Scientific Notation pt1

Summary

Today's lesson focuses on scientific notation, a method to express extremely large or small numbers concisely. The video explains the importance of Avagadro's number in chemistry and how scientific notation simplifies its representation. The key rules are highlighted: move the decimal point to have one non-zero digit to the left, and the exponent indicates the number of places moved. Examples demonstrate converting large and small numbers into scientific notation, emphasizing the coefficient and exponent. The video concludes with a preview of upcoming content on performing calculations using scientific notation.

Takeaways

• 🔢 Scientific notation is used to express very large or very small numbers in a compact form.
• 📏 The first rule of scientific notation is to move the decimal point so that there is only one non-zero digit to the left of the decimal.
• ✏️ Rule number two states that the exponent in scientific notation represents the number of places the decimal point has been moved.
• 🔑 The general form of a number in scientific notation is a coefficient multiplied by 10 raised to the power of an exponent.
• 🔄 For large numbers, the exponent is positive, and for small numbers, the exponent is negative.
• 🔎 Significant figures (sigfigs) can affect how you determine the coefficient in scientific notation.
• 🧮 Avogadro's number, a critical number in chemistry, is often expressed in scientific notation due to its large size.
• 📉 When dealing with small numbers, the decimal point is moved to the right, resulting in a negative exponent.
• 🔠 The coefficient in scientific notation is the non-zero digits left after adjusting the decimal point according to the first rule.
• 📚 In the next video, calculations using scientific notation will be discussed, often involving the use of a calculator.

Q & A

What is the purpose of scientific notation?

Scientific notation is used to condense very large or very small decimal numbers into a more manageable form for calculations and representation.

What is Avogadro's number and why is it important in chemistry?

Avogadro's number is the number of particles in a mole of a substance, which is approximately 6.02 × 10^23. It is crucial in chemistry as it helps in understanding the relationships between the macroscopic and microscopic properties of substances.

What is the first rule for expressing a number in scientific notation?

The first rule is to move the decimal point so that there is only one non-zero digit to the left of the decimal point.

Can you explain the role of the exponent in scientific notation?

The exponent in scientific notation indicates the number of places the decimal point has been moved. It is a power of 10 and is positive if the decimal point is moved to the left and negative if it is moved to the right.

What is meant by the coefficient in scientific notation?

The coefficient in scientific notation is the non-zero digit(s) that remain after the decimal point has been moved according to the first rule. It is multiplied by 10 raised to the power of the exponent.

How do you determine if the exponent should be positive or negative when expressing a number in scientific notation?

The exponent is positive if the original number is greater than 1 and the decimal point is moved to the left. It is negative if the original number is less than 1 and the decimal point is moved to the right.

What happens to the exponent in scientific notation when dealing with very small numbers?

For very small numbers, the exponent in scientific notation is negative, reflecting the movement of the decimal point to the right to satisfy the first rule.

How can you convert a number from scientific notation back to its standard form?

To convert from scientific notation to standard form, move the decimal point to the right for positive exponents and to the left for negative exponents by the number of places indicated by the exponent, filling in zeros where necessary.

Why is it important to know how to perform calculations with scientific notation even if a calculator can do it?

Understanding calculations with scientific notation is important because there are instances where a calculator may not be available, or it is necessary to understand the underlying mathematics for educational or professional purposes.

What is the significance of significant figures in scientific notation?

Significant figures are crucial in scientific notation as they determine the precision of the measurement. When expressing a number in scientific notation, only the significant figures are retained, and trailing zeros after the decimal point are not counted.

Outlines

00:00

🔬 Introduction to Scientific Notation

This paragraph introduces scientific notation, a method used to express very large or very small numbers in a more manageable form. The importance of Avogadro's number in chemistry is highlighted, emphasizing its significance in calculations involving moles. The speaker explains the need for a system like scientific notation due to the impracticality of inputting extremely long numbers into calculators. The rules for scientific notation are outlined: the decimal must be moved to the right of the first non-zero digit, and the exponent indicates the number of places the decimal has been moved. The speaker also discusses the concept of the coefficient in scientific notation, which is the number that remains after the decimal has been adjusted. The process of converting a large number, such as Avogadro's number, into scientific notation is demonstrated, with a focus on the movement of the decimal point and the calculation of the exponent.

05:06

📉 Scientific Notation for Small Numbers

The second paragraph continues the discussion on scientific notation, this time focusing on its application to small numbers. The speaker explains that the same principles apply: moving the decimal point to the right of the first non-zero digit and counting the number of places moved to determine the exponent. Since the number is small, the exponent will be negative. The speaker provides an example of converting a small number into scientific notation, emphasizing the importance of counting the correct number of decimal places moved. The paragraph also touches on the process of expanding scientific notation back into standard form, illustrating this with examples. The speaker clarifies that when expanding, one should fill in any empty spaces with zeros and provides a method to check the accuracy of the expanded form by converting it back into scientific notation. The paragraph concludes with a teaser for the next video, which will cover calculations involving scientific notation, hinting that while calculators can handle most calculations, understanding the manual process is still valuable.

Keywords

💡Scientific Notation

Scientific notation is a way to express very large or very small numbers in a compact form. It is a mathematical notation that simplifies calculations and expressions involving such numbers. In the video, scientific notation is introduced as a method to condense large numbers, such as Avogadro's number, which is crucial in chemistry but too large to be conveniently typed into calculators without this notation. The video script explains how to convert numbers into this form, emphasizing the importance of moving the decimal point to the right position.

💡Avogadro's Number

Avogadro's number is a fundamental constant in chemistry, approximately equal to 6.022 x 10^23, representing the number of particles (atoms, molecules, ions, etc.) in one mole of a substance. The video script uses Avogadro's number to illustrate the application of scientific notation, as it is an example of a number that is too large to be conveniently expressed without it.

💡Coefficient

In scientific notation, the coefficient is the first part of the expression, which consists of a single non-zero digit followed by any additional digits and is multiplied by a power of ten. The video script explains that when converting a number to scientific notation, the decimal point is moved so that only one non-zero digit remains to the left of the decimal point, forming the coefficient. For example, in the number 6.022 x 10^23, '6.022' is the coefficient.

💡Exponent

The exponent in scientific notation indicates the power to which 10 is raised. It is determined by the number of places the decimal point is moved to achieve the coefficient. The video script explains that if the decimal point is moved to the left (for large numbers), the exponent is positive, and if it is moved to the right (for small numbers), the exponent is negative. For instance, in the number 6.022 x 10^23, '23' is the exponent.

💡Decimal Point Movement

In the context of the video, moving the decimal point is a crucial step in converting a number into scientific notation. The script describes how to move the decimal point to the right or left to ensure only one non-zero digit is to the left of the decimal point, which is a requirement for scientific notation. This movement determines the value of the exponent.

💡Significant Figures

Significant figures are the digits in a number that carry meaning contributing to its precision. The video script mentions that when converting to scientific notation, the number of significant figures might determine how many digits are kept after the decimal point. However, for the example of Avogadro's number, the script simplifies to the first few significant digits for demonstration.

💡Expansion

Expansion in scientific notation refers to the process of converting a number from its scientific form back to its standard decimal form. The video script provides examples of how to expand numbers by moving the decimal point in the opposite direction of how it was moved during the conversion to scientific notation and filling in zeros where necessary.

💡Calculations

The video script hints at the topic of performing calculations using scientific notation, which will be covered in a subsequent video. It suggests that while calculators often handle these calculations, understanding how to manually compute with scientific notation is still valuable. This includes operations like multiplication, division, and exponentiation involving numbers in scientific form.

💡Mole

A mole is a unit of measurement used in chemistry to express amounts of a chemical substance, defined as the number of atoms in 12 grams of carbon-12. The video script introduces Avogadro's number in the context of moles, as it represents the number of particles in one mole of a substance, which is a foundational concept in understanding chemical reactions and stoichiometry.

💡Placeholder Zeros

Placeholder zeros are zeros that are added in a number's decimal part to maintain the scale of the number when it is in scientific notation. The video script explains that when expanding numbers, any empty spaces that would be created by moving the decimal point are filled with zeros to reflect the original magnitude of the number accurately.

Highlights

Scientific notation is used to condense large or very small decimal numbers.

Avogadro's number is a crucial number in chemistry, representing the number of particles in a mole.

Large numbers in scientific notation require the decimal to be moved so only one non-zero digit is to the left of the decimal point.

The exponent in scientific notation indicates the number of decimal places moved.

The coefficient is the part of the number left after collapsing it down to one non-zero digit to the left of the decimal.

For large numbers, the exponent is positive when the decimal is moved to the left.

For small numbers, the exponent is negative when the decimal is moved to the right.

Avogadro's number in scientific notation is 6.02 x 10^23.

When dealing with small numbers, the process of converting to scientific notation involves moving the decimal to the right and using a negative exponent.

The process of converting a number from scientific notation to standard form involves moving the decimal in the opposite direction of the exponent.

Empty spaces in standard form notation are filled with zeros when converting from scientific notation.

Checking your work by converting the number back to scientific notation ensures accuracy.

Calculators are typically used for calculations involving scientific notation, but it's important to understand the manual process.

The next video will cover calculations with scientific notation.

Transcripts

hey kiddos today we're going to talk

about scientific notation now what

scientific notation is for is that we

have to have some way to sort of

condense down an abnormally large or

really small decimal number um so for

instance this number is one of the most

important numbers in all of chemistry

it's avagadro's number this is how many

particles are in a mole of something

that doesn't mean anything at all to you

now but it will um in a will in a few

weeks or a month or whatever um so this

number is really important now in a lot

of calculators you wouldn't even be able

to put this number in as it is because

there's not 20s something digits worth

of number space to type in the numbers

so we need some way to express large

numbers and we'll see here in a second

really small numbers um in a way that

will make a little bit more sense um and

so that's what we're going to do in this

video is show you how to put things in a

scientific notation um and and what

scientific notation means then in our

next video we'll talk about how to do

calculations with it

all right so rule number one for putting

a number a big number or a small number

we'll see here in a minute um into

scientific notation is that we need the

decimal to be moved so that we only have

one nonzero digit left of the decimal

place if you have any more than one

nonzero digit left of the decimal um

then that means that you're not in

scientific notation so you're going to

have to do something to put everything

back into scientific notation so what

that means is that we have a decimal

place here right okay that's unspoken

decimal we know it's there and we need

to move it until it's here so that

that's rule number one I'm not going to

do that just yet because I want you to

see what rule number two is first and

then we'll put everything into the

correct setup okay so rule number two

says that my exponent is going to be the

number of spaces um that we move the

decimal and you might be saying to

yourself what do you mean by exponent

okay so every scientific notation number

has the same general form so there's a

Time

10 raised to some exponential power and

the exponential power is what rule two

applies to however many spaces I move

the decimal that's going to become my

exponent okay so that's the exponential

part of it the other part of the number

is called the coefficient and that's

essentially whatever is left after you

collapse the number back down that that

how much you leave might be dependent on

significant figures or like in the case

that we're going to do here might be

just dependent on hey we're going to cut

it off here and everything else was

zeros and therefore placeholders and so

I'm not as worried about those okay so

let's go ahead and do this then and then

put this number into scientific notation

so we're going to count our

spaces we've got this is where the

decimal is and I'm going to probably

speed up um the counting a little bit

because it's boring to watch me do it

[Music]

so okay so that took us 23 spaces I'm

going to write that down just so that I

don't forget it okay so what that means

is that when I'm going to put my stuff

in scientific notation I automatically

know that my exponent is 23 now it could

be a negative or a

positive um and so there's a couple of

ways to remember essentially if you move

the decimal place to the left you're

going to have a positive number if you

move the decimal place to the right to

obey rule one then that gives you a

negative EXP exponent I always find that

really hard to remember I don't know why

I mean know I probably learned this

stuff 30 something years ago so what

always makes more sense in my brain to

remember is that if it's a big number

like this is then that's a positive

exponents for a really small number like

the next one I'll show you then that's a

negative exponent okay so that takes

care of the exponential part of the

exponential or scientific notation so

what about the rest of it well in this

case I don't really know anything about

my sigfigs so I'm just going to take the

nonzero digits there and I'm I'm going

to count that so

6.02 3 * 10 23rd that's what you're

going to use for avagadro's number in a

wide variety of calculations all right

so we worked a really big number what do

we do with a small number well the same

rules apply I'm going to want one

nonzero to the left of the decimal so

that means my decimal is here now and

I'm going to want it to be right here

okay so I've got to move and count that

many spaces the number of spaces that I

count that's what's going to be my

exponent um and then since I've got a

small number and since I'm moving the

decimal to the right this time my

exponent is going to be negative instead

of the positive that we had before so

real quick just so that I don't forget

anything in the process of doing it I'm

going to go ahead and write down my

times 10 um I can already sort of tell

what my coefficient is but I'm just

going to wait until I count and then

we'll walk through that once we get

there so let's count the numbers now I

sort of divided this out where your

commas would be so that's going to make

it a little bit easier but again I'm

going to go ahead and count them so

it'll be SP sped up a little

okay so we had

34 and so my exponent here is going to

be 34 I move 34 spaces I move the

decimal to the right so therefore it's a

negative also remember it's a small

number so it's going to be a negative

and then we're going to write down what

is left remember that my new decimal is

right here okay so that becomes 6 point

um actually I wrote that number down

wrong that should be

6.62607004 expand them back out just to

make sure that you're clear on that then

in the next video we'll talk about how

to do the calculation part of it okay so

some real quick examples just to make

sure that we're all on board and can do

this so decimal point is here one two

three four spaces so that's going to

give me time 10 to the

4th and then I've got

2.24 * 10 4th okay pretty

straightforward and this case and

remember I move to the left in this case

I'm going to move to the right 1 2 3 4

five remember I have to get this number

to the left of the decimal so that means

I'm left with 6.3 I moved at five

spaces and I moved it to the right which

means it's got a negative in front of it

okay so how about these These are

already in scientific notation so then

what do I do with them then um typically

speaking what you would be asked to do

is probably to expand them back out into

to normal notation in other words

without the times 10 so how does that

work well here it it's pretty

straightforward you go the opposite way

that you did to put it into scientific

notation if I put it into scientific

notation this way then to undo it I go

back the other way so what that means is

if I go left to get a positive exponent

then when I'm expanding a positive

exponent I take the decimal and go back

right and so that's exactly what we're

going to do here I'm going to go one two

three space is right so that would give

me

929 and then you're like well there was

nothing there what do I do well any

empty spaces there you fill in with a

zero okay and how do you know that you

got the right answer well check it put

it back in scientific notation make sure

you got the same number if I move the

decimal here one two three spaces to the

right I get 9.29 * 10 the 3 that's

exactly what I should get so for the

last one again you go the opposite

direction if I go to the right to get a

negative number then I go back to the

left when I have a negative number and

I'm trying to expand it out so in that

case that means I'm going to go one two

three four

spaces okay again what do I do with

those empty spots well you're going to

fill those in with zeros okay and I'm

going to clean that up so you can

actually see it

00178 would be my correct answer now

typically speaking you would sort of put

that other zero there just to let you

know that the decimals there that's not

absolutely necessary and I certainly

wouldn't Mark you wrong for that and

that zero does not count as one of your

spaces remember you're counting the

spaces to get to the correct decimal all

right guys next video we will talk about

how do we do calculations with

scientific notation although I'll tell

you the secret right now most of the

time you're just going to put them in

your calculator and let it do all the

work but you should still know how to do

it um because there are some occasions

when you won't have it thanks kiddos