Significant Figures and Scientific Notation
Summary
TLDRThe video explains how to handle problems involving significant figures and scientific notation. It starts by solving a multi-step significant figure problem involving subtraction, division, and multiplication, emphasizing the importance of rounding based on the least significant figures. The video then covers scientific notation, focusing on how to convert numbers to and from this format while maintaining the correct number of significant digits. It also includes examples of adding, subtracting, multiplying, and dividing numbers in scientific notation, highlighting the need to adjust exponents and apply the proper rounding rules.
Takeaways
- 🧪 When adding or subtracting numbers with significant figures, the result should match the least number of decimal places in the input numbers.
- 🧮 Multiplying or dividing by 100 to get a percentage involves an unlimited number of significant figures, so 100 is treated as having infinite sig figs.
- 🔢 When performing multi-step operations like subtraction and division, calculate significant figures at each step and round accordingly.
- 🔬 Scientific notation helps express very large or small numbers. The coefficient must be between 1 and 10, while the power of 10 adjusts the scale.
- 📐 Scientific notation doesn't affect the number of significant figures in the coefficient, but the number of sig figs must be consistent in calculations.
- 📏 When taking numbers out of scientific notation, ensure that the number of significant figures in the coefficient matches the original value.
- 🧮 In multiplication or division of numbers in scientific notation, use the least number of significant figures from the input values for rounding.
- 🔢 Addition and subtraction in scientific notation require making the exponents the same before performing the operation, then rounding based on the least number of decimal places.
- ✏️ In cases where you need to move a decimal to match exponents in scientific notation, shift to the left when increasing the exponent and right when decreasing it.
- 📊 When subtracting or adding in scientific notation, always move to the larger exponent and adjust the other number accordingly, before calculating the final result.
Q & A
What is the first step in solving a multi-step problem with significant figures?
-The first step is to apply the rules of significant figures to the operation, starting with subtraction in this example. You look at the least number of decimal places to determine how many decimal places the result should have.
How do you determine the number of decimal places to retain in subtraction?
-You look at the numbers involved and check which one has the least number of decimal places. For example, if both numbers have one decimal place, the result should also have one decimal place.
What happens when multiplying by 100 in a significant figures calculation?
-When multiplying by 100 to get a percentage, the number 100 has an infinite amount of significant digits, meaning it does not affect the number of significant figures in the result.
Why is it important to round results to the correct number of significant figures?
-Rounding to the correct number of significant figures ensures accuracy based on the precision of the numbers used. In the example, the least number of significant figures in the numerator and denominator determines how the result should be rounded.
What is scientific notation and why is it used in chemistry?
-Scientific notation consists of a coefficient and a power of 10, used to simplify working with very large or very small numbers in chemistry. The coefficient must be between 1 and less than 10.
How do you maintain the correct number of significant figures when converting numbers into scientific notation?
-You ensure that the number of significant figures in the coefficient matches the number of significant figures in the original number. The power of 10 does not affect the number of significant figures.
What is the rule for handling zeros in significant figures?
-Zeros count as significant figures when they are after a decimal point or between non-zero digits. For example, in the number 0.002560, the final zero is significant because it follows a decimal point.
How do you perform multiplication and division with numbers in scientific notation?
-Use a scientific calculator to enter the numbers, considering the exponent buttons. After calculating, round the result based on the number with the least significant figures.
What is the first step when adding or subtracting numbers in scientific notation?
-The first step is to make the exponents the same. You usually adjust the smaller exponent to match the larger exponent by moving the decimal point accordingly.
How do you round results after adding or subtracting numbers in scientific notation?
-Once the exponents are the same, perform the addition or subtraction and round the result to match the number with the least decimal places in the original values.
Outlines
🔬 Scientific Notation and Significant Figures: A Multi-Step Example
In this paragraph, the speaker walks through a detailed multi-step problem involving significant figures in subtraction and division. The example uses 25.0 milliliters minus 23.2 milliliters, applying the rules for decimal places in significant figures. After the subtraction, the resulting value is divided by another number, with a final multiplication by 100 to convert to a percentage. The speaker emphasizes that multiplying by 100 does not reduce the number of significant digits, and the result is rounded to two significant figures. This part also provides a brief review of significant digits rules from a previous video.
📐 Introduction to Scientific Notation and Examples
The speaker introduces scientific notation, explaining its purpose in handling large and small numbers in chemistry. The explanation includes the structure of scientific notation, which consists of a coefficient and a power of 10. The coefficient must be between 1 and less than 10, with examples provided. A strong emphasis is placed on maintaining the correct number of significant figures when converting numbers into scientific notation. Several examples demonstrate this, ensuring that both the original number and the coefficient have matching significant figures, while the power of 10 does not affect the count of significant digits.
🧪 Significant Figures in Addition and Subtraction Using Scientific Notation
This paragraph demonstrates how to handle addition and subtraction when using scientific notation, starting with making the exponents the same for both numbers. The speaker first rewrites the smaller exponent number by shifting its decimal, allowing both numbers to have the same exponent. After aligning the exponents, the speaker shows how to add the coefficients and emphasizes rounding the result based on the least number of decimal places. A clear example is provided, illustrating the process and reinforcing the importance of rounding to the correct number of significant figures.
Mindmap
Keywords
💡Scientific Notation
💡Significant Figures (Sig Figs)
💡Addition and Subtraction in Scientific Notation
💡Multiplication and Division in Scientific Notation
💡Rounding
💡Coefficients in Scientific Notation
💡Decimal Places
💡Percentage
💡Subtraction in Multi-step Problems
💡Calculator Usage in Scientific Notation
Highlights
Introduction to the video, covering examples of addition, subtraction, multiplication, and division in scientific notation.
Review of a multi-step problem involving subtraction, division, and multiplication of significant figures.
Explanation of subtracting significant figures, focusing on keeping the least number of decimal places.
Introduction of division in significant figures and using a calculator for division followed by multiplying by 100 to obtain percentages.
Clarification that multiplying by 100 has an infinite number of significant figures, which can be confusing to students.
Calculation result with significant figures: after subtraction and division, the final answer is rounded to 7.8% based on the least significant figures.
Introduction to scientific notation and its use in chemistry to deal with large and small numbers, consisting of a coefficient and 10 raised to a power.
Detailed example of converting a number into scientific notation with a coefficient between 1 and 10 and ensuring consistency with significant figures.
Explanation of converting numbers from scientific notation back to standard form while preserving significant figures.
Multiplication in scientific notation: using the calculator’s exponent button to input numbers correctly and maintaining significant figures in the final result.
Division example in scientific notation: after dividing, the result is rounded to the correct number of significant figures based on the input numbers.
Explanation of the different rules for addition and subtraction in scientific notation, emphasizing the need to align exponents before performing the operation.
Example of adding numbers in scientific notation by adjusting one of the exponents and then rounding the result to the correct decimal place.
Final example of subtracting numbers with negative exponents in scientific notation by adjusting one exponent and rounding to the correct significant figures.
Closing remarks: Encouragement to seek help if further questions arise regarding scientific notation and significant figures.
Transcripts
okay in this short video I wanted to go
through some examples using scientific
notation when you have to add and
subtract and also when you have to
multiply and divide in scientific
notation before we do that I wanted to
go through a multi-step significant
figure problem just to kind of review
from the last video that we had on
significant digits let's apply all the
rules to a problem where you have both
subtraction and division and
multiplication all combined so if you
look at the numerator here you have an
example of 25.0 milliliters - 23.2
milliliters the first thing you have to
look at is the least number of decimal
places so for these two numbers what I'm
gonna look at is that 25.0 has one
decimal place 23.2 has one decimal place
so according to the rules of significant
figures when I do the subtraction I need
one place after the decimal and that
would be one point eight milliliters now
the next thing I'm going to do is the
division so 1.8 milliliters divided by
twenty three point two milliliters if I
plug that into my calculator and then
multiply by 100 to get a percentage I
then have to look at significant figures
for the one that has the least amount
now the 100 in this case is just telling
me to move the decimal place over two
times so that acts as an integer this
has an infinite amount of significant
digits anytime you're multiplying by a
hundred to get a percentage which is the
case for this example you need to have
an unlimited number of sig figs in that
number 100 don't round it to one sig fig
that's a confusing point but anytime you
multiply by a hundred it is going to be
an unlimited amount of sig figs when you
want to get a percentage so if I pop
this in on my calculator it
spits out to me seven point seven five
eight six blah blah blah blah keeps
going so I need to round to the correct
number of sig figs the numerator as a
result of the subtraction has two sig
figs the denominator has three I'll just
write SF for sig figs the 100 has
unlimited so my least amount of
significant figures in this problem is
two so I'm gonna round my answer to
seven point eight percent that's just a
quick review of a multi-step problem
let's talk a little bit about scientific
notation
in chemistry we use scientific notation
to help us with very large numbers and
very small numbers scientific notation
consists of a coefficient and 10 raised
to a specific power the coefficient has
to be between 1 and less than 10 so 6.02
in our example fits nicely between 1 and
less than 10 you cannot write the
coefficient as sixty point two or 602
and then of course in blue here you have
10 raised to a certain power the power
could be negative or the power could be
positive in this example it's 10 raised
to the positive power of 23
let's take some examples of scientific
notation and let's put it into proper
scientific notation and let's take
examples and take them out of scientific
notation so the important thing to
remember here is keep your number of sig
figs consistent so in our first example
this has one two three four significant
figures when I change this into proper
scientific notation my coefficient is
going to have four significant figures
so let's write that down seven point
three two five times ten to the six and
let's write the unit ions don't forget
to write your unit we have one two three
four sig figs in our number and one two
three four sig figs and our coefficient
so the number and the coefficient have
to match as far as significant figures
goes the power of 10 does not come into
play when we are calculating number of
significant digits so in 576 grams there
are 1 2 3 significant figures so let's
put this into scientific notation five
point seven six times 10 to the 2 grams
and make sure you have one two three
significant figures in your coefficient
let's keep going we're going to take
this number two point five six Oh times
10 to the negative three and we're going
to take it out of scientific notation so
that translates into zero point zero
zero two five six zero molecules let's
check our sig figs
the coefficient has one two three four
sig figs that zero counts because it is
after a decimal
let's go and check out our number one
two three four sig figs in our actual
number if you get confused about whether
the zero counts or not here's what I say
to my students if you see a number as
well as a decimal start counting from
the number so number and a decimal start
counting from the number everything
after that two is significant that's
kind of a saying that will really help
you see a number see a decimal start
counting from the number let's go and do
the next example so taking it out of
scientific notation this would come out
to be five nine nine point eight
particles and we're going to make sure
we have the correct amount of sig figs
our coefficient has one two three four
see a number see a decimal start
counting from the number see a number
see a decimal start counting from the
number one two three four and they match
alright let's do some examples with
multiplication and division with
scientific notation in multiplication
and division use your scientific
calculator and your button or your
exponent button and plug in the numbers
just as they appear so on my calculator
I am hitting seven point two times 10
squared which can be represented as
seven point two exponent to the two
times 5.0 to exp EE or exponent to the
negative three and my calculator spits
out this number it is three point six
one four four don't forget your units a
meter times a meter is going to be a
meter squared
okay this is not correct you have to
look at each of the numbers and see the
least amount amount of significant
figures okay so on our first number this
has two significant figures in our
second number this has one two three
significant figures so this number would
actually be rounded to three point six
because that's our least amount of sig
figs in one of our numbers meter squared
that is our correct answer let's go to
the bottom problem here one point four
eight three punch that in on your
calculator that has four significant
figures and divided by one point three
milliliters that has two significant
figures my calculator never gives me the
right amount of sig figs I have to know
that and it's going to spit out eleven
point four zero seven six nine blah blah
blah blah too much okay so here we have
to round that to two significant figures
and I get 11 grams per milliliter
don't forget your unit two sig figs
one of the hardest things to remember is
the different rule for addition and
subtraction with scientific notation the
first thing that you must do when you're
adding and subtracting in scientific
notation is make both of the exponents
the same what I typically do is I go to
the larger exponent so look at your
first number here it is six point seven
times ten to the six that I'm gonna
leave alone I'm not going to change the
exponent I'm not going to change that
exponent so let me just rewrite that six
point seven times ten to the six grams
and I'm going to then change the three
point four times ten to the third grams
into ten to the six so the difference
between three and six is three places I
must move that decimal if you're always
going to the larger exponent you're
gonna always move to the left so watch
what happens here the three point four
you're gonna move that three point four
decimal three places to the left so you
have zero point zero zero three four
times ten to the six grams now when both
of these numbers have the same exponent
then you can count the least amount of
decimal places if you add up all of
these you get six point seven oh three
four just bring down the power of ten
ten to the six grams the first number
has one place after the decimal the
second number has one two three four
places after the decimal so you must
round your answer to one place after the
decimal the correct answer six point
seven times ten to the six grams
let's try one more this is confusing
this is a hard problem
our final problem nine point two eight
times ten to the minus three and two
point eight times ten to the minus four
first thing I'm gonna do to subtract
addition and subtraction make your
exponents the same I always go to the
larger exponent so believe it or not the
larger exponent is negative three so
nine point two eight times ten to the
minus three liters I'm going to change
the to point eight times ten to the make
negative four I'm gonna make that a
negative three watch what happens
you're gonna move it one place to the
left you're always gonna move it to the
left so my new number is zero point two
eight times ten to the minus three
leaders perform your subtraction so you
get nine point zero zero times ten to
the minus three leaders count your
number of decimal places for each
problem one two one two the least amount
of decimal places is two and your answer
comes out to be nine point zero zero two
places after the decimal times ten to
the minus three liters I hope this has
helped you and you can see us if you
have any further questions take care
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