Download Math 302A section 5 - University of Arizona Math

April 21, 2009
0011 0010 1010 1101 0001 0100 1011
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“Energy and persistence conquer all things.”
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~Benjamin Franklin
April 21, 2009
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• Section 5.4 – Decimals
• Exploration 5.16
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5.4 – Decimals
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• Many decimal numbers are rational
numbers, but some are not.
• A decimal is a rational number if it can be
written as a fraction. So, those are decimals
that either terminate (end) or repeat are
rational numbers.
• Repeating decimals: 7.6666…; 0.727272…
• Terminating decimals: 4.8; 9.00001; 0.75
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5.4 (cont’d)
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• A decimal like
3.56556555655556555556…
is not rational because although there is a
pattern, it does not repeat. It is irrational.
• Compare this to
3.556556556556556556…
It is rational because 556 repeats.
• All rational numbers can be represented by
terminating or repeating decimals!
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5.4 (cont’d)
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Another look at place value:
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We know that the number 123,456 represents
one 100,000 plus two 10,000’s plus three
1,000’s plus four 100’s plus five 10’s plus
six 1’s or
1×100,000 + 2×10,000 + 3×1,000 + 4×100 +
5×10 + 6×1
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5.4 (cont’d)
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What does the number 1.234 mean?
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1 is in the ones place
2 is in the “tenths” place (not the tens place!)
3 is in the “hundredths” place
4 is in the “thousandths” place
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5.4 (cont’d)
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From our work with fractions, you should
recognize the difference between “tens” and
“tenths”:
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Each “ten” represents 10 ones, while each
“tenth” is one of the 10 equal pieces that
one whole was divided into.
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5.4 (cont’d)
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Let
=1
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Then the shaded area below represents a
“tenth”:
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And the figures on the next slide make a
“ten”:
5.4 (cont’d)
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10 =
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5.4 (cont’d)
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In symbols, a “tenth” is 1/10.
In decimal form, 1/10 = 0.1
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Similarly, a “hundredth” is 1/100, and a
“thousandth” is 1/1000.
In decimal form, 1/100 = 0.01 and 1/1000 =
0.001
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Can you see how base 10 blocks can be used
to visualize these?
5.4 (cont’d)
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When decimals are equal:
3.56 = 3.56000000
But, 3.056 ≠ 3.560.
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To see why, examine the place values.
3.056 = 3 + 0 × .1 + 5 × .01 + 6 × .001
whole tenths hundredths thousandths
3.560 = 3 + 5 × .1 + 6 × .01 + 0 × .001
5.4 (cont’d)
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Ways to compare decimals:
• Write them as fractions and compare the fractions
as we did in the last section.
• Use base-10 blocks.
• Write them on a number line.
• Line up the place values.
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Exploration 5.16
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Do #1, 2, 4, 7 and 8.
For #8, draw a picture of blocks to represent each
decimal.
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5.4 (cont’d)
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Rounding
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3.784: round this to the nearest hundredth.
• Look at the hundredths.
• Well, 3.784 is between 3.78 and 3.79. On the
number line, which one is 3.784 closer to?
• 3.785 is half way in between.
3.78
3.785
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3.79
5.4 (cont’d)
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Rounding
So, is 3.784 closer to 3.78 or 3.79?
3.78
3.785
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3.79
5.4 (cont’d)
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Practice Rounding:
• Round to the nearest tenth: 5.249
– Closer to 5.2 or 5.3?
• Round to the nearest hundredth: 5.249
– Closer to 5.24 or 5.25?
• Round to the nearest whole: 357.82
– Closer to 357 or 358?
• Round to the nearest hundred: 357.82
– Closer to 300 or 400?
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5.4 (cont’d)
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Practice Rounding:
• Round to the nearest thousandth: 5.0099
– 5.010 Must have the last 0 for the thousandths
place!
• Round to the nearest hundredth: 64.284
– 64.28
• Round to the nearest tenth: 10.957
– 11.0 Must have the last 0 for the tenths place!
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5.4 (cont’d)
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Adding and subtracting decimals:
• Same idea as with fractions: the
denominator (place values) must be
common.
• So, 3.46 + 2.09 is really like
3 + 2 ones +
4 + 0 tenths +
6 + 9 hundredths = 5.55
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5.4 (cont’d)
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Multiplying decimals:
(Easiest to see with the area model)
2.1 × 1.3
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Where is 2 × 1?
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2 × 0.3?
1 × 0.1?
0.1 × 0.3?
+
.3
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+ .1
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5.4 (cont’d)
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Dividing decimals:
Find the quotient
7.8 ÷ 3.12
What steps did you take?
Why do they work?
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Homework
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Link to online homework list:
http://math.arizona.edu/~varecka/302AhomeworkS09.
htm
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*Note: approximate grades from before test 3 are
posted on D2L; I will update as soon as I can.