Powers of 10
Before going through this page, you need to know how to write and understand numbers in index form as discussed in the Indices part 1 page.
Multiplying by powers of 10 is not hard at all!
5 x 10 = 50 5 x 100 = 5005 x 1000 = 5000
Instead of writing 10, 100, 1000, 10,000, 100,000 and so on, we can write in index form. It is easier to write this way, as it is much smaller and neater. Imagine writing 1,000,000 and 106, which one is faster and takes up less space? It is 106 .By the way 106 = 1,000,000
SO LETS ONLY USE INDEX FORM NOW.
10 = 101100 = 102 1000 = 103
You can see any WHOLE NUMBER multiplied by 10n, where n is positive and a WHOLE NUMBER, is the same as adding n zeros to the end of the number.
e.g.
5 x 105 = 50000060 x 103 = 6000075 x 102 = 7500
If the number to be multiplied by 10n is a decimal and n is a whole number, then it is just a matter of moving the decimal point forwards n times (from a visual perspective).
e.g.5.67 x 102 = 567.0 = 5676.7689 x 103 = 6768.90.0045 x 101 = 00.045 = 0.045
00.045 equals 0.045 because there is no need to put the extra 0 in the front. For example, we would not write 04.5, we would write 4.5 only; another example is, we would not write 020, we would write just 20.
5/10 = 0.55/100 = 0.055/1000 = 0.005
Another way of writing 5/10 = 5 x (1/10). Another way of writing 5/100 is 5 x (1/100).
So
5/10 = 5 x (1/10) = 5 x 10-1 = 0.55/100 = 5 x (1/100) = 5 x 10-2 = 0.055/1000 = 5 x (1/1000) = 5 x 10-3 = 0.005
HOW DID WE GET FROM 1/10 to 10-1. Well, when you have a number a to the power of -n, a-n is equal to 1/an . We learnt this in the Indices part 1 page.
e.g. 5-4 = 1/54. So that is why we can rewrite 1/10 as 10-1 as they are equal.
The same idea as mentioned above can be applied here, what is easier to write 1/1,000,000 or 10-6? It is the index form, so from now on we write in index form.
1/10 = 10-11/100 = 10-21/1000 = 10-3
You can see any whole number multiplied by a power of 10, where the n is negative and a WHOLE NUMBER, is the same as moving the decimal point backwards n times (from a visual perspective).
e.g.
5 x 10-2 = 5.0 x 10-2 = 0.050 = 0.05 67 x 10-1 = 67.0 x 10-1 = 6.70 = 6.7678.045 10-4 = 0.067804567.8 x 10-3 = 0.0678
You might be thinking that in these above examples, why is 5 written as 5.0 and why 67 is written as 67.0. Remember that 5 and 5.0 are equal, so by writing it in decimal form IT IS EASIER TO VISUALISE THE DECIMAL POINT MOVING.
MULTIPLYING
MULTIPLYING
Multiplying by powers of 10 is not hard at all!
5 x 10 = 50 5 x 100 = 5005 x 1000 = 5000
Instead of writing 10, 100, 1000, 10,000, 100,000 and so on, we can write in index form. It is easier to write this way, as it is much smaller and neater. Imagine writing 1,000,000 and 106, which one is faster and takes up less space? It is 106 .By the way 106 = 1,000,000
SO LETS ONLY USE INDEX FORM NOW.
10 = 101100 = 102 1000 = 103
You can see any WHOLE NUMBER multiplied by 10n, where n is positive and a WHOLE NUMBER, is the same as adding n zeros to the end of the number.
e.g.
5 x 105 = 50000060 x 103 = 6000075 x 102 = 7500
If the number to be multiplied by 10n is a decimal and n is a whole number, then it is just a matter of moving the decimal point forwards n times (from a visual perspective).
e.g.5.67 x 102 = 567.0 = 5676.7689 x 103 = 6768.90.0045 x 101 = 00.045 = 0.045
00.045 equals 0.045 because there is no need to put the extra 0 in the front. For example, we would not write 04.5, we would write 4.5 only; another example is, we would not write 020, we would write just 20.
DIVIDING
DIVIDING
5/10 = 0.55/100 = 0.055/1000 = 0.005
Another way of writing 5/10 = 5 x (1/10). Another way of writing 5/100 is 5 x (1/100).
So
5/10 = 5 x (1/10) = 5 x 10-1 = 0.55/100 = 5 x (1/100) = 5 x 10-2 = 0.055/1000 = 5 x (1/1000) = 5 x 10-3 = 0.005
HOW DID WE GET FROM 1/10 to 10-1. Well, when you have a number a to the power of -n, a-n is equal to 1/an . We learnt this in the Indices part 1 page.
e.g. 5-4 = 1/54. So that is why we can rewrite 1/10 as 10-1 as they are equal.
The same idea as mentioned above can be applied here, what is easier to write 1/1,000,000 or 10-6? It is the index form, so from now on we write in index form.
1/10 = 10-11/100 = 10-21/1000 = 10-3
You can see any whole number multiplied by a power of 10, where the n is negative and a WHOLE NUMBER, is the same as moving the decimal point backwards n times (from a visual perspective).
e.g.
5 x 10-2 = 5.0 x 10-2 = 0.050 = 0.05 67 x 10-1 = 67.0 x 10-1 = 6.70 = 6.7678.045 10-4 = 0.067804567.8 x 10-3 = 0.0678
You might be thinking that in these above examples, why is 5 written as 5.0 and why 67 is written as 67.0. Remember that 5 and 5.0 are equal, so by writing it in decimal form IT IS EASIER TO VISUALISE THE DECIMAL POINT MOVING.
Try answering these questions:
What is 5.678 x 10-3?
What is 0.0456 x 104?
Answers:
0.005678
456