Scientific Notation

Summary

This 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

• 🔢 Scientific notation is used to express very large or very small numbers more easily.
• 💡 To convert a large number into scientific notation, move the decimal point so that the number is between 1 and 10.
• ⬅️ For large numbers, move the decimal point to the left and count the number of spaces moved.
• 🔍 The number of spaces moved becomes the exponent in scientific notation.
• 🔢 The number in front of the scientific notation (e.g., 1.9 in 1.9 x 10^13) is called the mantissa.
• ➖ Negative exponents are used for very small numbers, indicating how many places the decimal point was moved to the right.
• 🔄 The rule of thumb: positive exponents for large numbers, negative exponents for small numbers.
• 📈 Moving a decimal point to the left results in a positive exponent; to the right results in a negative exponent.
• 📝 Scientific notation simplifies calculations and is easier to input into calculators.
• 📊 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

00:00

🔢 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.

05:02

📚 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.

Keywords

💡Scientific Notation

Scientific notation is a way of expressing very large or very small numbers in a compact form. It is defined as a number expressed as the product of a number between 1 and 10 and a power of 10. In the video, scientific notation is introduced as a tool to handle large numbers like 19 trillion, which is represented as 1.9 times 10 to the 13th. This notation simplifies the representation and manipulation of such numbers in scientific calculations.

💡Exponent

In the context of scientific notation, the exponent refers to the power to which the number 10 is raised. It is a key component of the notation that indicates the number of places the decimal point has been moved. For instance, in the number 1.9 times 10 to the 13th, the exponent '13' signifies that the decimal point has been moved 13 places to the right, making it easier to express large numbers.

💡Mantissa

The mantissa is the part of the scientific notation that is a number between 1 and 10. It represents the significant figures of the number being expressed in scientific notation. For example, in the scientific notation of 19 trillion as 1.9 times 10 to the 13th, '1.9' is the mantissa. It is crucial for ensuring that the number is correctly represented without unnecessary zeros.

💡Decimal Point Movement

Moving the decimal point is a fundamental action in converting a number to scientific notation. The direction and number of places the decimal point is moved determine the value of the exponent. In the video, moving the decimal point to the left for large numbers results in a positive exponent, while moving it to the right for small numbers results in a negative exponent.

💡Large Numbers

Large numbers, such as 19 trillion, are difficult to write out in full due to the many zeros involved. The video explains how scientific notation simplifies these numbers by using a mantissa and an exponent. This method is particularly useful in scientific contexts where such large values are common.

💡Small Numbers

Small numbers, like 0.000000063, are also challenging to work with due to the many leading zeros. Scientific notation addresses this by using a negative exponent, which indicates the decimal point has been moved to the right. The video demonstrates converting such a small number into scientific notation as 6.3 times 10 to the negative 8th.

💡Positive Exponent

A positive exponent in scientific notation indicates that the original number is greater than 1. It shows the number of places the decimal point has been moved to the right to reach a number between 1 and 10. For example, in the video, 8 billion is expressed as 8 times 10 to the 9th, indicating a large number.

💡Negative Exponent

A negative exponent in scientific notation signifies that the original number is less than 1. It represents the number of places the decimal point has been moved to the left to achieve a mantissa between 1 and 10. The video uses the example of 0.000000063, which becomes 6.3 times 10 to the negative 8th, to illustrate this concept.

💡Placeholder Zeros

Placeholder zeros are zeros added in the decimal part of a number when converting from scientific notation to standard form. They ensure the decimal point is in the correct position. The video shows how, after moving the decimal point, placeholder zeros are added to complete the number, as seen when converting 5.3 times 10 to the negative 6th into 0.0000053.

💡Coefficient

Although not explicitly mentioned in the script, the coefficient is closely related to the mantissa. It is the number that is multiplied by the power of 10 in scientific notation. The coefficient is typically a number greater than or equal to 1 and less than 10, which is the same range as the mantissa.

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