Principles of Technology/Physics in Context (PT/PIC)

1
Principles of Technology/Physics in Context
(PT/PIC)

• Arithmetic Overview
• Decimals

2
Principles of Technology/Physics in Context
(PT/PIC)

• During the lecture assessment questions will be
  asked.
• Indicate your answer on the scantron sheet
  provided. (21-30)

3
Decimals

• There are two different ways to express numbers
  that are not integers (whole numbers)
• As fractions and as decimals.
• Now it's time to talk about decimals.

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Decimals

• When a number is expressed in decimal form, it
  has two parts
• The whole number part
• The decimal fraction part.
• The two parts are separated by the decimal point
  (.)
• (.)

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Decimals and Money

• You are certainly familiar with decimals from
  dealing with money
• 12.45 is an example of a decimal.
• The part to the left of the decimal point is the
  whole number part
• 12 in this case. The decimal fraction part is
  0.45 in this case.

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Decimals and Money

• Let's see how decimals work by looking at dollars
  and cents. What fraction of a dollar is 1 cent?
  That's
• 0.01 dollars. Well there are 100 cents in a
  dollar, so 1 cent must represent 1/100 of a
  dollar. So we know
• that the fraction 1/100 is equivalent to the
  decimal 0.01.

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Decimals and Money

• Now, what fraction of a dollar is 10 cents, or
  0.10 dollars?
• That's equal to a dime, and there are 10 dimes in
  a dollar, so 10 cents must be 1/10 of a dollar.
• So we also know that 1/10 is equivalent to the
  decimal 0.10.

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Assessment Question 1

• All of the following are true EXCEPT
• A decimal has two parts, the whole number and
  decimal, separated by a decimal point (.).
• 2 quarters is equivalent to 0.5 dollar.
• 2.17 the whole number is 17 and the decimal is
  2.
• 0.05 dollar is equivalent to a nickel.
• 1 penny is equivalent to 0.01 dollar.

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Got the Digits

• You can tell the value of a digit in a number by
  its place value.
• For instance, in the number 27,465
• The number 6 has a value of 60, because it's in
  the tens' place.
• Each place is worth ten times as much as the
  place to its right

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Got the Digits
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Got the Decimals

• Decimals work the same way.
• You know the value of each digit in a decimal by
  its place relative to the decimal point.
• The first place to the right of the decimal point
  is worth 1/10 (thus we call it the tenths
  place).

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Got the Decimals

• The second place is worth 1/100 (thus we call it
  the hundredths' place).
• The third place is worth 1/1000 (thus we call it
  the thousandths' place).
• The fourth place is worth 1/10,000 (thus we call
  it the ten-thousandths' place), and so on.

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Assessment Question 2

• All of the following are true EXCEPT
• A decimal increases in value the further from the
  decimal point the non-zero digit is.
• You can tell the value of a digit in a number by
  its place value.
• Each place is worth ten times as much as the
  place to its right.
• 0.02 is equivalent to 2/100
• 3/1000 is equivalent to 0.003

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Got the Decimal Point

• The decimal point is small and has a habit of
  getting lost (is that a decimal or a bug?)
• For this reason, it's best to put the 0 before
  the decimal.
• That doesn't change the value
• 0.01 is the same as .01.

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Got the Decimal Point

• Also, zeros to the right of a decimal don't
  change the value
• 0.10 is the same as 0.1.
• They're both 1/10
• Similarly, 7.59 and 7.59000 have the same value.

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Change that Fraction to a Decimal

• It's easy to change a fraction into a decimal-
• All you do is divide the denominator of the
  fraction into the numerator.

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Fractions
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Change that Fraction to a Decimal
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Change that Fraction to a Decimal

• First write the fraction as long division.

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Change that Fraction to a Decimal

• 3,220 is much bigger than 415.
• So what we do is add a zero to the 415 to make
  the division work out.
• The only way we can do this without changing the
  value of 415 is if we add a decimal point after
  the 5.
• Then we're just changing 415 to 415.00-and those
  zeros don't change the value of anything.
• We divide normally, but we put a decimal point in
  the quotient (the answer) directly above the
  decimal point in 415.

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Change that Fraction to a Decimal
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Change that Fraction to a Decimal

• How far we should go depends on how much accuracy
  we need, but at this point, we can tell that the
  answer is going to be close to 0.13.

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Assessment Question 3

• All of the following are true EXCEPT
• 4/7 0.57
• 17/49 0.35
• 39/100 0.39
• 0.25 is equivalent to 25/100
• 35/1000 is equivalent to 0.35

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Addition and Subtraction of Decimal

• You add and subtract decimals the same way you
  add and subtract whole numbers.
• Just make sure the decimal points are lined up,
  and add.
• In the answer, put the decimal point directly
  below the other decimal points.

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Addition and Subtraction of Decimal
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Addition and Subtraction of Decimal
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Assessment Question 4

• All of the following are true EXCEPT
• 4.7 1.590 6.29
• 1.749 2.331 4.08
• 0.39 1.00 0.13900
• 0.25 1.825100 2.0751
• 3.51000 0.35 3.86

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Addition and Subtraction of Decimal

• If one of the terms you are adding or subtracting
  is longer than another (has more digits to the
  right of the decimal point)
• It helps to add zeros to the shorter number.

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Addition and Subtraction of Decimal
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Addition and Subtraction of Decimal
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Assessment Question 5

• All of the following are true EXCEPT
• 4.7 - 1.590 3.11
• 2.749 - 2.331 4.08
• 10.39 -1.00 9.39
• 7.25 - 1.825100 5.4249
• 3.51000 - 0.35 3.16

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Multilplication of Decimal

• As with addition and subtraction, you multiply
  decimals as if they were whole numbers and worry
  about the decimal points later.
• You don't need to add zeros to make the numbers
  the same length when you multiply, however.

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Multilplication of Decimal

• 4.5 x 3.2

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Multilplication of Decimal
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Multiplication of Decimals

• To place the decimal point in the answer, count
  the number of digits to the right of the decimal
  point in each number.
• Here we have 1 decimal place in 4.5
• And 1 in 3.2
• For a total of 1 1 or 2 places.
• Put the decimal point 2 places from the right in
  the answer 14.40.

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Multiplication of Decimals

• It's a good idea when you get the answer to see
  whether it makes sense and to check that you put
  the decimal point in the right place.
• Here the answer should be a little bigger than 4
  x 3 or 12.
• So 14.40 should be about right.
• If you placed the decimal point incorrectly, and
  ended up with 144, you would know that was wrong.

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Assessment Question 6

• All of the following are true EXCEPT
• 4.7 x 1.59 7.473
• 2.74 x 2.3 6.302
• 10.39 x 1.00 103.9
• 7.5 x 1.8 13.5
• 3.5 x 0.3 1.05

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Division of Decimals

• It's easiest to discuss division of decimals if
  we express the division in fractional form.

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Division of Decimals

• Example
• 4.15 32.2

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Division of Decimals

• Example

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Division of Decimals

• Make both the numerator and the denominator of
  the fraction whole numbers
• To do this, multiply both top and bottom by a
  sufficient power of 10.

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Division of Decimals

• In our example, we need to multiply by 100
• This will make the denominator 3,220 and the
  numerator 415.

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Division of Decimals

• Now divide 3,220 into 415.

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Assessment Question 7

• 10.8 / 1080.00
• 0.11
• 1.74
• 0.01
• 1.8
• 1.05

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Rounding Decimals to the Nearest Place

• To round a decimal to the nearest place, look at
  the digit immediately to the right of that place.
  
• If that digit is 5, 6, 7, 8, or 9, then round up
  the place you are rounding to.

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Rounding Decimals to the Nearest Place

• If the digit immediately to the right of the
  place you are rounding to is 0, 1, 2, 3, or 4,
  then don't change the digit at the place you are
  rounding to.
• In either case, in the rounded-off number, there
  will be no digits to the right of the place you
  are rounding off to.

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Rounding Decimals to the Nearest Place

• Example
• Round 0.5827 to the nearest hundredth.

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Rounding Decimals to the Nearest Place

• Round 0.5827 to the nearest hundredth.
• The digit in the hundredths' place is 8.
• Immediately to the right of the 8 is a 2.
• Because 2 is among the digits 0 through 4, we
  keep the digit in the hundredths' place the same.
  
• So, 0.5827 rounded to the nearest hundredth is
  0.58.
• In general we round measurements and calculations
  to the hundredth in science lab.

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Assessment Question 8

• Round the following to the nearest hundredth
  458.79499
• 460
• 459
• 458.8
• 458.79
• 458.795

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Assessment Question 9
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Assessment Question 10