standard form

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

How far is it from the Earth to the Sun

BOOM!

How far?

92 000 000 miles

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




Add x 10

To change to STANDARD FORM 2

x 10

3 8 2 0 0

3 8 21


234


0 0

56

907.5 to STANDARD FORM 3

x 10

9 0 7 5

9 0 7




Add x 10

5

To change to STANDARD FORM 3

x 10

9 0 7 5

9 0 71


2


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


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


You have to multiply by 10 twice Instead of adding 2 zeros put 2 underlines


2

5

5.084 x 10

0 8 4=


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


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=


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

One Light Year

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