Math Unit

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Math Unit
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Measurement

• When making any measurement, always estimate one
  place past what is actually known.

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Example

• For example, if a meter stick has calibrations
  (markings) to the 0.1 cm, the measurement must be
  estimated to the 0.01 cm.
• If you think its perfectly on a line, estimate
  the last digit to be zero
• For example, if you think its on the 2.1 cm
  line, estimate it to 2.10 cm.

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Electronic Devices

• When making a measurement with a digital readout,
  simply write down the measurement. The last
  digit is the estimated digit.

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Significant Digits

• Significant digits are all digits in a number
  which are known with certainty plus one uncertain
  digit.

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5 Rules for Counting Significant Digits in a
Measurment

• 1. All nonzero numbers are significant.
• 132.54 g has 5 significant digits.

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• 2. All zeros between nonzero numbers are
  significant.
• 130.0054 m has 7 significant digits

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• Zeros to the right of a nonzero digit but to the
  left of an understood decimal point are not
  significant unless shown by placing a decimal
  point at the end of the number.
• 190 000 mL has 2 significant digits
• 190 000. mL has 6 significant digits

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• All zeros to the right of a decimal point but to
  the left of a nonzero digit are NOT significant.
• 0.000 572 mg has 3 significant digits

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• All zeros to the right of a decimal point and to
  the right of a nonzero digit are significant.
• 460.000 dm has 6 significant digits

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Shortcut

• If the number contains a decimal point, draw an
  arrow starting at the left through all zeros and
  up to the 1st nonzero digit. The digits
  remaining are significant.

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Exact Numbers

• Exact numbers have an infinite (8) number of
  significant digits.
• 3 types of numbers with (8) number of sig
  digs
• 1. Definitions (12 eggs 1dozen)
• 2. Counting numbers (there are 24 desks in
  this room
• 3. Numbers in a formula (2pr)

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Try these

• 0.002 5
• 1.002 5
• 0.002 500 0
• 14 100.0

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• If the quantity does not contain a decimal point,
  draw an arrow starting at the right through all
  zeroes up to the 1st nonzero digit. The digits
  remaining are significant.

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Try these

• 225
• 10 004
• 14 100
• 103

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Remember Atlantic Pacific

• Decimal Point Present, start at the Pacific.
• Decimal Point Absent, start at the Atlantic.

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How many significant digits do these have?

• 1.050
• 20.06
• 13
• 0.303 0
• 373.109

• 420 000
• 970
• 0.002
• 0.007 80
• 145.55

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Rounding Rules

• Round up if the digit immediately to the right of
  the digit you are rounding to is
• Greater than 5
• Round 0.236 to 2 significant digits
• 5 followed by another nonzero number
• Round 0.002351 to 2 significant digits
• Round 0.00235000000001 to 2 significant digits

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• Kepp the digit the same if the digit immediately
  to the right of the digit you are rounding to is
• Less than 5
• round 1.23 to 2 significant digits

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What if the digit to the right of the number you
are rounding to is 5 and theres nothing after it?

• That means you are perfectly in the middle.
• Half of the time you must round up and half of
  the time you must round down.
• There are 2 rules for this

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Look to the digit to the right of the number you
are rounding to.

• If it is even keep the same.
• Round 0.8645 to 3 significant digits
• If it is odd round up.
• Round 0.8675 to 3 significant digits.

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Round These to 3 significant digits

• 279.3
• 32.395
• 18.29
• 42.353
• 0.008 752

• 18.77
• 7.535
• 32.25
• 5 001

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Applying significant digits to arithmetic
operations
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Addition and Subtraction

• Look at the numbers being added or subtracted and
  identify which one has the lowest number of
  decimal places. Calculate the answer. Round the
  answer to the lowest number of decimal places.

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Examples

• 14.565 7.32 21.885
• 7.32 has only 2 decimal places, so the answer
  should be rounded to 21.88
• 143.52 100.6 42.92
• 100.6 has only 1 decimal place, so the answer
  should be rounded to 42.9

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Multiplication and Division

• Look at the numbers being multiplied or divided
  and identify which one has the lowest number of
  significant digits. Calculate the answer. Round
  the answer to the lowest number of significant
  digits.

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Examples

• 172.6 x 24.1 4159.66
• 24.1 has only 3 significant digits, so the answer
  should be rounded to 4160
• 172.6 24.1 7.161 82
• 24.1 only has 3 significant digits, so the answer
  should be rounded to 7.16

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Practice

• Add 5.34 cm, 9.3 cm, and 12 cm.
• Subtract 4.31 cm from 7.542 cm.
• Subtract 1.512 g from 16.748 g.
• Add 2.572 5 m, 14.55 m and 0.035 m.
• Multiply 176.335 and 0.003 2.
• Divide 475.90 by 35.
• Multiply 0.000 565, 1.579 52, and 45.006 86.
• Multiply 1 456.00 by 0.035 0 and divide that by
  17.070.

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Percent Error

• This is a way of expressing how far off an
  experimental measurement is from the
  accepted/true value.
• Final Exam Question

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Formula

• X 100

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Scientific Notation

• It is used for extremely large or small numbers.
• The general form of the equation is
• m x 10n
• With the absolute value of m 1 and lt 10

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Practice

• 12 300
• -1 456
• 0.005 17
• -0.000 6

• 6.650 x 102
• 3.498 x 105
• -2.208 x 10-3
• 1.1650 x 10-4

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Arithmetic Rules for Scientific Notation

• Follow the same rules for math operations with
  scientific notation as you would with standard
  notation.

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Addition and Subtraction

• (3.37 x 104) (2.29 x 105)
• (9.8 x 107) (3.2 x 105)
• (8.6 x 104) (7.6 x 103)
• (2.238 6 x 109) (3.335 7 x 107)

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Multiplication and Division

• (1.2 x 103) x (3.3 x 105)
• (7.73 x 102) x (3.4 x 10-3)
• (9.9 x 106) ? (3.3 x 103)
• (1.55 x 10-7) ? (5.0 x 10-4)

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Temperature Conversion

• Temperature is defined as the average kinetic
  energy of the particles in a sample of matter.
• The units for this are oC and Kelvin (K). Note
  that there is no degree symbol for Kelvin.

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Kelvin Scale

• The Kelvin scale is based on absolute zero.
• This is the theoretical temperature when motion
  stops.

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• Heat is a measurement of the total kinetic energy
  of the particles in a sample of matter.
• The units for this are the calorie (cal) and the
  Joule (J).

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Formulas

• T(K) t(oC) 273.15
• t(oC) T(K) - 273.15

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Dimensional Analysis

• Dimensional analysis is the algebraic process of
  changing from one system of units to another.

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You must develop the habit of including units
with all measurements in calculations. Units are
handled in calculations as any algebraic symbol

• Numbers added or subtracted must have the same
  units.

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• Units are multiplied as algebraic symbols. For
  example 10 cm x 10 cm 10 cm2

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• Units are cancelled in division if they are
  identical.
• For example, 27 g 2.7 g/cm3 10 cm3.
  Otherwise, they are left unchanged. For example,
  27 g/10. cm3 2.7 g/cm3.

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Conversion Factor

• These are fractions obtained from an equivalence
  between two units.
• For example, consider the equality 1 in. 2.54
  cm. This equality yields two conversion factors
  which both equal one
• and

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Convert 5.08 cm to inches

• 5.08 cm x 2.00 in

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Convert 6.53 in to cm

• 6.53 in x 16.6 cm

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Here are some common English/metric conversions.
You will not need to memorize these.

• 1 mm 0.039 37 in 1 in 2.54 cm
• 1 cm 0.393 7 in 1 yd 0.914 40 m
• 1 m 39.37 in 1 mile 1.609 Km
• 1 Km 0.621 4 mi. 1 pound 453.6 g
• 1 quart 946 ml 1 ounce 28.35 g
• 1 quart 0.946 L

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Converting within metric units

• In section 2-5 of your textbook, you learned the
  relationship between metric prefixes and their
  base units. You need to have these relationships
  memorized to do these problems.
• When you write your conversions factor, always
  use the number 1 with the unit with the prefix
  and meaning of the prefix with the base unit.

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Examples
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Try to set up these conversion factors