Fishlock Lesson One Standard Form
- 1. 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
- 2. 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 …
- 3. 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
- 4. 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 ,
- 5. 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 ,
- 6. 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
- 7. 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
- 8. 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 =
- 9. 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 =
- 10. 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.
- 11. 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 =
- 12. 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 =
- 13. Which number is incorrect?
- 14. 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
- 16. 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
- 17. 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
- 18. 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?
- 19. 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
- 20. 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
- 21. 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!
Editor's Notes
- #3: Discuss the use of index notation to describe numbers like 10, 100 and 1000 as powers of 10. Be aware that pupils often confuse powers with multiples and reinforce the idea of a power as a number, in this case 10, repeatedly multiplied by itself. Make sure that pupils know that 10 3 , for example, is said as “ten to the power of three”. Explain that the index tells us how many 0s will follow the 1 (this is only true for positive integer powers of ten).
- #4: Talk through the use of negative integers to represent decimals. This is discussed in the context of the place value system in N4.1 Decimals and place value.
- #5: Different models of calculator may show the answer in different ways. Many will leave out the ×10 and will have EXP before the power or nothing at all. Discuss how many zeros there will be in the answer. 4 × 5 is 20. There are 7 zeros in 40 000 000 and 7 zeros in 50 000 000. That means that the answer will have 14 zeros plus the zero from the 20, making 15 zeros altogether.
- #6: Point out that if we include the 0 before the decimal point the answer has 12 zeros altogether.
- #7: Point out that the numbers between 1 and 10 do not include the number 10.
- #9: Discuss how each number should be written in standard form. Notice that for large numbers the power of ten will always be one less than the number of digits in the whole part of the number.
- #10: Discuss how each number written in standard form should be written in full.
- #11: The image of a shelled amoeba has been reproduced with the kind permission of Wim van Egmond © Microscopy UK http://www.microscopy-uk.org.uk/index.html
- #12: Notice that the power of ten is always minus the number of zeros before the first significant figure including the one before the decimal point..
- #13: Again, notice that the power of ten tells us the number of zeros before the first significant figure including the one before the decimal point.
- #14: Ask pupils how the number that is incorrectly written can be expressed correctly in standard form before revealing the answer.
- #16: The diameter of each planet is given in standard form. Ask a volunteer to come to the board and put the in the correct order from smallest to biggest.
- #17: Remind pupils that indices are added when we multiply. Point out that 14.4 × 10 8 is not in standard form and discuss how it can be converted into the correct form.
- #18: Remind pupils that indices are subtracted when we divide. Discuss how 0.25 × 10 -13 can be converted into the correct form.
- #20: Remind pupils that 10 7 ÷ 10 3 = 10 4 because the indices are subtracted when dividing.
- #21: Make sure that pupils are able to enter numbers given in standard form into their calculators.