Standard Form

Revision Powers: 2 is multiplied by itself 3 times On calculator: Try these: 65 54 36 25 94 104 Answers: 10 000 7776 625 729 32 6561

Powers of 10 Powers of 10 are easy e.g.103 = 10 x 10 x 10 =1000 3 zeros Look at these sequences Divide by 10 From this we see that

Working with big numbers Scientists often have to work with big numbers. Example: The distance the earth travels in one orbit is 558 000 000 miles. We can express this number in a more concise form as follows. 558 000 000 = 5.58 x 100 000 000 = 5.58 x 108 5.58 x 108 is said to be in Standard Form or Scientific Notation 558 000 000 is said to be in Normal Form

Look at this number which is in Standard Form 6.3 x 105 the power is a positive or negative whole number exactly one digit before the point 6.3 x 105 = 6.3 x 10 x 10 x 10 x 10 x 10 = 630 000 watch this Standard form Ordinary Number x 105 5 jumps 6.3 63. 1 630. 2 6300. 3 63000. 4 630000. 5

Example: Write 2.45 x 104 as an ordinary number. Method: Write the question List the digits without the point 4 jumps from position of “old” point Write the answer Add zeros as required 2 4 5 2 1 3 4 Now try these: Write as ordinary numbers: 1.47 x 102 9.08 x 106 1.3 x 100 4 x 103 4.88 x 101 147 9080000 1.3 4000 48.8

Ordinary Number Standard form watch this 630 000 = 6.3 x 100 000 630000. 63000. 6300. 630. 63. 6.3 1 2 4 5 3 x 105 5 jumps Therefore 630 000 = 6.3 x 105

Positive power since large number Example: Write 2706 in standard form. Method: Write the number Insert “new” decimal point Count jumps to position of “old” decimal point Positive power since large number Make the number “look” like standard form Put in a decimal point to make the number “look” like a number between 1 and 10. 2 7 0 6 3 1 2 = 2 7 0 6 x 10 3 Now try these: Write in standard form: 34560 1023.6 12.8 4.6 230000 3.456 x 104 1.0236 x 103 1.28 x 101 4.6 x 100 2.3 x 105

Numbers less than 1 Standard form Ordinary Number Consider the number 2.03 x 10-3 exactly one digit before the point the power is a positive or negative whole number This number is in standard form but is different from previous examples. Look again at powers of 10 Notice the power is negative 2.03 x 10-3 = 2.03 x 0.001 = 0.00203 watch this Standard form Ordinary Number x 10-3 3 jumps 2.03 x 10-3 .203 .0203 .00203 = 0.00203 1 2 3

Example: Write 1.07 x 10-4 as an ordinary number. Write the question List the digits without the point 4 jumps from position of “old” point Write the answer Method: Add zeros as required Note: When the power is negative the ordinary number is always less than 1. That means the answer starts with 0.----- 2 3 1 4 1 0 7 Now try these: Write as ordinary numbers: 1.47 x 10-2 9.08 x 10-5 1.3 x 10-1 4 x 10-3 4.081 x 10-4 0.0147 0.0000908 0.13 0.004 0.0004081

Ordinary Number Standard form watch this 0.00631 = 6.31 x 0.001 0.0631 0.631 6.31 1 2 3 x 10-3 3 jumps Therefore 630 000 = 6.31 x 10-3

Negative power since tiny number Example: Write 0.0027 in standard form. Method: Write the number Insert “new” decimal point Count jumps to position of “old” decimal point Negative power since tiny number Put in a decimal point to make the number “look” like a number between 1 and 10. 0 0 0 2 7 3 1 2 Make the number “look” like standard form 3 = 2 7 x 10 - Now try these: Write in standard form: 0.3456 0.00102 0.0128 0.000046 0.0000000023 3.456 x 10-1 1.02 x 10-3 1.28 x 10-2 4.6 x 10-5 2.3 x 10-9

Standard Form on the Calculator We use EXP button as shown on the calculator to enter numbers already in standard form. Examples: 2.6 x 108 2 6 EXP 8 = 260 000 000 check this is correct 3.5 x 1012 3 5 EXP 1 2 = 3.5 12 Calculator’s way of writing 3.5 x 1012 – does not mean 3.5 to the power 12! Too large to convert

Calculations Giving your Answer in Standard Form calculator display Example 1: (2.6 x 108) x (4.2 x 106) = 1.092 x 1015 1.092 15 calculator display Example 2: (2.484 x 107) (4.6 x 10-4) = 5.4 x 1010 5.4 10 Note: Only enter EXP before power, never x 10 EXP See calculators again For negative powers use the ( - ) key not the key, we are not subtracting. Remember to change number on display to ---- x 10-- Click button to end presentation

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