Transcript Scientific Notaion Standard Form

Standard Form
What is Standard Form
Six Questions
Interactive
43620
12.7 Million to Standard Form
Standard Form to Large
4.362 x 10
7
5.084 x 10
2
Interactive
Six Questions
Expressing Small Numbers in Standard Form
0.0005362
0.781
Small SF to normal
Light Year Intro
Light Year Calculation
How far is it from the Earth to the Sun
BOOM!
How far?
92 000 000 miles
Some calculations result in very Large answers
Happy 70th Birthday!
SPLAT!
Howyears
many= seconds
in 70
years?
70
2 200 000
000
seconds!
Using millions to understand the size
Dinosaurs roamed the earth
228 000 000 years ago
Dinosaurs roamed the earth
228 million years ago
Number is different formats
CostSavers made a profit of £7 500 000
MegaSales made a profit of £ 1 230 000
or
CostSavers made a profit of £7 .5 Million
MegaSales made a profit of £ 1.23 Million
Do you need to write all the ZEROS to make sense of the number
Standard Form
A number in STANDARD FORM has two parts
1.01
5.038
1.01
2.79
1.01
9.999
1.01
Number between 1 and
9.999999999….
x
x
x
10
10
10
7
12
-8
18
x
10
x
Power of 10
Making sense of the code
10
 10
1
100 = 10 x 10
 10
2
 10
4
10 000 = 10 x 10 x 10 x 10  10
5
100 000 = 10 x 10 x 10 x 10 x 10  10
1 000 = 10 x 10 x 10
3
1 000 000 = 10 x 10 x 10 x 10 x 10 x 10
 10
6
Not beginning with 1
200 = 2 x 10 x 10
2 x
=
4 000 = 4 x 10 x 10 x 10
70 000 = 7 x 10 x 10 x 10 x 10
10
= 4 x
= 7
3 000 000 = 3 x 10 x 10 x 10 x 10 x 10 x 10
2
x
10
10
3
4
= 3 x 10
This is also known as
Scientific Notation.
6
Whole No to SF
(2) 20 000
(3) 500
 2 10
= 2 x 10 x 10 x 10
(1) 2 000
= 2 x 10x10x10x10
3
 2 10
= 5 x 10 x 10
(4) 800 000
= 8x10x10x10x10x10
4
 5 10
2
 8 10
5
(5) 9 000 000 = 9x10x10x10x10x10x10  9 10
6
43620 to STANDARD FORM
4
3
6
2
0
4
3
6
2
x 10
Move your finger, from point, until you get a
whole number less than 10
Copy figure then add a point
Copy other figure until all that is to be copied is zeros
Add x 10
To change to STANDARD FORM
4
4
3
3
6
6
2
2
0
x 10
4
213
Count number of places from new position to old position
This number goes above the 10 to indicate how often
you multiply by 10
3820000 to STANDARD FORM 2
3
8
2
3
8
2
0
0
0
0
x 10
Move your finger, from point, until you get a
whole number less than 10
Copy figure then add a point
Copy other figure until all that is to be copied is zeros
Add x 10
To change to STANDARD FORM 2
3
3
8
8
2
0
2 x 10
0
0
0
6
5
4
2
13
Count number of places from new position to old position
This number goes above the 10 to indicate how often
you multiply by 10
907.5 to STANDARD FORM 3
9
0
7
5
9
0
7
5
x 10
Move your finger, from point, until you get a
whole number less than 10
Copy figure then add a point
Copy other figure until all that is to be copied is zeros
Add x 10
To change to STANDARD FORM 3
9
9
0
0
7
7
5
5
x 10
21
Count number of places from new position to old position
This number goes above the 10 to indicate how often
you multiply by 10
2
907.5 = 9.075 x 10
Large To SF Interactive
8
9
1
0
0
0
0
x 10
New Example
0
Test
√ ² C x
÷
^
5 6 7 8 9
+
-
On
0 1 2 3 4
.
=
Ans
(-)
7.53x10^5
753000
Exp
Large to SF Examples
(a) 52700
(b) 6650000
(c) 240000000
(d) 6890000
(e) 4510
x
x
6.65
x
2.4
6.89
4.51
(f) 805000
New
5.27
8.05
7.84x10^6
7840000
x
x
4
10
10
10
10
10
6
8
6
3
5
x
10
(-)
√ ² C x
÷
^
5 6 7 8 9
+
-
On
0 1 2 3 4
.
=
Ans
Exp
12.7 Millions to SF
Change 12.7 million to Standard Form
1
1
2
2
7
7
0
0
x 10
0
0
0
7
45
1236
12 million would have 6 zeros .
Write down 12 then an underline where the zeros would have been
Add any figures after the point above the underline then fill
remainder with zeros
NOW CHANGE TO STANDARD FORM
7
12.7 million = = 1.27 x 10
Millions to SF
Change 7¾ million to Standard Form
7
7
7
7
5
5
0
0
x 10
0
7¾ million = 7.75 million
0
45
1236
7 million would have 6 zeros .
Write down 7 then an underline where the zeros would have been
Add any figures after the point above the underline then fill
remainder with zeros
NOW CHANGE TO STANDARD FORM
6
7¾ million = 7.75 x 10
Millions to SF : Examples
(a) 5.3 million
(b) 1.3 million
(c) 49 million
(d) 78 million
(e) 7 million
(f) 16 million
x
5.3
10
x
1.3
x
4.9
7.8
x
7
x
1.6
6
7
10
10
10
7
6
7
√ ² C x
÷
^
5 6 7 8 9
+
-
On
0 1 2 3 4
.
=
Ans
(-)
0
10
10
x
New
6
Exp
Standard Form to Normal
To multiply a whole number by 10 just add a zero
In Standard form the power of 10 ( small number above the 10 )
tells you how often to multiply by 10.
Click the arrow to see some simple examples
1
x 10
9
= 1
0
0
0
0
Normally the 1st number includes a point.
0
0
0
0
0
What is …… ?
(1)
(2)
(3)
(4)
(5)
(6)
3
= 600
3
= 8 000
5
= 400 000
2
= 700
8
= 300 000 000
6
= 9 000 000
6 x 10
8 x 10
4 x 10
7 x 10
3 x 10
9 x 10
1
3.715x10
3.715x10
3.715x10
3.715x10
2
3
4
.
5
7
1
= 3
7
. 1
5
= 3
7
1
. 5
= 3
7
1
5
= 3
7
1
5
3
x 10
4
SF to Normal T
0
=
=
=
=
3.715 x 10
Next
4
3
Finish
7
1
5
0
Multiplying by a +ve power of 10
moves the point to the right
Back to Normal
=
4
3
6
2
4
3
6
2
x 10
0
0
7
0
0
Start as if there were no figures after point
You have to multiply by 10 seven times.
Instead of adding 7 zeros put 7 underlines
The “lines” show where the point should go
Copy other figure until all that is to be copied is zeros
Fill remaining places with zeros
Back to Normal
=
5
0
8
5
0
8
x 10
4
2
4
Start as if there were no figures after point
You have to multiply by 10 twice
Instead of adding 2 zeros put 2 underlines
The “lines” show where the point should go
Copy other figure until all that is to be copied is zeros
5.084 x 10
2
= 502.4
Large SF to normal
3
New Example
6
x 10
8
Test
0
Place Point
√ ² C x
÷
^
5 6 7 8 9
+
-
On
0 1 2 3 4
.
=
Ans
(-)
6
0
Exp
SF to Normal Examples
(a)
(b)
(c)
(d)
(e)
(f)
New
4.73
x
x
9.62
x
6.47
8.73
3.16
2.77
x
x
x
3
10
10
10
10
10
10
3
5
5
3
5
4730
9620
647000
873000
3160
277000
√ ² C x
÷
^
5 6 7 8 9
+
-
On
0 1 2 3 4
.
=
Ans
(-)
7.84x10^6
7840000
Exp
Small Numbers
How wide is an atom?
0.000 000 000 1 metres wide!
Small numbers like this will have negative powers of 10
.
3
3
3.9x10
3.9x10
3.9x10
3.9x10
3.9x10
3.9x10
3.9x10
Next
9
0
=
3
9
0
=
3
9
=
3
. 9
0 .
3
9
0 .
0
3
9
0
-1
=
-2
-3
-4
Introducing SmallT
3
1
3.9 x 10
9
8
=
2
3.9x10
x 10
=
=
0 .
0
0
3
9
= 0 .
0
0
0
3
9
8
0
3
Number
>=1 …….. Power of 10 will be positive
Number between 0 and 1 …….. Power of 10 will be negative
Small to SF
0.15
0.00132
0.0000042
0.0000042
0.0000846
0.39
=
0.97
0.1
4.42
x
0.0596
10
0.0000194
-6
0.0000035
0.00007
0.012
Click number to convert. Numbers from 0 to 1 have negative powers.
Compare number of zeros at front to the power.
Next Examples
Small Numbers
0
0
0
0
5
5
3
3
6
6
2
2
x 10
-2
-3
-4
-1
Ignore the zeros at the front then cover
until you get a number less than 10
Copy figure, add a point then other figure until all that is left are zeros
Add x 10
The original position of the point is to THE LEFT
Count …. 1 to left …. -1 …. 2 to left ….. -2 and so on
-4
0.0005362 = 5.363 x 10
Small Numbers
0
7
7
8
8
1
1
x 10
-1
Ignore the zeros at the front then cover
until you get a number less than 10
Copy figure, add a point then other figure until all that is left are zeros
Add x 10
The original position of the point is to THE LEFT
Count …. 1 to left …. -1 …. 2 to left ….. -2 and so on
0.781 = 7.81 x 10
-1
Small To SF Interactive
0
0
0
0
0
8
7
9
5
x 10
Click top no then click destination.
New Example
0
Use arrows to set power
√ ² C x
÷
^
5 6 7 8 9
+
-
On
0 1 2 3 4
.
=
Ans
(-)
Test
0
Exp
Small to SF Examples
(a) 0.00000142
(b) 0.00627
(c) 0.00075
(d) 0.0012
(e) 0.00933
(f) 0.0000000388
1.42
6.27
x
10
x
x
7.5
1.2
x
9.33
x
3.88
0
10
10
10
10
x
10
-3
-4
-3
-3
-8
√ ² C x
÷
^
5 6 7 8 9
+
-
On
0 1 2 3 4
.
=
Ans
(-)
New
-6
Exp
8
8.437x10
8.437x10
8.437x10
8.437x10
8.437x10
8.437x10
8.437x10
8.437x10
3
=
8
4
3
7
=
8
4
3 .
7
=
8
4 .
3
7
=
8
. 4
3
7
0 .
8
4
3
7
0 .
0
8
4
3
7
2
1
0
-1
-2
-3
-4
T
4
Small
SF
to
Normal
x
10
. 4
7
3
=
=
=
0 .
0
0
8
4
3
7
= 0 .
0
0
0
8
4
3
7
4When the power is negative the point moves to left
8
8.437 x 10
Next
There will be the same no of zeros at front as power
Small SF to Normal
7
=
0
0
0
0
0
0
0
0 7
8
8
Start as if there were no figures after point
0
0
3
x 10
-8
3
0
Positive means GO RIGHT ….. Negative means GO LEFT
Need 8 underline going left starting under the 7 …. -8 …. 8 to LEFT
The “lines” show where the point should go
Copy other figure after the 7
Small SF to normal
2
New Example
-6
x 10
8
Test
0
Place Point
√ ² C x
÷
^
5 6 7 8 9
+
-
On
0 1 2 3 4
.
=
Ans
(-)
6
0
Exp
Small SF to Normal Examples
(a)
(b)
(c)
(d)
7.42
7.01
1.03
x
x
x
3.21
x
(e) 8.91
x
(f) 2.38
x
-5
10
10
10
10
10
10
-5
-8
0.0000742
0.0000701
0.0000000103
-6
0.00000321
-4
0.000891
-4
0.000238
√ ² C x
÷
^
5 6 7 8 9
+
-
On
0 1 2 3 4
.
=
Ans
(-)
New
0
Exp
Positive Powers
(a)
(b)
(c)
x
1.2
9.24
x
x
7.1
10
10
10
(d) 562000
(e) 418000
(f)
New
3420
6
5
6
1200000
924000
7100000
5.62
4.18
3.42
x
x
x
7840000
10
10
5
3
√ ² C x
÷
^
5 6 7 8 9
+
-
On
0 1 2 3 4
.
=
Ans
(-)
7.84x10^6
10
5
Exp
Negative Powers
(a)
(b)
(c)
7.82
6.58
5.43
x
x
x
10
10
10
(d) 0.0000000913
(e) 0.00206
(f)
New
0.00000239
-5
-5
-4
0.0000782
0.0000658
0.000543
9.13
2.06
2.39
x
x
x
7840000
10
10
-3
-6
√ ² C x
÷
^
5 6 7 8 9
+
-
On
0 1 2 3 4
.
=
Ans
(-)
7.84x10^6
10
-8
Exp
Mixed Powers
(a)
(b)
(c)
7.36
8.84
6.49
x
x
x
10
10
10
(d) 5280
(e) 8000000
(f)
New
0.000311
4
-3
8
73600
0.00884
649000000
5.28
x
8
x
3.11
x
7840000
10
10
6
-4
√ ² C x
÷
^
5 6 7 8 9
+
-
On
0 1 2 3 4
.
=
Ans
(-)
7.84x10^6
10
3
Exp
Light Year
How far is it to the Town Centre?
5 minutes by car or about 20 minutes walk.
How far is it to Glasgow?
About 20 minutes by car?
How far is it to London ?
About 400 miles by road taking about 7 hours or about 5
hours by train
Sometimes the time a journey takes is a better indication of
the distance.
Light Years
Distances in the solar system are vast.
Distance from the Sun to Pluto is
59 400 000 000 km or 5.95 x 1010 km
To make sense of distance people often use
time. For these extremely large distances
scientists use the time that Light takes to go
from one point to another
As a comparison light takes about 1.27 seconds to go from
the Moon to Earth.
Light takes 4 Hours and 2 minutes to go from Pluto to the Earth
About 22 000 times the time so about 22000 the distance
Distances around the Universe
The distance from the Sun to the Earth is
150 000 000 km or 1.5 x 108 km
It is hard to make sense of this distance
The distance round the equator is about 38 000 km or
380 hours (nearly 16 days ) by car.
The distance from Sun to Earth is about 4 000 times a
journey round the equator or 174 year by car
Scientist need to compare these distances and as
there are no roads in space they use one quantity that
can move there LIGHT
Time for light to travel from
Earth to the Sun
8 min 20 sec
Earth to Mercury
5 min 10 sec
Earth to Venus
2 min 20 sec
Earth to the moon
1.268 sec
Earth to Mars
4 min 10 sec
Earth to Jupiter
35 min
Earth to Saturn
71 min
Earth to Uranus
2 hr 31 min
Earth to Pluto
4 hrs 2 min
If a spacecraft could travel at the speed of light it would take
about 1.3 seconds to get to the moon
The distance from Earth to Pluto is like 3 600 RETURN
journeys to the moon
4.22 Light Years
Proxima Centauri is the closest star to the Solar
System. It is 4.22 Light years away
If it was possible to build a craft which could travel at the speed of
light it would take over 4 years to get to this star. 4 hours into the
journey it would pass Pluto having completed about 1 / 365 part of
the journey
1 Light Year is 9.467 x 1012 km
Distance to Proxima Centauri
is 4.22 Light Year
4.22x9.47 x 1012 = 4.22 x 9.47
4.22 x 9.467
x 10x
12
4.22
E
12
12
Light
Year
9.467
x 10
km
4.22 x
=3.995 x 1013 km
7 Light Years
7 Light Year
7x9.47 x 1012 =
7 x 9.467
7
7 x 9.47
x 10x
E
12
12
32 x 9.467
7x
=6.629 x 1013 km
32 Light Year
32x9.47 x 1012 =
32
32 x 9.47
x 10x
E
12
12
Light
Year
9.467
x 10
km
12
12
Light
Year
9.467
x 10
km
32 x
=3.0304 x 1014 km
Light Year Calculations
1 Light Year is 9.467 x 1012 km
6.1
Light Year = 6.1
On
6.1
x
x ( 9.467 x 1012 ) km
9.467
12
Exp
=
√ ² C x
÷
^
5 6 7 8 9
+
-
On
0 1 2 3 4
.
=
Ans
(-)
0
=
=
Click after using calculator
Next
x 10
Exp
Light Year
Speed of Light is 299 792 458 m/s
In 1 second Light travels a distance of 300 million
metres or 300 000 km
In 1 hour Light travels a distance of 1 080 000 000
km or 1 080 Million Km or 1.08 x 109 km
In 1 year Light travels a distance of 9 467 280 000 000 km or
1 080 Million Km or 9.47 x 1012 km
To make sense it may be better to relate this to distances in the
Universe
One Light Year