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Chapter 3. scientific notation, area, and volume

# 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...

Dorm Inspection

Your dorm room is becoming hazardous

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

56   Chapter 3

scientific notation

Head First U Depar tment of

Dorm Inspection

Your dorm room is becoming haz

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?

you are here 4   57

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...

58   Chapter 3

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.

you are here 4   59

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.

60   Chapter 3

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.

you are here 4   61

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

said in here.

Number = 1 x 236 =

What’s that great big

’E’ doing in the middle

Huh?! This doesn’t

make sense at all!

It’s important to understand the

62   Chapter 3

Don’t just copy

and move on.

scientific notation

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.

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

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.

you are here 4   63

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

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?!

64   Chapter 3

scientific notation

Does writing our answers with scientific

notation really help us keep track of the digits?

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

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

10

(3 sd)

to 3 significant digits without making mistakes.

you are here 4   65

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:

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

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.

66   Chapter 3