Objective: To convert numbers into standard index form

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

Reminder About Powers of 10 10 = 101 100 = 10 X 10 = 102 1000 = 10 X 10 X 10 = 103 10000 = 10 X 10 X 10 X 10 = 104 100000 = 10 X 10 X 10 X 10 X 10 = 105 Rule: Count the number of zeros

Reminder about Place Value Hundreds Tens Units Tenths Hundredths Thousandths 2 3 5 7 1 9 2 3 5 7 1 9 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 Hundreds Tens Units Tenths Hundredths Thousandths 2 3 5 7 1 9 2 3 5 7 1 9 If you multiply by 10 all the digits move 1 place to the left. 200 becomes 2000 30 becomes 300 5 becomes 50 . So 235.719 X 10 = 2 3 5 7 1 9

Reminder about Place Value Hundreds Tens Units Tenths Hundredths Thousandths 2 3 5 7 1 9 2 3 5 7 1 9 If you multiply by 100 all the digits move 2 places to the left. 200 becomes 20000 30 becomes 3000 5 becomes 500 . So 235.719 X 100 = 2 3 5 7 1 9

Reminder about Place Value Hundreds Tens Units Tenths Hundredths Thousandths 2 3 5 7 1 9 2 3 5 7 1 9 If you divide by 10 all the digits move 1 place to the right. 200 becomes 20 30 becomes 3 .5 5 becomes . So 235.719  10 = 2 3 5 7 1 9

Reminder about Place Value Hundreds Tens Units Tenths Hundredths Thousandths 2 3 5 7 1 9 2 3 5 7 1 9 If you divide by 100 all the digits move 2 places to the right. 200 becomes 2 .3 30 becomes .05 5 becomes . So 235.719  100 = 2 3 5 7 1 9

Reminder about Place Value Hundreds Tens Units Tenths Hundredths Thousandths 2 3 5 7 1 9 2 3 5 7 1 9 Instead of moving the digits most people think it easier to move the point . 235.719 x 10 = 2 3 5 7 1 9 But it is actually the digits that move . 235.719 x 100 = 2 3 5 7 1 9 . 235.719  10 = 2 3 5 7 1 9 . 235.719  100 = 2 3 5 7 1 9

Let’s investigate! Converting large numbers How could we turn the number 800,000,000,000 into standard index form? We can break numbers into parts to make it easier, e.g. 80 = 8 x 10 and 800 = 8 x 100 Size given by first number 8 and the index 11 800,000,000,000 = 8 x 100,000,000,000 And 100, 000,000,000 = 1011 So, 800,000,000,000 = 8 x 1011 in standard index form

So 30,000 = 3 x 104 in standard index form Try it out! How can we convert 30,000 into standard index form? Break into easier parts: 30000 = 3 x 10,000 And, 10,000 = 104 So 30,000 = 3 x 104 in standard index form The number is now easier to use

Now it’s your turn: 500 = 5 x 100 = 5 x 102 4000 60,000 Copy down the following numbers, and convert them into standard index form. 500 4000 60,000 900,000 7000,000 = 5 x 100 = 5 x 102 = 4 x 1000 = 4 x 103 = 6 x 10,000 = 6 x 104 = 9 x 100,000 = 9 x 105 = 7 x 1000,000 = 7 x 106

The first number must be a value between 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 Pupils should copy down rule in green For example, 39 x 106 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 106 = 3.9 x 10 x 106 = 3.9 x 107 Add powers

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 3.5 x 100 = 350, x by 1,000,000 makes 350,000,000 100,000,000 = 108 350,000,000 = 3.5 x 108 in standard index form

Try it out! How can we convert 67,000 into standard index form? 10,000 = 104 67,000 = 6.7 x 104 in standard index form

Now it’s your turn: Copy out the following numbers and convert them into standard index form. 940 8,600 34, 000 570,000 1,200,000 = 9.4 x 100 = 9.4 x 102 = 8.6 x 1000 = 8.6 x 103 = 3.4 x 10,000 = 3.4 x 104 = 5.7 x 100,000 = 5.7 x 105 = 1.2 x 1000,000 = 1.2 x 106

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 Pupil book G3 page 137 4.5000000000 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 1010

Using the quick method Example 237000000 Place the decimal point between the 2 and 3 (2.37000000) Then count the number of places that the decimal point has moved. 237000000 = 2.37 x 108 8 places

Reminder about Negative Powers 0.1= 10–1 0.01 = 10–2 0.001 = 10–3 0.0001= 10–4

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 Remember to divide by 10 move the digits right But 0.23 = Using the rules of powers 10–1 So 0.23 = = 2.3 x 10–1

Converting Very Small Numbers into Standard Form 0.056 is not in standard form as the 1st digit is NOT between 1 and 9 Remember to divide by 100 move the digits 2 places right But 0.056 = Using the rules of powers 10–2 So 0.056 = = 5.6 x 10–2

Converting Very Small Numbers into Standard Form 0.00039 is not in standard form as the 1st digit is NOT between 1 and 9 Remember to divide by 10000 move the digits 4 places right But 0.00039 = Using the rules of powers 10–4 So 0.00039 = = 3.9 x 10–4

Now it’s your turn: Copy out the following numbers and convert them into standard index form. 0.94 0.086 0.00034 0.0057 0.000012 = 9.4  10 = 9.4 x 10–1 = 8.6  100 = 8.6 x 10–2 = 3.4  10000 = 3.4 x 10–4 = 5.7  1000 = 5.7 x 10–3 = 1.2  100000 = 1.2 x 10–5

Converting to Normal Numbers

To convert standard form to ordinary numbers: Positive powers 1.31 x 105 Remember 105 means multiply by 10000 Write the digits 1 31 Hundreds Tens Units Tenths Hundredths Thousandths 1 3 Now move all the digits 5 places left But this is the same as moving the decimal point 5 places right

Which ever way you think about it the number gets BIGGER To convert standard form to ordinary numbers: Positive powers 1.31 x 105 Write the digits 1 31 Hundreds Tens Units Tenths Hundredths Thousandths 1 3 Which ever way you think about it the number gets BIGGER 1.310000= 1 3 1 0 0 0 1.310000= 1 3 1 0 0 0

To convert standard form to ordinary numbers: Negative powers Remember 10–2 means 1.31 x 10–2 Write the digits 1 31 So DIVIDE by 100 Hundreds Tens Units Tenths Hundredths Thousandths 1 3 Now move all the digits 2 places right But this is the same as moving the decimal point 2 places left

Which ever way you think about it the number gets Smaller To convert standard form to ordinary numbers: Negative powers Remember 10–2 means 1.31 x 10–2 Write the digits 1 31 Hundreds Tens Units Tenths Hundredths Thousandths 1 3 Which ever way you think about it the number gets Smaller 01.31= 0 1 3 1 1.31= 0 1 3 1

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

Multiplying standard form 1.27 x 105 x 2.36 x 104 Separate the calculation into 2 parts as follows: (1.27 x 2.36) x (105 x 104) Rule of index says we ADD the powers! = 2.9972 X 109

Sometimes when we multiply out the first part we can get more than one digit before the decimal point. 23.742 x 106 This can be rewritten as 2.3742 x 10 x 106 which becomes 2.3742 x 107 Rule of indices says we ADD powers

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

Division in Standard Form 4.8 x 107 ÷ 1.5 x 103 Separate into two parts (4.8 ÷ 1.5) x (107 ÷ 103) x 3.2 10(7-3) 3.2 x 104 Rule of indices says subtract the powers!!

Sometimes we get a number that has a zero before the decimal point…. 0.742 x 1012 This can be rewritten as: 7.42 x 10-1 x 1012 Rule of indices says ADD the powers 7.42 x 1011

Standard Form These are numbers of the type a x 10n 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: 30000 = 3 x 10,000 And, 10,000 = 104 So 30,000 = 3 x 104 in standard index form 500 = 5 x 100 = 5 x 102 4000 = 4 x 1000 = 4 x 103 60,000 = 6 x 10,000 = 6 x 104 900,000 = 9 x 100,000 = 9 x 105 7000,000 = 7 x 1000,000 = 7 x 106

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 106 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 106 = 3.9 x 10 x 106 = 3.9 x 107 940 = 9.4 x 100 = 9.4 x 102 8,600= 8.6 x 1000 = 8.6 x 103 34, 000= 3.4 x 10,000 = 3.4 x 104 570,000 = 5.7 x 100,000 = 5.7 x 105 1,200,000 = 1.2 x 1000,000 = 1.2 x 106 Using the quick method Example 237000000 Place the decimal point between the 2 and 3 (2.37000000) Then count the number of places that the decimal point has moved. 237000000 = 2.37 x 108 8 places

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 0.94 = 9.4  10 = 9.4 x 10–1 0.086 = 8.6  100 = 8.6 x 10–2 0.00034 = 3.4  10000 = 3.4 x 10–4 0.0057 = 5.7  1000 = 5.7 x 10–3 0.000012 = 1.2  100000 = 1.2 x 10–5 To convert standard form to ordinary numbers: Positive powers 1.31 x 105 Remember 105 means multiply by 10000 1.31 x 105 = 131000 To convert standard form to ordinary numbers: Negative powers 1.31 x 10–2 Remember 10–2 means So DIVIDE by 100 1.31 x 10–2 = 0.0131 So if the power is positive, the ordinary number is BIG If the power number is negative the ordinary number is small

Multiplying standard form 1.27 x 105 x 2.36 x 104 Separate the calculation into 2 parts as follows: (1.27 x 2.36) x (105 x 104) = 2.9972 x 109 Rule of indices says we ADD the powers! Division in Standard Form 4.8 x 107 ÷ 1.5 x 103 Separate into two parts (4.8 ÷ 1.5) x (107 ÷ 103) = 3.2 x 10 (7-3) = 3.2 x 104 Rule of indices says subtract the powers!! Sometimes we get a number that has a zero before the decimal point….0.742 x 1012 This can be rewritten as: 7.42 x 10-1 x 1012 = 7.42 x 1011 Rule of indices says ADD the powers