Introduction and Mathematical Concepts

Introduction and Mathematical Concepts Chapter 1

Objectives • To express real numbers in scientific notation and solve using a calculator • To determine the number of sig figs in a given number • To solve equations using only variables

Scientific Notation • Scientific notation: useful for small and large #’s • Use a coefficient raised to the power of 10 • Coefficient must be between 1 and 10 • Use a power of 10 depending on how many times you moved the decimal place

Scientific Notation • Example: • 10,000,000 becomes 1 x107 • 0.0000025 becomes 2.5 x10-6 • Greater than 1= positive power • Less than 1= negative power

Scientific Notation Examples • Express in scientific Notation • .00000000001874 • 1.874 x 10-11

Scientific Notation Examples • Express in scientific Notation • 200,000,000,000 • 2 x 1011

Scientific Notation Examples • Express in scientific Notation • .00076321 • 7.6321 x 10-4

Scientific Notation Examples • Express in scientific Notation • 984.73 • 9.8473 x 102 • Explanation: Why is this example odd?

Scientific Notation Examples • Express as real numbers • 1.34 x 105 • 134,000

Scientific Notation Examples • Express as real numbers • 6.0 x 10-3 • .0060

Scientific Notation Examples • Express as real numbers • 5.223 x 108 • 522,300,000

Scientific Notation and Calculators • When using a calculator DO NOT type in x10 meaning DO NOT Push the buttons [x], [1] & [0]. • Depending on your calculator you will use the [EE], [Exp] or [x10n] button when a number is in scientific notation

Scientific Notation • What is the answer to… • 642 x (4.0 x 10-5) • 2.6 x 10-2 or .02568

Scientific Notation • What is the answer to… • 1.7 x 105 / 3.88 x 107 • 4.4 x 10-3 or .0044

Scientific Notation • What is the answer to… • 2.9 x 10-5 x (8.1 x 102) • 2.3 x 10-2 or .023

Scientific Notation • What is the answer to… • 6.02 x 1023 / (7.2 x 108) • 8.4 x 1014

Scientific Notation • What is the answer to… • 5.40 x10-18 / 769 • 7.02 x 10-21

Scientific Notation • What is the answer to… • (1.0 x 107) x (1.0 x 10-6) • 10

Scientific Notation Examples • Express as real numbers • 1.59 x 10-2 • .0159

Scientific Notation Sample Problems • Put the following numbers in scientific notation: • 4500000 • 0.00234 • 168000000000 • 0.00000000036

Scientific Notation Sample Problems • Answers: • 4.5 x106 • 2.34 x10-3 • 1.68 x1011 • 3.6 x10-10

Significant Figures • Significant figures: (also called significant digits) of a number are those digits that carry meaning contributing to its precision. This includes all digits except: • leading and trailing zeros which are merely placeholders to indicate the scale of the number. • spurious digits introduced

which instrument gives us more sigfigs? • that’s the one you want to use (but it probably costs a lot more!)

Rules for Significant Figures • 1) The #’s 1-9 are always significant • Examples: • 2467 • 4 sig. figs. • 2.344678, • 7 sig. figs.

Rules for Significant Figures • 2) A zero between 2 significant figures is always significant • Examples: • 23045, • 5 sig. figs. • 450000001, • 9 sig. figs. Because all zeros are between significant figures

Rules for Significant Figures • 3) Place holders are not significant, so a zero in a decimal before a significant figure is not counted • Example: • 0.023, • 2 sig. figs. Zeroes come before a significant figure • 0.00000004, • 1 sig. fig., all zeroes come before a significant figure

Rules for Significant Figures • 4) Zeroes after a significant figure and after a decimal are counted. • Example: • 0.00230, • 3 sig. figs. The zero after the 3 is after a significant number and a decimal so it is counted • 23.34000, • 7 sig. figs.

Rules for Significant Figures • 5) Zeroes after a significant figure when there is no decimal are place holders so they are not counted. • Example: • 1200 • 2 sig. figs. Zeros are place holders • 145670, • 5 sig. figs., last decimal is a place holder

Rules for Significant Figures • 6) If you count to have an exact quantity, you can use unlimited significant figures. • Example: • You counted 18 sheep, you could write the number as 18.0000000 and have 9 significant figures • You can use as many significant figures as you want because you know you have exactly 18 sheep

Significant Figure Sample Problem • How many significant figures do each of these have? • 15.008 • 146000 • 0.00025760 • 2357 • 6.0 x 105 • Answers: • 15.008 = 5 • 146000 = 3 • 0.00025760 = 5 • 2357 = 4 • 6.0 x 105 = 2

Solving Equations Using Variables • 2 Rules: The variable you are solving for must… • Be by itself on one side of the equal sign • In the numerator • Look to worksheet for examples

Objectives • We will be able to convert between different metric units given a conversion chart • We will be able to convert between units using dimensional analysis • We will be able to solve right triangle problems using sine, cosine, and tangent

1.2 Units SI units Le SystèmeInternational d’Unités (What the rest of the world uses Length: meter (m) Mass: kilogram (kg) Time: second (s)

1.2 Units The Standard Platinum-Iridium Meter Bar kept at 0°C

1.2 Units The standard platinum-iridium kilogram

1.3 The Role of Units in Problem Solving REMEMBER THIS! The Great Mighty Kids Have Dropped Over Dead Converting Metrics Many Nights Past Friday

1.2.8. How many meters are there in 12.5 kilometers? a) 1.25m b) 125m c) 1250m d) 12 500m e) 125 000m

1.3 The Role of Units in Problem Solving Reasoning Strategy: Converting Between Units 1. In all calculations, write down the units explicitly. 2. Treat all units as algebraic quantities. When identical units are divided, they are eliminated algebraically.

1.3 The Role of Units in Problem Solving • Example 1The World’s Highest Waterfall • The highest waterfall in the world is Angel Falls in Venezuela, with a total drop of 979.0 m. Express this drop in feet. • Since 3.281 feet = 1 meter, it follows that

1.3 The Role of Units in Problem Solving Example 2Interstate Speed Limit Express the speed limit of 65 miles/hour in terms of meters/second. Use 5280 feet = 1 mile and 3600 seconds = 1 hour and 3.281 feet = 1 meter.

1.4 Trigonometry The sides of a right triangle

1.4 Trigonometry Sine, Cosine and Tangent Abbrev. SOH CAH TOA

1.4.3. Referring to the triangle with sides labeled A, B, and C as shown, which of the following ratios is equal to the sine of the angle ? • a)

1.4.4. Referring to the triangle with sides labeled A, B, and C as shown, which of the following ratios is equal to the tangent of the angle ? • a)

1.4 Trigonometry Trig Example: Solve for ho

Introduction and Mathematical Concepts