Standard Form Large Numbers

Standard form – writing large numbers We can write very large numbers using standard form. To write a number in standard form we write it as a number between 1 and 10 multiplied by a power of ten. For example, the average distance from the earth to the sun is about 150 000 000 km. Discuss the use of standard form to write large numbers. Discuss whether 10×105 and 1×104 are correct examples of standard form. The first is not whereas the second is; if the discussion is interesting, you might want to discuss how to use inequality notation to show exactly what is meant by ‘between 1 and 10’ in the definition of standard form. We can write this number as 1.5 × 108 km. A number between 1 and 10 A power of ten

Powers of ten Our decimal number system is based on powers of ten. We can write powers of ten using index notation. 10 = 101 100 = 10 × 10 = 102 1000 = 10 × 10 × 10 = 103 10 000 = 10 × 10 × 10 × 10 = 104 100 000 = 10 × 10 × 10 × 10 × 10 = 105 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 103, 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). The standard meaning of a billion is 109 (one thousand million). In the past, a British billion was one million million, or 1012. Links: N4 Powers and roots – powers 1 000 000 = 10 × 10 × 10 × 10 × 10 × 10 = 106 . . . . . . 1 000 000 000 = 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 = 109

6 2 6 2 Multiplying by 10, 100 and 1000 What is 6.2 × 10? Let’s look at what happens on the place value grid. Thousands Hundreds Tens Units tenths hundredths thousandths 6 2 6 2 This is the first of three slides illustrating the effect of multiplying a decimal number by 10, 100 and 1000. Start by checking that pupils can multiply whole numbers by 10, 100 and 1000 and divide whole numbers ending in 0, 00 and 000 by 10, 100 and 1000. Tell pupils that we often need to be able to multiply and divide numbers by 10, 100 and 1000 in every day life, for example when converting between metric units. Give examples as necessary. Explain that we are now going to look at the effect of multiplying decimals by 10, 100 and 1000. Look at the question on the slide. Most pupils will be able to solve this without reference to the place value grid. Reveal each step on the slide and encourage the idea that the digits are moved one place to the left to make the number ten times bigger (rather than the decimal point being moved one place to the right). Links: N9 Mental methods – multiplication and division S7 Measures – converting units When we multiply by ten the digits move one place to the left. 6.2 × 10 = 62

3 1 3 1 Multiplying by 10, 100 and 1000 What is 3.1 × 100? Let’s look at what happens on the place value grid. Thousands Hundreds Tens Units tenths hundredths thousandths 3 1 3 1 When we multiply by one hundred the digits move two places to the left. Again, stress that it is the digits that are being moved two places to the left and not the decimal point that is being moved two places to the right. We then add a zero place holder. 3.1 × 100 = 310

7 7 Multiplying by 10, 100 and 1000 What is 0.7 × 1000? Let’s look at what happens on the place value grid. Thousands Hundreds Tens Units tenths hundredths thousandths 7 7 When we multiply by one thousand the digits move three places to the left. We then add zero place holders. 0.7 × 1000 = 700

Standard form – writing large numbers How can we write these numbers in standard form? 80 000 000 = 8 × 107 230 000 000 = 2.3 × 108 724 000 = 7.24 × 105 Discuss how each number should be written in standard form. Notice that the power of ten will always be one less than the number of digits in the number. 6 003 000 000 = 6.003 × 109 371.45 = 3.7145 × 102

Standard form – writing large numbers These numbers are written in standard form. How can they be written as ordinary numbers? 5 × 1010 = 50 000 000 000 7.1 × 106 = 7 100 000 4.208 × 1011 = 420 800 000 000 Discuss how each number written in standard form should be written in full. 2.168 × 107 = 21 680 000 6.7645 × 103 = 6764.5

Use you calculator to work out the answer to 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: 2 ×10 15 , 2 E 15 or 2 15 What does the 15 mean? 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. The 15 means that the answer is 2 followed by 15 zeros or: 2 × 1015 = 2 000 000 000 000 000