division, trigonometric functions, etc., the number of significant digits in an answer should equal the least number of significant digits in any one of the numbers being multiplied, divided etc. Thus in evaluating sin(kx), where k = 0.097 m-1 (two significant digits) and x = 4.73 m (three significant digits), the answer should have two significant digits. Significant Digits in Addition and Subtraction When quantities are being added or subtracted, the number of decimal places (not significant digits) in the answer should be the same as the least number of decimal places in any of the numbers being added or subtracted. Example: 5.67 J (two decimal places) 1.1 J (one decimal place) 0.9378 J (four decimal place) 7.7 J (one decimal place) Keep One Extra Digit in Intermediate Answers When doing multi-step calculations, keep at least one more significant digit in intermediate results than needed in your final answer. For instance, if a final answer requires two significant digits, then carry at least three significant digits in calculations