Scientific Notation
Summary
TLDRThis educational video script introduces scientific notation, a method for expressing very large or small numbers in a more manageable form. It explains how to convert numbers like 19 trillion or 0.000000063 into the format of a mantissa (a number between 1 and 10) multiplied by 10 raised to an exponent. The script clarifies the use of positive exponents for large numbers and negative exponents for small numbers, providing examples and a clear explanation to help viewers understand and apply scientific notation effectively.
Takeaways
- đą Scientific notation is used to express very large or very small numbers more easily.
- đĄ To convert a large number into scientific notation, move the decimal point so that the number is between 1 and 10.
- âŹ ïž For large numbers, move the decimal point to the left and count the number of spaces moved.
- đ The number of spaces moved becomes the exponent in scientific notation.
- đą The number in front of the scientific notation (e.g., 1.9 in 1.9 x 10^13) is called the mantissa.
- â Negative exponents are used for very small numbers, indicating how many places the decimal point was moved to the right.
- đ The rule of thumb: positive exponents for large numbers, negative exponents for small numbers.
- đ Moving a decimal point to the left results in a positive exponent; to the right results in a negative exponent.
- đ Scientific notation simplifies calculations and is easier to input into calculators.
- đ When converting from scientific notation to regular notation, move the decimal point based on the exponent's value and direction.
Q & A
What is scientific notation and why is it used in science?
-Scientific notation is a way of expressing very large or very small numbers in a compact form. It is used in science to simplify the representation of numbers that would otherwise be cumbersome to write or read, such as numbers with many zeros.
How do you express a large number like 19 trillion in scientific notation?
-You express 19 trillion as 1.9 times 10 to the 13th power. This is done by moving the decimal point 13 places to the left, making the number between 1 and 10, which is 1.9, and then indicating the number of places moved as the exponent of 10.
What is the term for the part of a scientific notation number that represents the power to which 10 is raised?
-The term for the power of 10 in scientific notation is 'exponent'. It indicates how many places the decimal point has been moved to adjust the number between 1 and 10.
What is the term for the number in scientific notation that is between 1 and 10?
-The number in scientific notation that is between 1 and 10 is called the 'mantissa' or sometimes referred to as the 'coefficient'. It is the number that multiplies the power of 10.
How do you determine whether to use a positive or negative exponent in scientific notation?
-You use a positive exponent for large numbers and a negative exponent for small numbers. If the original number is greater than 1, the exponent is positive, and if it is less than 1, the exponent is negative.
Can you provide an example of expressing a small number in scientific notation?
-An example of a small number in scientific notation is 0.000000063, which can be expressed as 6.3 times 10 to the negative 8th power. This is done by moving the decimal point 8 places to the right.
What is the significance of the mantissa being between 1 and 10 in scientific notation?
-The mantissa being between 1 and 10 standardizes the format of scientific notation, making it easier to compare and perform calculations on numbers regardless of their magnitude.
How many places does the decimal point need to be moved to express 78,000 in scientific notation?
-The decimal point needs to be moved 4 places to the left to express 78,000 in scientific notation, resulting in 7.8 times 10 to the 4th power.
What is the result of converting the scientific notation 8.22 times 10 to the negative 5th power back to standard notation?
-Converting 8.22 times 10 to the negative 5th power back to standard notation results in 0.0000822.
How do you convert a scientific notation number with a positive exponent back to standard notation?
-To convert a number with a positive exponent back to standard notation, you move the decimal point to the right by the number of places indicated by the exponent, adding zeros as placeholders where necessary.
Outlines
đą Introduction to Scientific Notation
This paragraph introduces the concept of scientific notation, which is essential for handling very large or very small numbers in science. It explains that scientific notation simplifies numbers by expressing them as a mantissa (a number between 1 and 10) multiplied by 10 raised to an exponent. The example of 19 trillion is used to demonstrate how to convert a large number into scientific notation by moving the decimal point 13 places to the left, resulting in 1.9 times 10 to the 13th power. The paragraph also discusses the importance of the exponent, which indicates the number of decimal places moved, and the mantissa, which is the number that multiplies the power of 10. Additionally, it touches on how scientific notation is used for small numbers, using the example of 0.000000063, which is written as 6.3 times 10 to the negative 8th power after moving the decimal point 8 places to the right.
đ Examples and Practice with Scientific Notation
This paragraph provides examples to further illustrate how to convert numbers into scientific notation. It starts with the number 78,000, which is converted by moving the decimal point four places to the left, resulting in 7.8 times 10 to the 4th power. The paragraph then contrasts this with smaller numbers, such as 0.0000826, which requires moving the decimal point five places to the right, resulting in 8.26 times 10 to the negative 5th power. Another example is the number 8 followed by nine zeros (eight billion), which is converted to 8 times 10 to the 9th power. The paragraph also reverses the process, showing how to convert scientific notation back to standard form, using examples like 5.3 times 10 to the negative 6th, which becomes 0.0000053, and 3.99 times 10 to the 4th, which becomes 39,900. The summary emphasizes the importance of understanding whether the number is large or small to determine the correct sign of the exponent and the process of moving the decimal point to convert between scientific and standard notation.
Mindmap
Keywords
đĄScientific Notation
đĄExponent
đĄMantissa
đĄDecimal Point Movement
đĄLarge Numbers
đĄSmall Numbers
đĄPositive Exponent
đĄNegative Exponent
đĄPlaceholder Zeros
đĄCoefficient
Highlights
Scientific notation is essential for handling very large or very small numbers.
A large number like 19 trillion is expressed as 1.9 times 10 to the 13th in scientific notation.
The decimal point is moved to position the number between 1 and 10 for scientific notation.
The exponent in scientific notation indicates the number of places the decimal point has been moved.
The mantissa is the number that is positioned between 1 and 10 in scientific notation.
A positive exponent indicates a large number, while a negative exponent indicates a small number.
Scientific notation simplifies typing numbers into a calculator and makes them easier to read.
An example of converting a small number, 0.0000000826, to scientific notation is 8.26 times 10 to the negative 8.
The mantissa must be a number greater than or equal to 1 but less than 10 for scientific notation to be correct.
The exponent's sign (positive or negative) corresponds to whether the original number is large or small.
78,000 is an example of a large number and is written as 7.8 times 10 to the 4th power in scientific notation.
Small numbers are represented with a negative exponent; for example, 0.0000826 is 8.26 times 10 to the negative 5th.
Eight billion is expressed as 8 times 10 to the 9th in scientific notation, indicating a large number.
Scientific notation numbers with negative exponents represent small numbers and are converted by moving the decimal point to the left.
To convert scientific notation back to standard form, move the decimal point to the right for positive exponents and to the left for negative exponents.
5.3 times 10 to the negative 6th is an example of a small number in scientific notation, converting to 0.0000053.
39,900 is the standard form of the scientific notation 3.99 times 10 to the 4th.
0.0000822 in standard form is represented as 8.22 times 10 to the negative 5th in scientific notation.
The next lesson will teach how to input scientific notation into calculators, a crucial skill in scientific work.
Transcripts
very often in science we have to use
very large numbers or sometimes very
small numbers and that's why we use
scientific notation and this lesson is
going to show you how to do that now
let's say we have a very large number
for example maybe we have this this
number right here and it looks like we
have one nine followed by looks to be
twelve zeros so it might be hard to read
that number if you're in the in the
United States we would probably read
that as 19 trillion if you live in
another country you might read that as
19 billion anyway it's a very large
number so how do we express that as a
scientific notation number well what you
have to do is you start by looking at
the understood decimal point there's
understood to be a decimal point right
here at that that spot right there and
we're going to move it we're gonna move
this so that the number is changed so
that it's in between 1 and 10 so I'm
going to have to move this decimal point
to the left and we start counting spaces
it's a lot of spaces there's six seven
we keep counting there's nine ten eleven
twelve and one more and if we move it to
that spot right there
that's thirteen places we just had to
move it thirteen places over to the left
and so the number is now 1.9 and we have
times 10 to the 13th and we used the
number 13 because we moved it 13 places
and so that's how you'd Express this
number in scientific notation so 19
trillion would be written as 1 point 9
times 10 to the 13th we don't have all
those zeros there that are running
together and so scientific notation is a
lot easier to type into a calculator or
to or to read and so we have two parts
of this scientific notation number the
first one is called the exponent and in
this case our exponent is 13
that's the power to which the number-10
is raised whenever you have a scientific
notation number the times is always
going to be the same the 10 there is
always going to be the same but that
exponent of course can change depending
upon how many places or how many spaces
you have to move that decimal point now
we also have a number out in front in
this case it's 1.9 and that's going to
be a number that's in between 1 and 10
we call that number the mantissa and
that's the number it sometimes looks
like a coefficient out in front
it's that multiplier that's always
greater than or equal to 1 but less than
10 and so if a number is incorrect
scientific notation it has to be in
between 1 and 10 and so in this case the
mantissa is 1.9 now we can also use
scientific notation to talk about very
small numbers and so let's say we have
this number point and then a bunch of
zeros 6 3 and this is a good example of
showing why we want to use scientific
notation because honestly those zeros
start to run together and it's it's a
hard number to read well we're going to
do the same thing we're going to take
that decimal point and we're going to
move it so that the number is in between
1 and 10
now this time we have to move the
decimal point to the right so it's 1 2 3
4 5 6 7 8 so it looks like we've just
moved that decimal point 8 places to the
right and so I move the decimal point
right there and so that's 6 point 3
we're going to call this times 10 to the
negative 8 and we have a negative
exponent when we have a very small
number and so once again the rule for
that is to take the decimal point and
move it so that the mantissa is in
between 1 and 10 so in this case we had
to move it to the right now the exponent
is going to be the number of places that
that decimal had to be moved so in this
case we moved at 8 places and so we had
an 8 up there right here in the last
example we had I believe it was a 13 now
some students get can
fused should you use a positive exponent
or a negative exponent and some
textbooks talk about if you move it to
the right it's negative the left is
positive but if you're going in the
other direction it's the opposite so I
tell students think about it as either a
small number or a big number if you have
a small number like this number we had
is obviously very small it's going to be
a negative exponent if you have a large
number like we had earlier that number
19 trillion we're going to have a
positive exponent so once again negative
exponents for small numbers positive
exponents for big numbers so let's try a
few examples here and we're going to try
to take some numbers and write them in
scientific notation so here's the first
example we have 78,000 so this time we
start with the decimal point which is
understood to be right there
and we're going to move it so that the
mantissa is in between 1 and 10 so we
move it to the left move it 1 2 3 4
places and so our mantissa is 7.8 times
10 to the 4th and notice it's positive 4
because this is a big number 78
thousands of pretty big numbers much
greater than 1 here's another example
looks like a small number this time
point oh-oh-oh-oh
8 to 6 so we have our decimal point
we're going to move it to the right this
time
how many places do we stop there no we
have to go one more right there and so
it's going to be eight point two six and
how many places did we have to move that
I hope you counted five places that's at
a positive five or a negative five for
the exponent well this is a pretty small
number so it's a negative five eight
point two six times 10 to the negative
fifth here's another example 8 followed
by nine zeroes and so in some parts of
the world they
that is 8,000 million in the United
States we read that as eight billion so
we take the understood decimal point
which is right there and we're going to
move that so that the mantissa is in
between one and ten so looks like we've
got to go to the left this time
how many places that's it so it's nine
places we moved it the answer is eight
times ten to the ninth is it positive
ninth or negative ninth well this is a
big number so it's positive nine eight
times 10 to the ninth is the right
answer now let's try going the other
direction let's take some scientific
notation numbers and write them as just
regular notation numbers so this time we
have five point six times ten I'm sorry
five point three times ten to the
negative sixth so the first thing you
want to do is think is that a big number
or a small number well it's a negative
exponent so it's a small number so we
start with the five point three and
we're going to move it to the left six
places so one and then two three four
five six we're gonna stick a decimal
point there and what goes in these other
places well we're gonna have to stick
some placeholder zeros so the answer is
point oh oh oh oh oh five three and if
you want you could rewrite that so that
it's a little bit neater and you don't
have all that I'll do the eraser here
and we'll erase all that and that
original decimal point so it's that's
our correct answer let's try the next
example here we have three point nine
nine times 10 to the fourth so once
again it's a positive exponent so it's a
big number we move the decimal point to
the right so it's three and then we had
our 0.99 move it to the right four
places so one two three four we need
some placeholder zeros there and so
that's our answer it looks like it's
39,900 so
erase all those and we have our answer
39,900 little comma there one more
example eight point two two times 10 to
the negative fifth well we have a
negative exponent so it looks like this
is a small number we're going to move
the decimal point to the left so we have
our eight point two two and we move it
to the left five places so there's one
two three four five so we need a decimal
point right there and we're going to
have some placeholder zeros in here and
so the answer is let me erase all that
extra ink on the slide and so the answer
is point
oooo 8 2 2 and so hopefully at this
point you have a pretty good feel for
how to deal with scientific notation in
our next lesson we're going to learn how
to input scientific notation numbers
into your calculator which is a very
important skill in science
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