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Scientific Notation & Significant Figures

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Scientific Notation Scientific Notation (also called Standard Form) is a special way of writing numbers that makes it easier to use big and small numbers. You write the number in two parts: Just the digits (with the decimal point placed after the first digit), followed by × 10 to a power that would put the decimal point back where it should be (i.e. it shows how many places to move the decimal point).

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Scientific Notation If the number is 10 or greater, the decimal place has to move to the left, and the power of 10 will be positive. If the number is smaller than 1, then decimal place has to move to the right, so the power of 10 will be negative. Example: 0.0055 would be written as 5.5 × 10 -3 Because 0.0055 = 5.5 × 0.001 = 5.5 × 10 -3

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Scientific notation 9.07 x 10 – 2 0.0907 5.06 x 10 – 4 0.000506 2.3 x 10 12 2 300 000 000 000 1.27 x 10 2 127 Scientific notationDecimal notation

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What is a significant figure? TThere are 2 kinds of numbers: EExact: the amount of money in your account. Known with certainty. AApproximate: weight, height—anything MEASURED. No measurement is perfect.

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Measuring and recording If you measured the length of a pen with your ruler you might record 4.72cm. To a mathematician 4.72, or 4.720 is the same.

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But, to a scientist 4.72cm and 4.720cm is NOT the same 4.720cm to a scientist means the measurement is accurate to within one thousandth of a cm. We know that the length of the pen is between 4.7 and 4.8 because these are our smallest markings. The last digit recorded is a guess.

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Rules for Significant Digits 1. Numbers All digits from 1 to 9 (nonzero digits) are significant. 5.87 = 3 significant digits 8981 = 4 significant digits

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Zeroes 2. All zeros which are between non-zero digits are always significant. Ex. 901 (3), 321.09 (5), 1011(4) 3. Zeroes to the left are NOT significant, and serve only to locate the decimal point. Ex. 0.0987(3), 0.00001(1) 4. Zeros to the right MAY be significant, if it is also to the right of the decimal place. To be significant, the zero must follow a non-zero number. Ex. 23400 (no, 3), 0.0670 (yes,3)

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Counted Numbers 5. Because counted numbers are not measured, they have an infinite number of significant figures. i.e. 3 test tubes, 5 pennies, 10 beakers.

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How many significant figures? 1.2 2100 56.76 4.00 0.0792 7,083,000,000 2 2 4 3 3 4

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Significant Digits Tips It is better to represent 100 as 1.00 x 10 2 Alternatively you can underline the position of the last significant digit. E.g. 100. This is especially useful when doing a long calculation or for recording experimental results Don’t round your answer until the last step in a calculation.

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Adding with Significant Digits When adding or subtracting, the answer can have no more places after the decimal than the LEAST of the measured numbers E.g. a) 13.64 + 0.075 + 67 b) 267.8 – 9.36 13.64 0.075 67. 80.71581 267.8 9.36 258.44 – + +

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Multiplication and Division Use the same number of significant digits as the value with the fewest number of significant digits. E.g. a) 608.3 x 3.45 b) 4.8 392 a) 3.45 has 3 sig. digits, so the answer will as well 608.3 x 3.45 = 2098.635 = 2.10 x 10 3 b) 4.8 has 2 sig. digits, so the answer will as well 4.8 392 = 0.012245 = 0.012 or 1.2 x 10 – 2

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Review Scientific Notation Know the 2 parts to writing scientific notation: The digits 10 to a power Know how to convert between scientific notation and decimal form (on your formulas handout). If the power of 10 is positive, the decimal place moves to the right that many times. If the power of 10 is negative, the decimal place moves to the left that many times.

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Review Significant Digits Know the 5 rules for determining significant digits 1. Digits 1-9 are always significant. 2. Zeroes between non-zero digits are significant. 3. Zeroes to the left of non-zero digits are not significant. They only serve as placeholders. 4. Zeroes to the right of non-zero digits are significant if they are also to the right of the decimal place. 5. Counted numbers have an infinite number of significant figures. Physics formula sheet

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Review Adding and Subtracting Your answer can only have as many decimal places as the measured number with the fewest decimal places. Multiplying and Dividing Your answer can only have as many significant figures as the measured number with the fewest significant figures.

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