1. Objective: To convert numbers into standard index form

2. 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.

3. 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 =
10 X 10 X 10 X 10 X 10
= 105
Rule: Count the number of zeros

4. 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

5. 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 X 10 =

6. 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 X 100 =

7. 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  10 =

8. 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  100 =

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 .
x 10 =
But it is actually the digits that move .
x 100 = .
 10 = .
 100 =

10. 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

11. 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

12. 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 = 4 x 103
= 6 x 10,000 = 6 x 104
= 9 x 100,000 = 9 x 105
= 7 x 1000,000 = 7 x 106

13. 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

14. 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

15. 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

16. 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

17. 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
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

18. 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 108
8 places

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

20. 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 =
Using the rules of powers
10–1
So =
= 2.3 x 10–1

21. 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 =
Using the rules of powers
10–2
So =
= 5.6 x 10–2

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

23. Now it’s your turn:
Copy out the following numbers and convert them into standard index form.
0.94
0.086
0.0057
= 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

24. Converting to Normal Numbers

25. 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

26. 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 = =

27. 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

28. 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=
1.31=

29. 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

30. 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! = X
109

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

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

33. 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!!

34. 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

35. 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 = 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

36. 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
Place the decimal point between the 2 and 3 ( )
Then count the number of places that the decimal point has moved.
= 2.37 x 108
8 places

37. 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 =
Remember to divide by 10 move the digits right
So =
= 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
= 3.4  = 3.4 x 10–4
= 5.7  1000 = 5.7 x 10–3
= 1.2  = 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 =
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 =
So if the power is positive, the ordinary number is BIG
If the power number is negative the ordinary number is small

38. 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) = 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… x 1012
This can be rewritten as:
7.42 x 10-1 x 1012 = 7.42 x Rule of indices says ADD the powers