Complete Math Skills Review for Science and Mathematics A review of key mathematical concepts for Physics, Chemistry, Biology, General Science and Mathematics Includes: Scientific Notation Significant Digits Rearranging Equations Metric Conversions Trigonometry Review

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Scientific Notation

Why we use scientific notation … Often in science numbers can be VERY large or VERY small. I.e. the distance to the nearest star besides our sun (Proxima Centauri) is 40 000 000 000 000 km And, the mass of the Earth is: 5970000000000000000000000 kg Scientific notation is a convenient way to make numbers easier to work with.

Scientific Notation Scientific notation expresses a large measurement or a small measurement in the form a x 10n a is greater than 1 and less than 10 _________________

The distance to Proxima Centauri in scientific notation (a x 10n) In decimal form - 40 000 000 000 000 km Where is the decimal point now? After the last zero Where would you put the decimal to make this number be between 1 and 10? ____________ __________

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Positive vs. Negative exponents (n) What’s the difference between these two numbers? 8,000,000,000 0.000000008 ____________ If we simply continue doing what we’ve learned so far, both would be 8.0 x 109 but that’s not right because they are vastly different numbers

Big Number, Small Number ___________ If you multiply something by x 10-1, you are essentially dividing it by 10, making it smaller We need to keep this in mind when writing our scientific notation If our original number was greater than 1, our _______________ If our original number was less than 1, our _______________

Find a and n for: 0.0000000902 Find a: Where would the decimal go to make the number be between 1 and 10? ______ Find n: How many places was the decimal moved? Was the original number greater than (+n) or less than (-n) 1? Therefore the answer is

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Check Your Understanding Write 289,800,000 in scientific notation

Write in PROPER scientific notation (Notice the number is not between 1 and 10) To solve these, pretend the x10n isn’t there and then add it in at the end. Example: Write 322.2 x 107 in scientific notation ____________ Convert 322.2 into scientific notation = 3.222 x 102 Add the exponents together to get your final exponent  7 + 2 = 9 Re-write your number  3.222 x 109

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Check Your Understanding Write 531.42 x 105 in scientific notation .53142 x 102 5.3142 x 103 53.142 x 104 531.42 x 105 53.142 x 106 5.3142 x 107 .53142 x 108

Going from Scientific Notation to Decimal Form Remember: if n is positive, your answer will be greater than 1 therefore, you move the decimal to the right and place zeros in any extra spaces __________ if n is negative, your answer will be less than 1, therefore you move the decimal to the left and place zeros in any extra spaces

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Significant Digits and Scientific Notation When trying to determine the number of significant digits within scientific notation we follow the same rules pretending the x 10n is not there. E.g. 0.01130x10-4 has _________ 5.3x1015 has __________

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Significant Digits (Figures)

Significant Digits Significant digits represent the amount of uncertainty in a measurement. _________ E.g. The length of something is between 5.2 and 5.3 cm. Suppose we estimate it to be 5.23 cm

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Significant Digit Rule #1 __________ Examples: 123 m=3 s.d. 23.56=4 s.d.

Significant Digit Rule #2 ___________ Examples: 1207 m = 4 s.d. 120.5 km/h = 4 s.d. In example 1 (1207), the zero is significant because to measure the 7, you must have measured the zero

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Significant Digit Rule #4 Zeros used to indicate the position of the decimal are not significant Examples: _________

Significant Digit Rule #5 All counting numbers have an infinite number of significant digits because we are certain of the exact number Example: _________

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Practice Complete the remaining pages from the worksheet – “Scientific Notation and Significant Digits Practice”

Rearranging Equations In almost every problem you do, you will need to isolate for a variable A variable is an unknown quantity you are trying to solve for

Rearranging Equations To isolate a variable you: Put your variable on one side of the equation and everything else on the other If there is a “+” or “-“ involved, do the opposite to eliminate it from the one side of the equation. What you do to one side you need to then do to the other If your variable is a multiple of another term, you must multiple or divide (again it’s the opposite) to eliminate everything but your variable from one side of the equation. What you do to one side of the equation, you must do to the other You may also need to find the root, third root, etc.

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Example Isolate x: 2x – 3 = 3 Add 3 to the left and right side of the equation and simplify 2x – 3 + 3 = 3 + 3 Simplify 2x = 6 Divide both sides by 2 and simplify 2x = 6 Simplify x = 3 2

Check Your Understanding Isolate a:

Metric Conversions When performing metric conversions keep the following chart in mind and simply multiply through until you get the units you need using the correct ratios Note: You may need to use two ratios to get your answer i.e. if you are converting km/h to m/s

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Example How many mm are in 3.5 m? Write out what you have x the ratio = what units you want Multiply what you have by the correct ratio in order to cross out unwanted units Cross out the unwanted units and multiply through to get your answer

Check Your Understanding How many mm’s are in 1 Gm?

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Example What is your speed in m/s if you are travelling 100.0 km/h? Write out what you have x the ratio = what units you want Multiply what you have by the correct ratio in order to cross out unwanted units Cross out the unwanted units and multiply through to get your answer

Trigonometry You need to be able to: find the angle between sides of a triangle use the Pythagorean theorem use sine and cosine law

Trig Ratios Sine Θ  Sin Θ = Opposite / Hypotenuse Cosine Θ  Cos Θ = Adjacent / Hypotenuse _________ Remember – _________

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Finding The Angle *Remember: All interior angles of a triangle = 180°* To find the angle you need to take the inverse trig ratio i.e. sin A = 3/5  A = sin-1 (3/5)  A = 37° Find each of the following to 3 significant digits sin R = 0.9 cos P = 0.343 Tan B = 4/3

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The Pythagorean Formula ____________ Example: What would be the length of b if a = 4 and c = 5? Solution: 42 + b2 = 52 Isolate b2  b2 = 52 – 42 = 25 – 16 = 9 Solve for b  b = 3

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The Sine Law The Sine Law can be used to solve a non-right triangle when given: the measures of two angles and any side the measures of two sides and an angle opposite one of these sides __________ You need 3 of the variables in 2 equations to solve the forth or

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The Cosine Law The Cosine Law Formulas, use the one that works best _________ Example: find the length of “a” Solution: a2 = b2 + c2 – 2ab cos(A) a2 = 182 + 212 – 2a(18) cos(61) a = 19.96 cm

Check Your Understanding Find the length of side b a2 = b2 + c2 – 2bc cos(A) b2 = a2 + c2 – 2ac cos(B) c2 = a2 + b2 – 2ab cos(C)