Chapter 3. scientific notation, area, and volume
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disgusting dormitory
A messy college dorm room
Well, actually a particularly filthy college dorm room - Matt and Kyle
probably wouldn’t know one end of a vacuum cleaner from the other,
and the idea of cleaning has never entered their heads.
But the Dorm Inspector has had enough...
Head First U Department of
Dorm Inspection
Your dorm room is becoming hazardous
to your health,
and this state of affairs must be
dealt with. We’ve
detected a single specimen of a
bug that doubles
itself every twenty minutes.
If the bugs grow to occupy more
than 6 × 10-5 m3 they’ll
take over your room, and you will
need to find a new
place to live while we fumigate your
living area.
Sincerely,
Dorm Inspection Team
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scientific notation
Head First U Depar tment of
Dorm Inspection
Your dorm room is becoming haz
ardous to your health,
and this state of affairs must be
dealt with. We’ve
detected a single specimen of a
bug that doubles
in number every twenty minutes.
If the bugs grow to occupy more
than 6 × 10—5 m3, they’ll
take over your room, and you will
need to find a new
place to live while we fumigate you
r living area.
Sincerely,
Dorm Inspection Team
So how long before things go really bad?
Do we have to clean up
tonight, or can we just
wait until tomorrow?
Yeah, how serious can
these little bugs be?
Every 20 minutes, the bugs will divide in two. So the
total number of bugs will double every 20 minutes.
How many do you think there will be by tomorrow
(12 hours later) - and how might you work that out?
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how many?
Is it true? Can these bugs
really get us evicted?
Kyle: Whether they can or not, just the thought of it makes me
queasy. Maybe we oughta just straighten this place up now.
Matt: I’m sooooooo tired. Can’t we wait until tomorrow?
Kyle: But that might be too late!
Matt: We can work it out, right? The bug doubles every 20 minutes,
and it’s 10 pm now. If we get up at 10 am, we’ve given the bug 12
hours to keep on doubling. Surely there can’t be that many by then?
Kyle: OK, let me sketch this out...
... if we start off with one bug ...
... after 20 minutes, there’ll be two of them ...
... 40 minutes = 4 bugs ...
... 1 hour = 8 bugs - not that many and
we only need to give it 12 hours ...
... 1 hour 20 minutes = 16 bugs ...
... 1 hour 40 minutes = ...
Kyle: ... hmmm, I’m not sure - my drawing’s getting messy!
Matt: Yeah, the drawing will take forever. There’s gotta be a math
way to figure out how many bugs there’ll be after 12 hours.
Kyle: Yeah, OK.
Matt: Hmmm. I can’t think of an equation for “the bugs double
every 20 minutes,” but we could just make a table to keep track of
things and keep on doubling until 12 hours are up. Then we’ll know
how many bugs there’ll be by the morning.
Kyle: I think that’ll work, but there’s still that funny phrase in the
note, “If the bugs grow to occupy more than 6 × 10—5 m3.” I don’t
know what that is, but it sure ain’t a number of bugs.
Matt: Why don’t we worry about that later, once we know how
many bugs there’ll be...
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scientific notation
You start off with 1 bug.
After 20 minutes, it’s doubled
once, and there are 2 bugs.
This is as
far as they
got with
their sketch.
Matt and Kyle have drawn up the table below and started
doubling the bugs. Your job is to finish off the table to see how
many bugs there’ll be after 12 hours.
Number of
doublings
Elapsed
time
Number of
bugs
1
20 min
2
2
40 min
4
3
1h
8
4
1 h 20 min
16
5
1 h 40 min
Number of
doublings
Elapsed
time
Number of
bugs
There are a lot of doublings in 12 hours, so
we’ve given you space to continue the table.
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sharpen solution
You start off with 1 bug.
After 20 minutes, it’s doubled
once, and there are 2 bugs.
Matt and Kyle have drawn up the table below and started
doubling the bugs. Your job is to finish off the table, to see how
many bugs there’ll be after 12 hours.
Number of
doublings
Elapsed
time
Number of
bugs
Number of
doublings
Elapsed
time
Number of
bugs
1
20 min
2
19
6 h 20 min
524288
2
40 min
4
20
6 h 40 min
1048576
3
1 h
8
21
7 h
2097152
4
1 h 20 min
16
22
7 h 20 min
4194304
5
1 h 40 min
32
23
7 h 40 min
8388608
6
2 h
64
24
7
2 h 20 min
128
25
8
2 h 40 min
256
9
3 h
10
This is as
far as they
got with
their sketch.
This is taking, like, forever. Isn’t
8h
16777216
there a button on my calculator I can
use instead of doing all that doubling?
8 h 20 min
33554432
26
8 h 40 min
67108864
512
27
9 h
134217728
3 h 20 min
1024
28
9 h 20 min
268435456
11
3 h 40 min
2048
29
9 h 40 min
536870912
12
4 h
4096
30
10 h
1073741824
13
4 h 20 min
8192
31
10 h 20 min 2147483648
14
4 h 40 min
16384
32
10 h 40 min 4294967296
15
5h
32768
33
11 h
8589934592
16
5 h 20 min
65536
34
11 h 20 min
17179869184
17
5 h 40 min
131072
35
11 h 40 min 34359738368
18
6 h
262144
36
There are a lot of doublings in 12 hours, so
we’ve given you space to continue the table.
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12 h
68719476736
scientific notation
Power notation helps you multiply
by the same number over and over
This whole term is
the same as
2x2x2x2x2
Number that
you’re multiplying
by lots of times.
If you want to multiply by the same number several times
over, you can write it down using power notation. This
means that 2 × 2 × 2 × 2 × 2 becomes 25, as there are five
instances of 2. When you say 25 out loud, you say “two to the
power of five” or sometimes just “two to the five.” The five
part is called the index.
How many times you’re
multiplying by it. This
is called the INDEX.
Your calculator’s power button gives you superpowers
You can use the power button on your calculator to multiply
by the same number lots of times without having to type it all
out. Usually, you type in the number you want to multiply by,
then press the power button, then type the number of times you
want to multiply by it.
Watch out though - different calculators have different things
written on the power button! Make sure you know what yours
looks like and how it works before you try to use it!
Number that
you’re multiplying
by lots of times.
Index
If your calculator doesn’t have a power button, then you’ll need
to get a scientific calculator. It’ll help you out in the long run
as you move onto solving more sophisticated and complicated
physics problems.
There’s space here to
explain what you’re doing.
(a) The number of bugs
doubles every 20 minutes.
How many times do you
need to multiply by 2 to get
the total number of bugs
after 12 hours?
(b) How many bugs will
there be after 12 hours?
Power
notation makes
multiplying
by the same
number over
and over
less prone to
mistakes.
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sharpen solution
Take time to jot down
what you’re doing and
why. It helps you to
stay on track..
(a) The number of bugs
doubles every 20 minutes.
How many times do you
need to multiply by 2 to get
the total number of bugs
after 12 hours?
(b) How many bugs will
there be after 12 hours?
(a) There are 3 lots of 20 minutes
in an hour. In 12 hours there are 3 x
12 = 36 periods of 20 minutes, so
they double 36 times.
(b) Number of bugs after 12 h =
number at the start x 36 groups of 2.
Write what
YOUR calculator
said in here.
Number = 1 x 236 =
What’s that great big
’E’ doing in the middle
of my answer?!
Huh?! This doesn’t
make sense at all!
It’s important to understand the
answers your calculator gives you.
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Don’t just copy
answers down
and move on.
scientific notation
Your calculator displays big numbers
using scientific notation
Sometimes, an answer has too many digits to fit on your
calculator’s screen. When that happens, your calculator displays
it using scientific notation. Scientific notation is an efficient
and shorter way of writing very long numbers.
The value of 236 has 11 digits in it, but a calculator doesn’t have
enough space to display all of the digits. So instead, they’ve
rounded the answer to the number of significant digits that
they can fit on the screen.
The first part of the number on the screen is for the part that
starts 6.87...
In math, scientific notation is
often called standard form.
Don’t worry, they’re the same thing.
Answers written
in scientific notation
have two parts.
But there were already 8 bugs after an hour. 8 is more than
6.87, so how can the answer to 236 possibly be that small?!
The second part of the number tells you
the size of the first part.
Numbers written in scientific notation have two parts.
The first part is a number with one significant
digit before the decimal point and the rest of the
number after the decimal point.
The second part tells you the number of 10’s you
have to multiply the first part by to make your
answer the correct size.
The first calculator’s given an answer
of 6.871947674 × 1010. It’s given you
10 significant digits, and the number is
the same as writing 6.871947674 × 10
× 10 × 10 × 10 × 10 × 10 × 10 × 10 ×
10 × 10, which is 68719476740.
This part tells
you how many 10’s
to multiply the
first part by.
The second calculator has displayed
6.871947E10, which is 68719470000it’s rounded the answer to seven
significant digit and used an E to
indicate the second part of the number
because of the limits of its display.
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the science behind scientific notation
Scientific notation uses powers of 10
to write down long numbers
So you don’t have to
write them out the long
way if they have 28
digits or something!
You calculator’s given you the answer 236 = 6.871947674 × 1010. You know
that to get this into the form you’re used to, you need to multiply the first part
of the number by ten groups of 10.
Each time you multiply by 10, the number’s digits shift along one place to
the left so that each digit is worth 10 times more than it was before.
But it’s quite hard for you to draw that, so practically speaking, you can get
to the same place by hopping the decimal point the correct number of
times to the right. Then the number becomes 68719476740.
You can work out
where the number’s
digits should lie
by ‘hopping’ the
decimal point.
Each time you multiply by 10, the decimal point hops along one place to make the number bigger.
This bit tells you how
7 8
1
6
9 10
2
3
4 5
many 10’s to hop along by.
6.871947674
68719476740
You should round your answers to
three significant digits, like
you did in Chapter 2.
× 1010
Decimal point
is now here.
You need to put a placeholding
zero in here for this hop.
But numbers like that are really
annoying to round. There are so many zeros to
write in at the end; I always lose count. Does it work
out as 6870000000 or 68700000000?!
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scientific notation
Does writing our answers with scientific
notation really help us keep track of the digits?
Scientific notation helps you round your answers.
If you have to round a long number to 3 significant digits (sd) as a
final answer, then the start is OK, but putting in the right number
of zeros is a real pain.
It’s a lot easier in scientific notation, as the tens are spelled out at
the end of the number. This lets you rewrite 6.871947674 x 1010 as
6.87 x 1010 without hopping the decimal point along.
10
9
8
7
6
5
4
3
2
1
68719476740
You can just start here, as this is
the answer your calculator gave you.
Strictly speaking, it’s
the digits that move,
not the decimal
point, but that’s
much harder for you
to draw!
To convert a normallywritten number into
scientific notation, count
how many hops until only
one digit is left in front
of the decimal point, and
multiply the number by
that many 10’s.
6.871947674 × 1010
3 significant digits
The less significant digits
- round to get rid of them.
6.87 × 10
You don’t need placeholding
zeros after a decimal point.
Or however many sd is
appropriate for your answer.
10
(3 sd)
Scientific notation helps you to round your answers
to 3 significant digits without making mistakes.
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ask... you know you want to
Q:
I thought what you’re calling
‘scientific notation’ is actually called
‘standard form.’ What gives?
A:
They’re both the same thing.
Scientists use the term ‘scientific notation’
and mathematicians ‘standard form.’
Q:
Why should I bother with scientific
notation when I’m really careful about
how I type numbers into my calculator?
A:
Q:
A:
How big are we talking about?
Well, the earth’s mass is
5.97 × 1024 kilograms. That’s a very big
number with a lot of zeros at the end if you
write it out longhand.
Q:
OK, I can see why I might not be
happy handling over 20 zeros at the
end. But why would I ever want to write
an answer I’ve worked out myself in
scientific notation?
Your calculator screen might not be
big enough to display an answer that’s
either really big or really small. So you
need to understand scientific notation, or it
won’t make sense.
A:
But I have a super duper flashy
calculator that’ll display lots and lots
of digits on its humongous screen. So
if I’m careful, why would I ever need
scientific notation?
You can just take the number your
calculator gives you, for example,
6.871947674 × 1010, and write 6.87 × 1010
without having to do anything else to it?
Q:
If you’re rounding your answer
to 3 significant digits (like you’ll do in
your exam), then it’s much easier to use
scientific notation than it is to scrawl a
whole lot of zeros across your page.
A:
Q:
So are you saying that scientific
notation isn’t just there because my
calculator’s screen isn’t big enough - it
helps me as well?
A:
Yep, scientific notation helps you to
write and round very long numbers in a
much shorter form. So it’s not just about
calculators - it’s about making your life
easier.
Q:
So which came first - small
calculator screens or scientific notation.
A:
Q:
Scientific notation came first by
several hundred years!
I have one more question. Are
numbers in scientific notation always
written with one digit in front of the
decimal point? Couldn’t you equally
write 6.87 x 1010 as 687 x 108?
A:
Conventionally, they’re written with
one digit in front of the decimal point. Your
brain will soon get used to estimating the
size of the number from the tens part, so
sticking to the convention is best.
You could be given a number in
scientific notation to work with - in an exam
question or when you look something up to
find out how big it is.
Scientific notation helps you to
handle very long numbers that
would otherwise have many digits,
even when you’ve round them.
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