Significant Figures and Scientific Notation

Summary

The 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

00:00

🔬 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.

05:03

📐 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.

10:05

🧪 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.

Keywords

💡Scientific Notation

Scientific notation is a way of expressing very large or very small numbers by using powers of ten. It consists of a coefficient between 1 and less than 10, multiplied by 10 raised to a power. In the video, this concept is fundamental when dealing with large and small numbers in chemistry, such as converting numbers like 7.325 × 10^6 ions or 5.76 × 10^2 grams into or out of scientific notation.

💡Significant Figures (Sig Figs)

Significant figures refer to the digits in a number that are reliable and necessary to indicate precision. In the video, significant figures play a key role in ensuring accuracy when performing mathematical operations. For example, during subtraction and division in the multi-step problem, the least number of significant figures determines how the final result is rounded.

💡Addition and Subtraction in Scientific Notation

When adding or subtracting numbers in scientific notation, the exponents must be made the same before performing the operation. In the video, this is demonstrated by adjusting the exponent of one number to match the other, such as converting 3.4 × 10^3 to 0.0034 × 10^6 to add it to 6.7 × 10^6. The answer is then rounded based on decimal places.

💡Multiplication and Division in Scientific Notation

Multiplication and division in scientific notation involve multiplying/dividing the coefficients and adding/subtracting the exponents. The number of significant figures determines the final answer's precision. In the video, this is illustrated with examples like multiplying 7.2 × 10^2 by 5.0 × 10^-3, resulting in an answer rounded based on the least number of significant figures.

💡Rounding

Rounding is the process of reducing the digits of a number while maintaining its value as close to the original as possible. In the video, rounding is done according to the number of significant figures, especially after performing operations like division or multiplication. For instance, 7.7586 is rounded to 7.8 based on the rules of significant figures.

💡Coefficients in Scientific Notation

The coefficient in scientific notation is the number between 1 and less than 10 that is multiplied by a power of ten. In the video, maintaining the number of significant figures in the coefficient is emphasized, as shown in examples where numbers like 576 grams are converted to 5.76 × 10^2 grams while keeping three significant figures.

💡Decimal Places

Decimal places are the number of digits to the right of the decimal point in a number. The video highlights their importance when performing subtraction and ensuring the correct number of decimal places based on the least precise measurement. For instance, in the subtraction of 25.0 and 23.2, the result is rounded to one decimal place.

💡Percentage

A percentage is a way of expressing a number as a fraction of 100. In the video, a multi-step problem involving division leads to the conversion of the result into a percentage by multiplying by 100, which, as noted, does not affect the significant figures of the final result. The video explains that the number 100 is treated as having an unlimited number of significant figures.

💡Subtraction in Multi-step Problems

Subtraction is one of the operations used in multi-step problems where significant figures and decimal places need to be carefully managed. In the video, this is demonstrated when subtracting 23.2 from 25.0 milliliters, yielding 1.8 milliliters. The result is rounded based on the least number of decimal places, a critical step in multi-step calculations.

💡Calculator Usage in Scientific Notation

Calculators are essential tools for performing operations in scientific notation, as they help to handle large and small numbers and their exponents. The video demonstrates how to use a calculator for operations like 7.2 × 10^2 × 5.0 × 10^-3, emphasizing the need to manually adjust the results for significant figures, as calculators do not always provide the correct rounding.

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