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**Significant Figures (digits)**

= reliable figures obtained by measurement = all digits known with certainty plus one estimated digit

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**Taking the measurement**

Is always some uncertainty Because of the limits of the instrument you are using

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EXAMPLE: mm ruler Is the length of the line between 4 and 5 cm? Yes, definitely. Is the length between 4.0 and 4.5 cm? Yes, it looks that way. But is the length 4.3 cm? Is it 4.4 cm? Let’s say we are certain that it is 4.3 cm or 43mm, but not at long as 4.4cm. So – we need to add one more digit to ensure the measurement is more accurate. Since we’ve decided that it’s closer to 4.3 than 4.4 it may be recorded at 4.33 cm.

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It is important to be honest when reporting a measurement, so that it does not appear to be more accurate than the equipment used to make the measurement allows. We can achieve this by controlling the number of digits, or significant figures, used to report the measurement.

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As we improve the sensitivity of the equipment used to make a measurement, the number of significant figures increases. Postage Scale 3 g 1 g 1 significant figure Two-pan balance 2.53 g 0.01 g 3 significant figures Analytical balance 2.531 g 0.001g 4 significant figures

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**Which numbers are Significant?**

5,551,213 55.00 mm Which numbers are Significant? How to count them! 9000 L 0.003g

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**Non-Zero integers Always count as significant figures**

has 4 significant digits

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**Zeros – there are 3 types Leading zeros (place holders)**

The first significant figure in a measurement is the first digit other than zero counting from left to right 0.0045g (4 is the 1st sig. fig.) “0.00” are place holders. The zeros are not significant

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Captive zeros Zeros within a number at always significant – g All digits are significant

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**Trailing zeros – at the end of numbers but to the right of the decimal point**

2.00 g - has 3 sig. digits (what this means is that the measuring instrument can measure exactly to two decimal places. 100 m has 1 sig. digit Zeros are significant if a number contains decimals

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**Exact Numbers Are numbers that are not obtained by measuring**

Referred to as counting numbers EX : 12 apples, 100 people

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**Exact Numbers Also arise by definition 1” = 2.54 cm or 12 in. = 1 foot**

Are referred to as conversion factors that allow for the expression of a value using two different units

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**Significant Figures Rules for sig figs.:**

Count the number of digits in a measurement from left to right: Start with the first nonzero digit Do not count place-holder zeros. The rules for significant digits apply only to measurements and not to exact numbers Sig figs is short for significant figures.

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**Determining Significant Figures**

State the number of significant figures in the following measurements: 2005 cm 4 0.050 cm 2 25,000 g 2 g 3 25.0 ml 3 50.00 ml 4 0.25 s 2 1000 s 1 mol 3 1000. mol 4

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**Rounding Numbers To express answer in correctly**

Only use the first number to the right of the last significant digit

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**Rounding Always carry the extra digits through to the final result**

Then round EX: Answer is rounds to 1.3 OR 1.356 rounds to 1.4

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**Rounding off sig figs (significant figures):**

Rule 1: If the first non-sig fig is less than 5, drop all non-sig fig. Rule 2: If the first sig fig is 5, or greater that 5, increase the last sig fig by 1 and drop all non-sig figs. Round off each of the following to 3 significant figures: 12.5 0.602 14,700 192

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**Math Problems w/Sig Figs**

When combining measurements with different degrees of accuracy and precision, the accuracy of the final answer can be no greater than the least accurate measurement.

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**Adding and Subtracting Sig. Figures**

This principle can be translated into a simple rule for addition and subtraction: When measurements are added or subtracted, the answer can contain no more decimal places than the least accurate measurement.

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Significant Figures Adding and subtracting sig figs - your answer must be limited to the value with the greatest uncertainty.

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**Line up decimals and Add**

g H2O (using significant figures) 0.507 g salt g solution g solution 150.0 is the least precise so the answer will have no more than one place to the right of the decimal.

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**Example Answer will have the same number of decimal places as the least precise measurement used.**

cm cm 1.013 cm cm 9.62 cm cm Correct answer would be 71.9 cm – the last sig fig is “8”, so you will round using only the first number to the right of the last significant digit which is “7”.

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Significant Figures Multiplication and division of sig figs - your answer must be limited to the measurement with the least number of sig figs. 5.15 X 2.3 11.845 3 sig figs 2 sig figs only allowed 2 sig figs so is rounded to 12 5 sig fig 2 sig figs

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**Multiplication and Division**

Answer will be rounded to the same number of significant figures as the component with the fewest number of significant figures. 4.56 cm x 1.4 cm = 6.38 cm2 = 6.4 cm2

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28.0 inches cm 1 inch Computed measurement is cm Answer is 71.1 cm x = 71.12 cm

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**When both addition/subtraction and multiplication/division appear in the same problem**

In addition/subtraction the number of significant digits is limited by the value of greatest uncertainty. In multiplication/division, the number of significant digits is limited by the value with the fewest significant digits. Since the rules are different for each type of operation, when they both occur in the same problem, complete the first operation and establish the correct number of significant digits. Then proceed with the second and set the final answer according to the correct number of significant digits based on that operation

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( )/7.5 Add = Then divide by 7.5 =

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