Standard Form What is Standard Form Six Questions Interactive 43620 12.7 Million to Standard Form Standard Form to Large Interactive 7 2 4.362 x 10 5.084 x 10 Six Questions Expressing Small Numbers in Standard Form 0.781 0.0005362 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! How many seconds in 70 years? SPLAT! 70 years = 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 7 1.01 1.01 x 10 12 1.01 5.038 x 10 -8 1.01 2.79 x 10 18 1.01 9.999 x 10 Number between 1 and 9.999999999…. x Power of 10

Making sense of the code 10 100 = 10 x 10 1 000 = 10 x 10 x 10 10 000 = 10 x 10 x 10 x 10 100 000 = 10 x 10 x 10 x 10 x 10 1 000 000 = 10 x 10 x 10 x 10 x 10 x 10

Not beginning with 1 10 2 x = 10 4 x = 10 x = 7 10 3 x = 2 3 4 6 3 000 000 = 3 x 10 x 10 x 10 x 10 x 10 x 10 = This is also known as Scientific Notation.

Whole No to SF 2 000 (2) 20 000 (3) 500 (4) 800 000 (5) 9 000 000 = 2 x 10 x 10 x 10 = 2 x 10x10x10x10 = 5 x 10 x 10 = 8x10x10x10x10x10 = 9x10x10x10x10x10x10

43620 to STANDARD FORM 4 3 6 2 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 3 6 2 4 2 1 3 4 3 6 2 x 10 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 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 8 2 5 6 4 2 1 3 3 8 2 x 10 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 7 5 9 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 7 5 2 1 9 7 5 x 10 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 5 8 9 1 5 x 10 4 6 x 6 New Example Test (-) √ ² C x ÷ ^ Exp 7.53x10^5 5 6 7 8 9 + - On 753000 1 2 3 4 . = Ans

Large to SF Examples 5 5 4 6 6 x 7.84x10^6 7840000 (a) 52700 5.27 x 10 (b) 5 6.65 x 10 5 6650000 8 (c) 240000000 2.4 x 10 4 6 6 6 x 6890000 6.89 x 10 (d) 3 4510 (e) 4.51 x 10 5 805000 8.05 x 10 (f) (-) √ ² C x ÷ ^ Exp New 7.84x10^6 5 6 7 8 9 + - On 7840000 1 2 3 4 . = Ans

12.7 Millions to SF Change 12.7 million to Standard Form 1 2 7 6 5 7 1 2 3 4 1 2 7 x 10 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 7 7 5 7 7 5 x 10 7¾ million = 7.75 million Change 7¾ million to Standard Form 7¾ million = 7.75 million 7 7 5 6 3 4 5 1 2 7 7 5 x 10 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 6 (a) 5.3 x 10 5.3 million 6 5 (b) 1.3 x 10 5 1.3 million 7 (c) 49 million 4.9 x 10 4 6 x 6 7 78 million (d) 7.8 x 10 6 7 million (e) 7 x 10 7 16 million 1.6 (f) x 10 (-) √ ² C x ÷ ^ Exp New 5 6 7 8 9 + - On 1 2 3 4 . = Ans

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 9 1 x 10 = 1 Normally the 1st number includes a point.

What is …… ? 6 x 10 3 = 600 (1) 8 x 10 3 = 8 000 (2) 4 x 10 5 = 400 000 (3) 7 x 10 2 = 700 (4) 3 x 10 8 = 300 000 000 (5) 9 x 10 6 = 9 000 000 (6)

4 SF to Normal T . 5 x 10 3 7 1 1 3.715x10 3 7 . 1 5 3 3 3 3 3 = 2 3.715x10 = 3 7 1 . 5 9 9 9 9 9 3 3.715x10 = 3 7 1 5 . 6 4 3.715x10 3 7 1 5 . = 5 7.218x10 = 7 2 1 8 . 6 6 6 7.218x10 7 2 1 8 . 6 6 = 7 7.218x10 7 2 1 8 . 6 = 8 7.218x10 7 2 1 8 . = 4 3 7 1 5 3.715 x 10 Multiplying by a +ve power of 10 moves the point to the right Finish Next

Back to Normal 7 4 3 6 2 x 10 = 4 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 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 2 5 8 4 x 10 4 = 5 8 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 2 5.084 x 10 = 502.4

Large SF to normal 6 3 8 6 4 x 10 5 3 6 5 4 6 x 6 6 . New Example Test 3 6 5 4 6 x 6 New Example Test Place Point (-) √ ² C x ÷ ^ Exp 6 5 6 7 8 9 + - On 1 2 3 4 . = Ans

SF to Normal Examples 5 5 4 6 x 6 7.84x10^6 7840000 (a) 4.73 x 10 4730 (b) 5 9.62 x 10 9620 5 5 (c) 6.47 x 10 647000 4 6 5 x 6 873000 (d) 8.73 x 10 3 3160 (e) 3.16 x 10 5 277000 2.77 x 10 (f) (-) √ ² C x ÷ ^ Exp New 7.84x10^6 5 6 7 8 9 + - On 7840000 1 2 3 4 . = Ans

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 T . x 10 3 9 3.9x10 7 6 3 9 . _ _ = 3.9x10 = 6 3 9 . 8 Introducing Small T . x 10 3 9 3 3.9x10 7 6 3 9 . _ _ = 2 3.9x10 = 6 3 9 . 7 _ _ 1 . 3.9x10 = 3 9 5 7 _ _ 3.9x10 3 . 9 5 7 _ _ = -1 3.9x10 = . 3 9 5 7 _ _ -2 3.9x10 . 3 9 5 7 _ _ = -3 3.9x10 . 3 9 5 7 _ _ = -4 . 3.9x10 3 9 5 7 _ = 8 Number >=1 …….. Power of 10 will be positive 3 1 3.9 x 10 Number between 0 and 1 …….. Power of 10 will be negative Next

Small to SF 0.15 0.00132 0.0000042 0.0000194 0.0000042 0.0000846 0.39 = -6 4.42 x 10 0.97 0.0000035 0.1 0.0596 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 5 3 6 2 -1 -2 -3 -4 5 3 6 2 x 10 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 7 8 1 -1 7 8 1 x 10 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 -1 0.781 = 7.81 x 10

Small To SF Interactive 5 8 7 9 5 5 x 10 4 6 x 6 Click top no then click destination. Use arrows to set power New Example Test (-) √ ² C x ÷ ^ Exp 5 6 7 8 9 + - On 1 2 3 4 . = Ans

Small to SF Examples -6 (a) 1.42 x 10 0.00000142 -3 5 (b) 6.27 x 10 5 0.00627 -4 (c) 0.00075 7.5 x 10 4 6 x 6 -3 0.0012 (d) 1.2 x 10 -3 0.00933 (e) 9.33 x 10 -8 0.0000000388 3.88 (f) x 10 (-) √ ² C x ÷ ^ Exp New 5 6 7 8 9 + - On 1 2 3 4 . = Ans

Small SF to Normal T . 4 3 7 x 10 8 8.437x10 7 6 8 4 3 7 . _ _ = 7 6 8 4 3 7 . _ _ = 2 8.437x10 = 6 8 4 3 . 7 _ _ 1 8.437x10 = 8 4 . 3 7 _ _ 8.437x10 8 . 4 3 7 _ _ = -1 8.437x10 = . 8 4 3 7 _ _ -2 8.437x10 . 8 4 3 7 _ _ = -3 8.437x10 . 8 4 3 7 _ _ = -4 8.437x10 . 8 4 3 7 _ = 4 When the power is negative the point moves to left 8 1 8.437 x 10 There will be the same no of zeros at front as power Next

Small SF to Normal -8 7 8 3 x 10 7 8 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 The “lines” show where the point should go Copy other figure after the 7

Small SF to normal -6 2 8 5 1 x 10 5 7 5 1 5 4 6 x 6 6 . New Example 7 5 1 5 4 6 x 6 New Example Test Place Point (-) √ ² C x ÷ ^ Exp 6 5 6 7 8 9 + - On 1 2 3 4 . = Ans

Small SF to Normal Examples -5 (a) 7.42 x 10 0.0000742 -5 5 (b) 7.01 x 10 0.0000701 5 -8 0.0000000103 (c) 1.03 x 10 4 6 -6 x 6 0.00000321 (d) 3.21 x 10 -4 0.000891 (e) 8.91 x 10 -4 0.000238 2.38 (f) x 10 (-) √ ² C x ÷ ^ Exp New 5 6 7 8 9 + - On 1 2 3 4 . = Ans

Positive Powers 5 5 4 6 x 6 7.84x10^6 7840000 (a) 1.2 x 10 1200000 (b) 9.24 x 10 924000 5 6 (c) 7.1 x 10 7100000 4 6 x 6 5 (d) 562000 5.62 x 10 5 (e) 418000 4.18 x 10 3 (f) 3420 3.42 x 10 (-) √ ² C x ÷ ^ Exp New 7.84x10^6 5 6 7 8 9 + - On 7840000 1 2 3 4 . = Ans

Negative Powers 5 5 4 6 x 6 7.84x10^6 7840000 (a) 7.82 x 10 0.0000782 -5 (a) 7.82 x 10 0.0000782 -5 5 (b) 6.58 x 10 0.0000658 5 -4 (c) 5.43 x 10 0.000543 4 6 x 6 -8 (d) 0.0000000913 9.13 x 10 -3 (e) 0.00206 2.06 x 10 -6 (f) 0.00000239 2.39 x 10 (-) √ ² C x ÷ ^ Exp New 7.84x10^6 5 6 7 8 9 + - On 7840000 1 2 3 4 . = Ans

Mixed Powers 4 (a) 7.36 x 10 73600 -3 5 (b) 8.84 x 10 0.00884 5 8 649000000 (c) 6.49 x 10 4 6 x 6 3 (d) 5280 5.28 x 10 6 (e) 8000000 8 x 10 -4 3.11 (f) 0.000311 x 10 (-) √ ² C x ÷ ^ Exp New 7.84x10^6 5 6 7 8 9 + - On 7840000 1 2 3 4 . = Ans

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 8 min 20 sec Earth to the Sun 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 4.22 Light Year 9.467 x 1012 km 4.22x9.47 x 1012 = 4.22 x 9.47 E 12 4.22 x x 10x 12 4.22 x 9.467 =3.995 x 1013 km

7 Light Years 7 Light Year 7 Light Year 9.467 x 1012 km 7x9.47 x 1012 = 7 x 9.47 E 12 7 x x 10x 12 7 x 9.467 =6.629 x 1013 km 32 Light Year 32 Light Year 9.467 x 1012 km 32x9.47 x 1012 = 32 x 9.47 E 12 32 x x 10x 12 32 x 9.467 =3.0304 x 1014 km

Light Year Calculations 1 Light Year is 9.467 x 1012 km 6.1 5 Light Year = 6.1 x ( 9.467 x 1012 ) km 5 On 6.1 x 9.467 Exp 12 = 4 6 x 6 (-) √ ² C x ÷ ^ Exp 5 6 7 8 9 + - On 1 2 3 4 . = Ans = = x 10 Click after using calculator Next

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