This
comes from a question on the HSTUAC home school discussion list.
"I
am not at all good at explaining nor being able to
explain scientific notation. Is there
somebody here who can explain scientific notation to me in an ultra simple
manner so that I can teach it? Thanks.
p.s.
0.16 X 10 (oh my gosh I need to make this 10 to the
sixth power and how do I do that on a keyboard???)"
Here is
my answer:
When
superscripts aren't available and you need exponents the usual notation is x^2
for "x squared."
OK,
here's a quick scientific notation lesson.
Scientific
notation is used when we have very large, or very
small, numbers and we want a nice way to write them. I'll start with a fairly small example.
Consider
240. For scientific notation, the convention
is that you look at the non-zero portion of the number. Write it as a number between 1 and 10. For this example, that would be 2.4. How does 2.4 relate to 240? Well, 240 = 24 x 10 = 2.4 x 10 x 10 = 2.4 x
10^2. That's the scientific notation.
Now
let's do 24,000,000,000.
24,000,000,000
= 2,400,000,000 x 10
=
240,000,000 x 10 x 10
=
24,000,000 x 10 x 10 x 10
=
2,400,000 x 10 x 10 x 10 x 10
=
240,000 x 10 x 10 x 10 x 10 x 10
= 24,000
x 10 x 10 x 10 x 10 x 10 x 10
= 2,400
x 10 x 10 x 10 x 10 x 10 x 10 x 10
= 240 x
10 x 10 x 10 x 10 x 10 x 10 x 10 x 10
= 24 x
10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10
= 2.4 x
10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10
= 2.4 x
10^10
Notice
the pattern of where the decimal point ended up compared to where it
started. In 240 (which with a decimal
point would be 240. ) the decimal point moved 2
places. That is how many powers of 10
were in the number. In 24,000,000,000 (which would be 24,000,000,000. ) it moved 10 places which is the number of powers of 10 that
we had in the number.
Now for a small number.
0.2 we know is 2/10. 1/10
is written as 10^(-1).
So 0.2 = 2 x 10^(-1).
Notice
that there is one power of 10 involved (don't worry about the negative for now)
and the decimal point moved one place.
0.0000000000024
= 0.000000000024/10
=
0.00000000024/10/10
=
0.0000000024/10/10/10
=
0.000000024/10/10/10/10
=
0.00000024/10/10/10/10/10
=
0.0000024/10/10/10/10/10/10
=
0.000024/10/10/10/10/10/10/10
=
0.00024/10/10/10/10/10/10/10/10
=
0.0024/10/10/10/10/10/10/10/10/10
=
0.024/10/10/10/10/10/10/10/10/10/10
=
0.24/10/10/10/10/10/10/10/10/10/10/10
=
2.4/10/10/10/10/10/10/10/10/10/10/10/10 (each "/10" is another -1
power)
= 2.4 x
10^(-12)
Let's do
three examples without doing one step at a time like we just did.
3,050,000,000,000,000
With a
decimal point it's 3,050,000,000,000,000.
The
non-zero part is 305. As a number
between 1 and 10 that would be 3.05. The
number of places the decimal point moves to get there is 15. If the number you started with is bigger than
or equal to 10, the exponent will be positive.
So this gives
3,050,000,000,000,000
= 3.05 x 10^15
0.0000000000000000153
The
non-zero part is 153 which will be written as 1.53. The decimal point moved 17 places. Since the number is smaller than 1, we make
the exponent negative. This gives
1.53 x
10^(-17)
What
about numbers between 1 and 10? That's
where the exponent 0 comes in.
Consider
9.5. The non-zero part is 9.5. Written as a number between 1 and 10 that is
9.5. The decimal point didn't move. So it is 9.5 x 10^0.
Why do
zero and negative exponents make sense?
Consider
the following.
10^3 = 10 x 10 x 10. If we divide
it by 10 we get 10^2 = 10 x 10. Notice
that dividing by 10 reduces the power of 10 by one. So 10^1 = 10. Divide it by 10 and the power will go down
1. That gives 10^0 = 1. Do it again.
10^(-1) = 1/10.
And again.
10^(-2) = 1/100 = 1/10^2.
To go
from scientific notation to regular notation we reverse the process.
3.52 x
10^12
Since
the power is positive, we'll make the number bigger than 10 so that means we'll
move the decimal 12 places to the right.
This gives 3,520,000,000,000.
5.2 x 10^(-16)
Since
the power is negative, we'll make the number smaller than 1 by moving the
decimal point to the left 16 places.
This gives 0.00000000000000052.