�ݺ�ߣ

N2.6 Standard form Contents A A A A A A N2.5 Surds N2.4 Fractional indices N2.2 Index laws N2.1 Powers and roots N2.3 Negative indices and reciprocals N2 Powers, roots and standard form

Powers of ten Our decimal number system is based on powers of ten . We can write powers of ten using index notation . 10 = 10 1 100 = 10 × 10 = 10 2 1000 = 10 × 10 × 10 = 10 3 10 000 = 10 × 10 × 10 × 10 = 10 4 100 000 = 10 × 10 × 10 × 10 × 10 = 10 5 1 000 000 = 10 × 10 × 10 × 10 × 10 × 10 = 10 6 …

Negative powers of ten Any number raised to the power of 0 is 1, so 1 = 10 0 Decimals can be written using negative powers of ten 0.01 = = = 10 -2 1 10 2 1 100 0.001 = = = 10 -3 1 10 3 1 1000 0.0001 = = = 10 -4 1 10000 1 10 4 0.00001 = = = 10 -5 1 100000 1 10 5 0.000001 = = = 10 -6 … 1 1000000 1 10 6 0.1 = = =10 -1 1 10 1 10 1

Very large numbers Use you calculator to work out the answer to 40 000 000 × 50 000 000. Your calculator may display the answer as: What does the 15 mean? The 15 means that the answer is 2 followed by 15 zeros or: 2 × 10 15 = 2 000 000 000 000 000 2 E 15 or 2 15 2 ×10 15 ,

Very small numbers Use you calculator to work out the answer to 0.0003 ÷ 200 000 000. Your calculator may display the answer as: What does the – 12 mean? The – 12 means that the 1.5 is divided by (1 followed by 12 zeros) 1.5 × 10 -12 = 0.000000000002 1.5 E – 12 or 1.5 – 12 1.5 ×10 – 12 ,

Standard form 2 × 10 15 and 1.5 × 10 -12 are examples of a number written in standard form . Numbers written in standard form have two parts: This way of writing a number is also called standard index form or scientific notation . Any number can be written using standard form, however it is usually used to write very large or very small numbers. A number between 1 and 10 × A power of 10

Standard form – writing large numbers For example, the mass of the planet earth is about 5 970 000 000 000 000 000 000 000 kg. We can write this in standard form as a number between 1 and 10 multiplied by a power of 10. 5.97 × 10 24 kg A number between 1 and 10 A power of ten

Standard form – writing large numbers How can we write these numbers in standard form? 8 × 10 7 2.3 × 10 8 7.24 × 10 5 6.003 × 10 9 3.7145 × 10 2 80 000 000 = 230 000 000 = 724 000 = 6 003 000 000 = 371.45 =

Standard form – writing large numbers These numbers are written in standard form. How can they be written as ordinary numbers? 50 000 000 000 7 100 000 420 800 000 000 21 680 000 6764.5 5 × 10 10 = 7.1 × 10 6 = 4.208 × 10 11 = 2.168 × 10 7 = 6.7645 × 10 3 =

Standard form – writing small numbers We can write very small numbers using negative powers of ten. We write this in standard form as: For example, the width of this shelled amoeba is 0.00013 m. A number between 1 and 10 A negative power of 10 1.3 × 10 -4 m.

Standard form – writing small numbers How can we write these numbers in standard form? 6 × 10 -4 7.2 × 10 -7 5.02 × 10 -5 3.29 × 10 -8 1.008 × 10 -3 0.0006 = 0.00000072 = 0.0000502 = 0.0000000329 = 0.001008 =

Standard form – writing small numbers 0.0008 0.0000026 0.00000009108 0.00007329 0.084542 These numbers are written in standard form. How can they be written as ordinary numbers? 8 × 10 -4 = 2.6 × 10 -6 = 9.108 × 10 -8 = 7.329 × 10 -5 = 8.4542 × 10 -2 =

Ordering numbers in standard form Write these numbers in order from smallest to largest: 5.3 × 10 -4 , 6.8 × 10 -5 , 4.7 × 10 -3 , 1.5 × 10 -4 . To order numbers that are written in standard form start by comparing the powers of 10. Remember, 10 -5 is smaller than 10 -4 . That means that 6.8 × 10 -5 is the smallest number in the list. When two or more numbers have the same power of ten we can compare the number parts. 5.3 × 10 -4 is larger than 1.5 × 10 -4 so the correct order is: 6.8 × 10 -5 , 1.5 × 10 -4 , 5.3 × 10 -4 , 4.7 × 10 -3

Calculations involving standard form What is 2 × 10 5 multiplied by 7.2 × 10 3 ? To multiply these numbers together we can multiply the number parts together and then the powers of ten together. 2 × 10 5 × 7.2 × 10 3 = (2 × 7.2) × ( 10 5 × 10 3 ) = 14.4 × 10 8 This answer is not in standard form and must be converted! 14.4 × 10 8 = 1.44 × 10 × 10 8 = 1.44 × 10 9

Calculations involving standard form What is 1.2 × 10 -6 divided by 4.8 × 10 7 ? To divide these numbers we can divide the number parts and then divide the powers of ten. (1.2 × 10 -6 ) ÷ ( 4.8 × 10 7 ) = (1.2 ÷ 4.8) × ( 10 -6 ÷ 10 7 ) = 0.25 × 10 -13 This answer is not in standard form and must be converted. 0.25 × 10 -13 = 2.5 × 10 -1 × 10 -13 = 2.5 × 10 -14

Travelling to Mars How long would it take a space ship travelling at an average speed of 2.6 × 10 3 km/h to reach Mars 8.32 × 10 7 km away?

Calculations involving standard form = 3.2 × 10 4 hours This is 8.32 ÷ 2.6 This is 10 7 ÷ 10 3 How long would it take a space ship travelling at an average speed of 2.6 × 10 3 km/h to reach Mars 8.32 × 10 7 km away? Time to reach Mars = 8.32 × 10 7 2.6 × 10 3 Rearrange speed = distance time time = distance speed to give

Calculations involving standard form Use your calculator to work out how long 3.2 × 10 4 hours is in years. You can enter 3.2 × 10 4 into your calculator using the EXP key: Divide by 24 to give the equivalent number of days. Divide by 365 to give the equivalent number of years. 3.2 × 10 4 hours is over 3 ½ years. 3 . 2 EXP 4

Physicists are a little more practical than the mathematicians! On your camera: 10 Mega Pixels Mega is 1 x 10 6 (1 000 000) The camera has 10 x10 6 pixels Often questions in Physics will give you values that can be conveniently expressed with a prefix if you fiddle the standard form a little!

�ݺ�ߣ

Fishlock Lesson One Standard Form

Recommended

More Related Content

What's hot (20)

Similar to Fishlock Lesson One Standard Form (20)

Recently uploaded (20)

Fishlock Lesson One Standard Form

Editor's Notes