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Standard FormWhat is Standard Form
43620 Interactive
12.7 Million to Standard Form
4.362 x 10
75.084 x 10
2Interactive
0.0005362 0.781
Light Year Intro Light Year Calculation
Six Questions
Standard Form to Large
Six Questions
Expressing Small Numbers in Standard Form
Small SF to normal
Some calculations result in very Large answers
How many seconds in 70 years?
70 years = 2 200 000 000 seconds!
Happy 70th Birthday!
SPLAT!
Using millions to understand the size
Dinosaurs roamed the earth 228 million years
ago
Dinosaurs roamed the earth 228 000 000
years ago
Number is different formats
CostSavers made a profit of £7 500 000
MegaSales made a profit of £ 1 230 000
CostSavers made a profit of £7 .5 Million
MegaSales made a profit of £ 1.23 Million
or
Do you need to write all the ZEROS to make sense of the number
Standard Form
A number in STANDARD FORM has two parts
1.011.01 1
0x
7
1.015.038 1
0x
12
1.012.79 1
0x
-8
1.019.999 1
0x
18
Number between 1 and 9.999999999…. x Power of 10
Making sense of the code
100 = 10 x 10
1 000 = 10 x 10 x 10
100 000 = 10 x 10 x 10 x 10 x 10
10 000 = 10 x 10 x 10 x 10
1 000 000 = 10 x 10 x 10 x 10 x 10 x 10
10 210
310410
510
110
610
Not beginning with 1
200 = 2 x 10 x 10
4 000 = 4 x 10 x 10 x 10
70 000 = 7 x 10 x 10 x 10 x 10
3 000 000 = 3 x 10 x 10 x 10 x 10 x 10 x 10
This is also known as
Scientific Notation.
2 x 102
=
4 x 103
104
x7=
=
3 x= 106
Whole No to SF
(1)2 000
(2) 20 000
(3) 500
(4) 800 000
(5) 9 000 000
= 5 x 10 x 10 25 10 = 8x10x10x10x10x10
58 10
= 2 x 10x10x10x10
42 10
= 9x10x10x10x10x10x10 69 10
= 2 x 10 x 10 x 10
32 10
43620 to STANDARD FORM
x 10
4 3 6 2 0
4 3 6 2
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
x 10
4 3 6 2 0
4 3 6 21
Count number of places from new position to old position
234
This number goes above the 10 to indicate how often you multiply by 10
0 0
3820000 to STANDARD FORM 2
x 10
3 8 2 0 0
3 8 2
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
x 10
3 8 2 0 0
3 8 21
Count number of places from new position to old position
234
This number goes above the 10 to indicate how often you multiply by 10
0 0
56
907.5 to STANDARD FORM 3
x 10
9 0 7 5
9 0 7
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
5
To change to STANDARD FORM 3
x 10
9 0 7 5
9 0 71
Count number of places from new position to old position
2
This number goes above the 10 to indicate how often you multiply by 10
5
907.5 = 9.075 x 10
2
Large To SF Interactive
0 1 2 3 4
5 6 7 8 97.53x10^5 C
.
÷x
753000+ On
²
-
Ans
=
√(-) ^ Exp
x 10
8 9 1 0 0 0 0
New Example
0Test
Large to SF Examples
0 1 2 3 4
5 6 7 8 97.84x10^6 C
.
÷x
7840000+ On
²
-
Ans
=
√(-) ^ Exp
5.27 x 10
4
(a)
52700
6.65 x 10
6
(b) 6650000
2.4 x 10
8
(c)
240000000
6.89 x 10
6
(d) 6890000
4.51 x 10
3
(e) 4510
8.05 x 10
5
(f) 805000
New
12.7 Millions to SFChange 12.7 million to Standard Form
1 2 7 0 0 0 0 0
x 101 2 7
12 million would have 6 zeros . Write down 12 then an underline where the zeros would have beenAdd any figures after the point above the underline then fill remainder with zeros
123456
NOW CHANGE TO STANDARD FORM
12.7 million = = 1.27 x 107
7
Millions to SFChange 7¾ million to Standard Form
7 7 5 0 0 0 0
x 107 7 5
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
123456
NOW CHANGE TO STANDARD FORM
7¾ million = 7.75 x 106
7¾ million = 7.75 million
Millions to SF : Examples
0 1 2 3 4
5 6 7 8 9
C
.
÷x
0+ On
²
-
Ans
=
√(-) ^ Exp
5.3 x 10
6
(a)
5.3 million
1.3 x 10
6
(b) 1.3 million
4.9 x 10
7
(c)
49 million
7.8 x 10
7(d) 78 million
7 x 10
6(e) 7 million
1.6 x 10
7(f) 16 million
New
Standard Form to Normal
1 109
x = 1 0 0 0 0 0 0 0 0 0
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
Normally the 1st number includes a point.
What is …… ?
= 6006 x 10
3(1)
= 8 0008 x 10
3(2)
= 400 0004 x 10
5(3)
= 7007 x 10
2(4)
= 300 000 0003 x 10
8(5)
= 9 000 0009 x 10
6(6)
SF to Normal
3.715 x 104
3 13.715x10 2
= 7 . 5
3 13.715x10
1
= 7 5.
3 13.715x10
3= 7 5
3 13.715x10
4= 7 5 0
=
=
=
=
x 104 T
Next
.3 7 1 5
3 7 1 5 0
Finish Multiplying by a +ve power of 10 moves the point to the right
Back to Normal
x 104 3 6 2
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
Copy other figure until all that is to be copied is zeros
7
4
Fill remaining places with zeros
3 6 2 0 0 0 0=
The “lines” show where the point should go
Back to Normal
x 105 0 8 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
Copy other figure until all that is to be copied is zeros
2
5
5.084 x 10
0 8 4=
The “lines” show where the point should go
2= 502.4
Large SF to normal
0 1 2 3 4
5 6 7 8 96 C
.
÷x
0+ On
²
-
Ans
=
√(-) ^ Exp
x 106
3 8
New Example
0Test Place Point
SF to Normal Examples
0 1 2 3 4
5 6 7 8 97.84x10^6 C
.
÷x
7840000+ On
²
-
Ans
=
√(-) ^ Exp
4.73 x 10
3
(a)
4730
9.62 x 10
3
(b) 9620
6.47 x 10
5
(c)
647000
8.73 x 10
5
(d) 873000
3.16 x 10
3
(e) 3160
2.77 x 10
5
(f)277000
New
Small Numbers
How wide is an atom?
0.000 000 000 1 metres wide!
Small numbers like this will have negative powers of 10
Introducing Small
3.9 x 108
3.9x10
2
= 3 9 0 0
3.9x10
3
= 3 9 0
3.9x101
= 3 9
3.9x100
= 3 9.
3.9x10-1
= 0 3 9.
03.9x10-2
= 0 3 9
03.9x10-3
= 0 0 3 9.
0 03.9x10-4
= 0 0 3 9.
x 108
.
T
Next
.3 9
3Number >=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 0.0000194
0.39
0.97
0.1 0.0596 0.00007 0.012
0.0000035
0.0000846
4.42 x 10
-6
0.0000042
=
Next Examples
Click number to convert. Numbers from 0 to 1 have negative powers. Compare number of zeros at front to the power.
6 2
Small Numbers
x 10
0 0 0 5 3
5 3 6 2
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 zerosAdd x 10
0
The original position of the point is to THE LEFT
Count …. 1 to left …. -1 …. 2 to left ….. -2 and so on
-1-2-3-4
0.0005362 = 5.363 x 10
-4
1
Small Numbers
x 10
0 7 8
7 8 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 zerosAdd x 10
The original position of the point is to THE LEFT
Count …. 1 to left …. -1 …. 2 to left ….. -2 and so on
-1
0.781 = 7.81 x 10
-1
Small To SF Interactive
0 1 2 3 4
5 6 7 8 9
C
.
÷x
0+ On
²
-
Ans
=
√(-) ^ Exp
x 10
0 0 0 0 0 8 7 9 5
New Example
0
Test
Click top no then click destination.
Use arrows to set power
Small to SF Examples
0 1 2 3 4
5 6 7 8 9
C
.
÷x
0+ On
²
-
Ans
=
√(-) ^ Exp
1.42 x 10
-6
(a)
0.00000142
6.27 x 10
-3
(b) 0.00627
7.5 x 10
-4
(c)
0.00075
1.2 x 10
-3(d) 0.0012
9.33 x 10
-3(e) 0.00933
3.88 x 10
-8(f) 0.0000000388
New
8.437 x 104
8.437x10 2
= 8 4 3 7
8.437x10
3
= .8 4 3 7
8.437x10
1= 8 .4 3 7
8.437x10
0= 8 4. 3 7
8.437x10
-1= 0 8 4 3. 7
08.437x10
-2= 0 8 4 3 7
08.437x10-3
= 0 0 8 4 3 7.
0 08.437x10
-4= 0 0 8 4 3 7.
x 104
.
T
Next
.8 4 3 7
8When the power is negative the point moves to leftThere will be the same no of zeros at front as power
Small SF to Normal
Small SF to Normal
x 107 8 0 3
Start as if there were no figures after point
Positive means GO RIGHT ….. Negative means GO LEFT Need 8 underline going left starting under the 7 …. -8 …. 8 to LEFT
Copy other figure after the 7
-8
7 8 0 30=
The “lines” show where the point should go
0
000000 0
Small SF to normal
0 1 2 3 4
5 6 7 8 96 C
.
÷x
0+ On
²
-
Ans
=
√(-) ^ Exp
x 10-6
2 8
New Example
0Test Place Point
Small SF to Normal Examples
0 1 2 3 4
5 6 7 8 9
C
.
÷x
0+ On
²
-
Ans
=
√(-) ^ Exp
7.42 x 10
-5
(a)
0.0000742
7.01 x 10
-5
(b) 0.0000701
1.03 x 10
-8
(c)
0.0000000103
3.21 x 10
-6(d) 0.00000321
8.91 x 10
-4(e) 0.000891
2.38 x 10
-4(f) 0.000238
New
Positive Powers
0 1 2 3 4
5 6 7 8 97.84x10^6 C
.
÷x
7840000+ On
²
-
Ans
=
√(-) ^ Exp
1.2 x 10
6
(a)
1200000
9.24 x 10
5
(b) 924000
7.1 x 10
6
(c)
7100000
5.62 x 10
5
(d) 562000
4.18 x 10
5
(e) 418000
3.42 x 10
3
(f) 3420
New
Negative Powers
0 1 2 3 4
5 6 7 8 97.84x10^6 C
.
÷x
7840000+ On
²
-
Ans
=
√(-) ^ Exp
7.82 x 10
-5
(a)
0.0000782
6.58 x 10
-5
(b) 0.0000658
5.43 x 10
-4
(c)
0.000543
9.13 x 10
-8
(d) 0.0000000913
2.06 x 10
-3
(e) 0.00206
2.39 x 10
-6
(f) 0.00000239
New
Mixed Powers
0 1 2 3 4
5 6 7 8 97.84x10^6 C
.
÷x
7840000+ On
²
-
Ans
=
√(-) ^ Exp
7.36 x 10
4
(a)
73600
8.84 x 10
-3
(b) 0.00884
6.49 x 10
8
(c)
649000000
5.28 x 10
3
(d) 5280
8 x 10
6
(e) 8000000
3.11 x 10
-4
(f) 0.000311
New
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 moon 1.268 sec
Earth to Mars 4 min 10 sec
Earth to Venus
5 min 10 secEarth to Mercury
2 min 20 sec
8 min 20 secEarth to the Sun
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
9.467 x 1012 km
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 kmDistance to Proxima Centauri is 4.22 Light Year 4.22 Light Year
4.22 x
4.22 x 9.467 =3.995 x 1013 km
x 10x 12
4.22 x 9.47 E 124.22x9.47 x 1012 =
7 Light Years
9.467 x 1012 km
7 Light Year7 Light Year
7 x
7 x 9.467 =6.629 x 1013 km
x 10x 12
7 x 9.47 E 127x9.47 x 1012 =
9.467 x 1012 km
32 Light Year 32 Light Year
32 x
32 x 9.467 =3.0304 x 1014 km
x 10x 12
32 x 9.47 E 1232x9.47 x 1012 =
Light Year Calculations
0 1 2 3 4
5 6 7 8 9
C
.
÷x
0+ On
²
-
Ans
=
√(-) ^ Exp
1 Light Year is 9.467 x 1012 km
Light Year =
6.1 x 9.467 Exp 12 =On
Next
6.1 x ( 9.467 x 1012 ) km6.1
=
x 10 =
Click after using calculator
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
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