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Objective: To convert numbers into standard index form Standard Index form

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999,999,999,999,999,999,999,999,999 Why is this number very difficult to use? Too big to read Too large to comprehend Too large for calculator To get around using numbers this large, we use standard index form.

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Reminder About Powers of = = 10 X 10 = = = = Rule: Count the number of zeros 10 X 10 X 10 = X 10 X 10 X 10 = X 10 X 10 X 10 X 10= 10 5

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This means 2 in the hundreds so its worth 200 This means 3 in the tens so its worth 20 This means 5 in the units so its worth 5 This means 7 in the tenths so its worth or.7 This means 1 in the hundreds so its worth or.01 This means 9 in the thousandths so its worth or.009 Reminder about Place Value HundredsTensUnitsTenthsHundredthsThousandt hs

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Reminder about Place Value HundredsTensUnitsTenthsHundredthsThousandt hs If you multiply by 10 all the digits move 1 place to the left becomes 30 becomes 5 becomes So X 10 =.

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Reminder about Place Value HundredsTensUnitsTenthsHundredthsThousandt hs If you multiply by 100 all the digits move 2 places to the left. So X 100 = becomes 30 becomes 5 becomes.

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Reminder about Place Value HundredsTensUnitsTenthsHundredthsThousandt hs If you divide by 10 all the digits move 1 place to the right becomes 30 becomes 5 becomes So 10 =.

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Reminder about Place Value HundredsTensUnitsTenthsHundredthsThousandt hs If you divide by 100 all the digits move 2 places to the right becomes 30 becomes 5 becomes So 100 =.

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Reminder about Place Value HundredsTensUnitsTenthsHundredthsThousandt hs Instead of moving the digits most people think it easier to move the point x 10 = x 100 = 10 = 100 = But it is actually the digits that move

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How could we turn the number 800,000,000,000 into standard index form? Let’s investigate! Converting large numbers 800,000,000,000 = 8 x 100,000,000,000 We can break numbers into parts to make it easier, e.g. 80 = 8 x 10 and 800 = 8 x 100 And 100, 000,000,000 = So, 800,000,000,000 = 8 x in standard index form

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Try it out! How can we convert 30,000 into standard index form? Break into easier parts: = 3 x 10,000 And, 10,000 = 10 4 So 30,000 = 3 x 10 4 in standard index form The number is now easier to use

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= 5 x 100 = 5 x 10 2 = 4 x 1000 = 4 x 10 3 = 6 x 10,000 = 6 x 10 4 = 9 x 100,000 = 9 x 10 5 = 7 x 1000,000 = 7 x 10 6 Now it’s your turn: Copy down the following numbers, and convert them into standard index form. a)500 b)4000 c)60,000 d)900,000 e)7000,000

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The first number must be a value between 1 and 9 One of the most important rules for writing numbers in standard index form is: For example, 39 x 10 6 does have a value but it’s not written in standard index form. The first number, 39, is greater than 10. But 39 = 3.9 x 10 So 39 x 10 6 = 3.9 x 10 x 10 6 = 3.9 x 10 7 Add powers

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How could we convert 350,000,000 into standard index form? Again, we can break the number into smaller, more manageable parts. 350,000,000 = 3.5 x 100,000, ,000,000 = ,000,000 = 3.5 x 10 8 in standard index form

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Try it out! How can we convert 67,000 into standard index form? 67,000 = 6.7 x 10,000 10,000 = ,000 = 6.7 x 10 4 in standard index form

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Now it’s your turn: Copy out the following numbers and convert them into standard index form. a)940 b)8,600 c)34, 000 d)570,000 e)1,200,000 = 9.4 x 100 = 9.4 x 10 2 = 8.6 x 1000 = 8.6 x 10 3 = 3.4 x 10,000 = 3.4 x 10 4 = 5.7 x 100,000 = 5.7 x 10 5 = 1.2 x 1000,000 = 1.2 x 10 6

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Can you find a quick method of converting numbers to standard form? For example, Converting 45,000,000,000 to standard form Place a decimal point after the first digit Count the number of digits after the decimal point. 10 This is our index number (our power of 10) So, 45,000,000,000 = 4.5 x 10 10

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Using the quick method Example Place the decimal point between the 2 and 3 ( ) Then count the number of places that the decimal point has moved = 2.37 x places

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Reminder about Negative Powers 0.01 = = = 10 –3 10 –4 0.1=10 –1 10 –2

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Converting Very Small Numbers into Standard Form 0.23 is not in standard form as the 1 st digit is NOT between the 1 and 9 But 0.23 = Remember to divide by 10 move the digits right Using the rules of powers10 –1 So 0.23 == 2.3 x 10 –1

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Converting Very Small Numbers into Standard Form is not in standard form as the 1 st digit is NOT between 1 and 9 But = Remember to divide by 100 move the digits 2 places right Using the rules of powers10 –2 So == 5.6 x 10 –2

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Converting Very Small Numbers into Standard Form is not in standard form as the 1 st digit is NOT between 1 and 9 But = Remember to divide by move the digits 4 places right Using the rules of powers10 –4 So = = 3.9 x 10 –4

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Now it’s your turn: Copy out the following numbers and convert them into standard index form. a)0.94 b)0.086 c) d) e) = 9.4 10 = 9.4 x 10 –1 = 8.6 100 = 8.6 x 10 –2 = 3.4 = 3.4 x 10 –4 = 5.7 1000 = 5.7 x 10 –3 = 1.2 = 1.2 x 10 –5

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Converting to Normal Numbers

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To convert standard form to ordinary numbers: Positive powers Write the digits x 10 5 HundredsTensUnitsTenthsHundredthsThousandths 131 Now move all the digits 5 places left But this is the same as moving the decimal point 5 places right Remember 10 5 means multiply by 10000

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Which ever way you think about it the number gets B I G G E R HundredsTensUnitsTenthsHundredthsThousandths 131 To convert standard form to ordinary numbers: Positive powers 1.31 x = = Write the digits 1 31

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To convert standard form to ordinary numbers: Negative powers Write the digits x 10 –2 HundredsTensUnitsTenthsHundredthsThousandths 131 Now move all the digits 2 places right But this is the same as moving the decimal point 2 places left Remember 10 –2 means So DIVIDE by 100

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Which ever way you think about it the number gets S m al l e r HundredsTensUnitsTenthsHundredthsThousandths 131 To convert standard form to ordinary numbers: Negative powers 1.31 x 10 – = = Write the digits 1 31 Remember 10 –2 means

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The ordinary number is: So if the power is positive, the ordinary number is BIG If the power number is negative the ordinary number is small

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Multiplying standard form 1.27 x 10 5 x 2.36 x 10 4 Separate the calculation into 2 parts as follows: (1.27 x 2.36) x (10 5 x 10 4 ) = X 10 9 Rule of index says we ADD the powers!

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Sometimes when we multiply out the first part we can get more than one digit before the decimal point x 10 6 This can be rewritten as x 10 x 10 6 which becomes x 10 7 Rule of indices says we ADD powers

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Division of Standard Form If when we multiply standard form, we add the powers, when we divide standard form we----- Subtract the powers

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4.8 x 10 7 ÷ 1.5 x 10 3 Separate into two parts (4.8 ÷ 1.5) x (10 7 ÷ 10 3 ) (7-3) x 3.2 x 10 4 Rule of indices says subtract the powers!! Division in Standard Form

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Sometimes we get a number that has a zero before the decimal point… x This can be rewritten as: 7.42 x x Rule of indices says ADD the powers 7.42 x 10 11

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Standard Form These are numbers of the typea x 10 n Where a is a decimal number with only one digit in front (left) of the decimal point n is a whole number that can be positive or negative How can we convert 30,000 into standard index form? Break into easier parts: = 3 x 10,000And, 10,000 = 10 4 So 30,000 = 3 x 10 4 in standard index form a)500 = 5 x 100 = 5 x 10 2 b)4000 = 4 x 1000 = 4 x 10 3 c)60,000 = 6 x 10,000 = 6 x 10 4 d)900,000 = 9 x 100,000 = 9 x 10 5 e)7000,000 = 7 x 1000,000 = 7 x 10 6

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a)940 = 9.4 x 100 = 9.4 x 10 2 b)8,600= 8.6 x 1000 = 8.6 x 10 3 c)34, 000= 3.4 x 10,000 = 3.4 x 10 4 d)570,000 = 5.7 x 100,000 = 5.7 x 10 5 e)1,200,000 = 1.2 x 1000,000 = 1.2 x 10 6 Using the quick method Example Place the decimal point between the 2 and 3 ( ) Then count the number of places that the decimal point has moved = 2.37 x places One of the most important rules for writing numbers in standard index form is The first number must be a value between 1 and 9 For example, 39 x 10 6 does have a value but it’s not written in standard index form. The first number, 39, is greater than 10. But 39 = 3.9 x 10 So 39 x 10 6 = 3.9 x 10 x 10 6 = 3.9 x 10 7

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0.94 = 9.4 10 = 9.4 x 10 – = 8.6 100 = 8.6 x 10 – = 3.4 = 3.4 x 10 – = 5.7 1000 = 5.7 x 10 – = 1.2 = 1.2 x 10 –5 To convert standard form to ordinary numbers: Positive powers 1.31 x 10 5 Remember 10 5 means multiply by x 10 5 = To convert standard form to ordinary numbers: Negative powers 1.31 x 10 –2 Remember 10 –2 means So DIVIDE by x 10 –2 = So if the power is positive, the ordinary number is BIG If the power number is negative the ordinary number is small Converting Very Small Numbers into Standard Form 0.23 is not in standard form as the 1st digit is NOT between the 1 and 9 But 0.23 = Remember to divide by 10 move the digits right So 0.23 = = 2.3 x 10 –1

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Multiplying standard form 1.27 x 10 5 x 2.36 x 10 4 Separate the calculation into 2 parts as follows: (1.27 x 2.36) x (10 5 x 10 4 ) = x 10 9 Rule of indices says we ADD the powers! Division in Standard Form 4.8 x 10 7 ÷ 1.5 x 10 3 Separate into two parts (4.8 ÷ 1.5) x (10 7 ÷ 10 3 ) = 3.2 x 10 (7-3) = 3.2 x 10 4 Rule of indices says subtract the powers!! Sometimes we get a number that has a zero before the decimal point… x This can be rewritten as: 7.42 x x = 7.42 x Rule of indices says ADD the powers

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