Scientific Notation - Fast Review!

Summary

This lesson offers a fundamental guide to scientific notation, ideal for expressing both very large and minute numbers. The instructor demonstrates converting numbers like 45,000 and 9.3 billion into scientific form, emphasizing the significance of positive and negative exponents. Examples are provided for converting both large and small numbers to scientific notation and vice versa, illustrating how shifting the decimal point left or right affects the exponent's sign. The tutorial aims to make the process of converting between scientific and standard notation clear and accessible.

Takeaways

• 🔢 Scientific notation is a method to express very large or very small numbers compactly.
• 📐 To convert a large number to scientific notation, move the decimal point to the right of the first non-zero digit and count the number of places moved to determine the exponent.
• 🔄 For large numbers, the exponent in scientific notation is positive, indicating the number is scaled up.
• 🔎 For small numbers, move the decimal point to the right to place it between the first and second significant digits, then count the moves to determine a negative exponent.
• 📉 A negative exponent in scientific notation indicates a very small number, less than one.
• 🔄 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 by the number of places indicated by the exponent.
• 📘 The exponent in scientific notation is a power of 10, which scales the number in front of it (the coefficient) up or down.
• 📋 Examples given include converting 45,000 to 4.5 × 10^4, 37,580,000 to 3.758 × 10^7, and 0.0023 to 2.3 × 10^-3.
• 🔠 The coefficient in scientific notation should be a number between 1 and 10 for non-zero values.
• 🔄 Moving the decimal point to the right increases the value of the number, while moving it to the left decreases it when converting between scientific and standard notation.

Q & A

What is scientific notation?

Scientific notation is a way to express very large or very small numbers in a compact form. It uses a base number between 1 and 10 multiplied by a power of 10.

How do you express the number 45,000 in scientific notation?

The number 45,000 is expressed in scientific notation as 4.5 x 10^4, by moving the decimal point four places to the left.

What does a positive exponent in scientific notation represent?

A positive exponent in scientific notation represents a very large number. It indicates how many places the decimal point must be moved to the right to return to the original number.

Can you provide an example of converting a large number to scientific notation?

Yes, the number 375,580,000 can be converted to scientific notation as 3.7558 x 10^8 by moving the decimal point eight places to the left.

How is the number 9.3 billion expressed in scientific notation?

9.3 billion is expressed as 9.3 x 10^9 in scientific notation, by moving the decimal point nine places to the left.

What does a negative exponent signify in scientific notation?

A negative exponent signifies a very small number, or a number less than 1. It indicates how many places the decimal point must be moved to the right to return to the original number.

How do you convert the number 0.0023 to scientific notation?

The number 0.0023 is converted to scientific notation as 2.3 x 10^-3 by moving the decimal point three places to the right.

What is the process of converting a number from scientific notation to standard notation?

To convert from scientific notation to standard notation, you move the decimal point to the right for positive exponents and to the left for negative exponents, the number of places indicated by the exponent.

Can you give an example of converting a number with a negative exponent from scientific notation to standard notation?

Yes, the number 2.4 x 10^-2 in scientific notation converts to 0.024 in standard notation by moving the decimal point two places to the left.

How do you determine whether to move the decimal point to the left or right when converting to scientific notation?

You move the decimal point to the left for large numbers (positive exponent) and to the right for small numbers (negative exponent) to position it between the first non-zero digits.

What is the significance of the exponent in scientific notation?

The exponent in scientific notation indicates the number of places the decimal point has been moved to reach its new position, which corresponds to the scale of the number.

Outlines

00:00

🔢 Introduction to Scientific Notation

This paragraph introduces scientific notation as a method for representing both very large and very small numbers. It explains how to convert a number like forty-five thousand into scientific notation by moving the decimal point between the first two digits and then multiplying by 10 raised to the power of the number of places moved. The paragraph also discusses how positive exponents relate to large numbers and negative exponents to small numbers, providing examples such as 37.55 billion and 0.0023 to illustrate the process of converting numbers to scientific notation.

05:02

🔄 Converting Scientific Notation to Standard Notation

This paragraph focuses on converting numbers from scientific notation back to standard notation. It emphasizes that positive exponents correspond to larger numbers, necessitating a move of the decimal point to the right to increase value, as demonstrated with the example of 2.4 times 10 to the power of 2. The explanation continues with examples of larger numbers and their conversion, such as 3.56 times 10 to the power of 3 and 4.27 times 10 to the power of 5, showing how to move the decimal point accordingly and add zeros to obtain the standard notation.

10:03

🔁 More Examples of Notation Conversion

The final paragraph provides additional examples of converting numbers from scientific notation to standard notation, including those with negative exponents which result in smaller numbers. It guides through the process of moving the decimal point to the left for negative exponents and to the right for positive exponents, filling in the necessary zeros. Examples given include 1.8 times 10 to the minus 3 and 2.7 times 10 to the power of 4, reinforcing the understanding of scientific notation and its conversion to standard form.

Keywords

💡Scientific Notation

Scientific notation is a method of expressing very large or very small numbers in a compact form. It is defined as a way to represent numbers as a product of a number between 1 and 10 and a power of 10. In the video, scientific notation is used to simplify the expression of large numbers like forty-five thousand (45,000) as 4.5 x 10^4, which is easier to read and manipulate mathematically.

💡Decimal Point

The decimal point is a punctuation mark used to indicate the division of an integer from a fractional part of a number. In scientific notation, moving the decimal point to the left or right changes the scale of the number. For large numbers, the decimal is moved to the left to create a coefficient between 1 and 10, while for small numbers, it is moved to the right. For instance, 45,000 is expressed as 4.5 x 10^4 by moving the decimal four places to the left.

💡Exponent

An exponent is a mathematical notation indicating the number of times a base number is multiplied by itself. In scientific notation, the exponent indicates the power of 10 that the coefficient is multiplied by. Positive exponents are used for large numbers, while negative exponents are used for small numbers. For example, in the number 3.75 x 10^3, the exponent 3 indicates that the decimal point is moved three places to the right.

💡Coefficient

The coefficient in scientific notation is the number that is multiplied by the power of 10. It is a number between 1 and 10 that represents the significant figures of the original number. For example, in the scientific notation of 45,000 (4.5 x 10^4), the coefficient is 4.5.

💡Positive Exponent

A positive exponent in scientific notation is used to represent large numbers. It indicates the number of places the decimal point has been moved to the right. The video explains that a positive exponent is associated with very large numbers, as shown when converting forty-five thousand to 4.5 x 10^4, where the exponent 4 signifies the decimal point's movement four places to the right.

💡Negative Exponent

A negative exponent is used in scientific notation to represent very small numbers. It indicates the number of places the decimal point has been moved to the left. The video illustrates this with the example of 0.0023, which is converted to 2.3 x 10^-3, showing the decimal point moved three places to the right to create a coefficient between 1 and 10.

💡Standard Notation

Standard notation, also known as decimal notation, is the common way of writing numbers without using scientific notation. The video discusses converting numbers from standard notation to scientific notation and vice versa. For example, the number 45,000 in standard notation is converted to 4.5 x 10^4 in scientific notation.

💡Conversion

Conversion in the context of the video refers to the process of changing a number from one form of notation to another. This includes converting from standard notation to scientific notation and from scientific notation back to standard notation. The video provides examples of both types of conversions, such as converting 4.2 x 10^5 to 420,000 in standard notation.

💡Significant Figures

Significant figures are the digits in a number that carry meaning contributing to its precision. In scientific notation, the significant figures are represented by the coefficient. The video emphasizes the importance of keeping track of significant figures when converting to and from scientific notation, ensuring that the precision of the number is maintained.

💡Power of 10

The power of 10 refers to the result of 10 raised to an integer power, which is used in scientific notation to scale the coefficient. The video explains that the exponent in scientific notation tells you how many times to multiply the coefficient by 10. For example, in 3.56 x 10^3, the power of 10 (10^3) means the coefficient 3.56 is multiplied by 1,000.

Highlights

Scientific notation is introduced as a method to represent very large or very small numbers.

Forty-five thousand is used as an example to demonstrate converting to scientific notation.

Moving the decimal point between the first two digits is key to expressing numbers in scientific notation.

A positive exponent indicates a very large number, while a negative exponent indicates a very small number.

Examples are given to convert large numbers like thirty-seven million five hundred eighty thousand into scientific notation.

9.3 billion is converted into scientific notation as an example.

The process of converting small numbers like 0.0023 into scientific notation is explained.

Negative exponents are associated with very small numbers, less than 1.

More examples are provided for converting decimal values to scientific notation with negative exponents.

The method for converting numbers from scientific notation to standard notation is explained.

An example shows how to convert 2.4 times 10 to the 2 into standard notation.

The importance of moving the decimal point to the right for positive exponents is highlighted.

Examples are given for converting numbers with positive exponents to standard notation.

The concept of converting numbers with negative exponents back to standard notation is covered.

A mixed review of converting numbers to scientific notation, including both large and small numbers, is provided.

The direction to move the decimal point is determined by whether the exponent is positive or negative.

Further examples are given to convert numbers with various exponents to standard notation.

The video concludes with a summary of how to convert between scientific and standard notation.

Transcripts

in this lesson i want to give you a

basic introduction into scientific

notation

scientific notation is a useful way to

represent very large numbers or very

small numbers

so

let's say if we have the number forty

five thousand

how can we express this number in

scientific notation

now you want to move the decimal

in between the first two numbers that is

between the four and five

so i'm going to move it four units to

the left one

two three four

so forty five thousand

is equal to four point five times ten to

the four

now it's important to understand that if

this number is positive

it's associated with a very large number

if this is a negative exponent it will

be associated with a very small number a

number between 0 and 1.

so let's work on some more examples

try these two examples

actually maybe more than two let's say

thirty seven fifty

five hundred eighty thousand

seventy two million

and let's say

9.3 billion

go ahead and convert these numbers

into scientific notation

feel free to pause the video

so let's start with this one

i'm gonna put the decimal between a

three and a seven so this is one two

three

since i moved it three spaces to the

left this is going to be 3.75

times 10 to the third power

and that's pretty straightforward

now let's move on to the next one

so i want the decimal point

to be between the 5 and the 8

so i'm going to move it 1 2 3 4

five units to the left

so therefore this is going to be 5.8

times 10 to the 5.

now for the next example

i want it to be between a 7 and a 2.

so this is going to be this is 3

6 and then 7.

so i move this 7 space to the left so

it's going to be 7.2 times 10 to the 7

and that's it for that one now for the

last one

i'm going to start here

3

6

9 units to the left

so this is going to be 9.3

times 10 to the ninth power

and that's a simple way to express very

large numbers using scientific notation

now what about some small numbers

for example 0.0023

we still want the decimal to be between

the two and a three

but this time i'm going to move it to

the right as opposed to the left

so i need to move it three spaces

to the right so therefore this is going

to be 2.3

times 10 to the negative 3.

now keep in mind a negative exponent

will always be associated with very

small numbers

a positive exponent will be associated

with very large numbers

here's some more examples that you can

try

so go ahead and try those examples

so this is three

four units so this is going to be 7.6

times 10 to the negative 4.

so anytime you have these decimal values

it's going to have a negative exponent

associated with the scientific notation

number

so for this one i've got to move it two

units to the right

and so that's going to be four point

nine

times ten to the minus two

now for the third example this is three

six

seven

actually not that far i needed to be

between the first two numbers so three

and six

so this is going to be five point four

one

times ten to the negative six

this is 3 6

9 and then 10 units to the right

so this is going to equal 8.35

times 10 to the negative 10.

and so now you know how to convert

a number in decimal notation or standard

notation

into scientific notation

now let's switch it up a bit let's work

on converting

a number from scientific notation

standard notation

so let's say if we have 2.4 times 10 to

the 2

what is this

equal to

now keep in mind that we said that if we

have a positive exponent it will be

associated with a larger number

so we need to increase the value of 2.4

so should we move the decimal to the

right or to the left

to increase the value we need to move it

to the right so we have the number 2.4

and let's add some zeros to it

so we're going to move it two units to

the right

so therefore this is going to change

to 240

and that's the answer

now if you think about what this

expression means

10 squared that's 10 times 10 which is

100

so this really means 2.4 times 100

which is 240. and so you could see it

that way if you want to as well

let's try this example 3.56

times 10 to the third power

so we need to move the decimal 3 units

to the right

so this is one two three so we need to

add another zero

so therefore this is going to be three

thousand five hundred and sixty

so ten to the third

means that well ten times ten times ten

that's a thousand with three zeros

and three point five six times a

thousand is thirty five sixty

go ahead and try these two examples

four point two seven times ten to the

five

and also

three point nine six

times ten to the seven

so let's start with this one four point

two seven

let's move the decimal point five units

to the right

so that's two three four five

so we need to add three zeros

so this is going to be

four two seven

zero zero zero or four hundred twenty

seven thousand

ten to the fifth is basically a hundred

thousand

so a hundred thousand times uh four

point two seven that's four hundred

twenty seven thousand

now let's try this one so we have three

point nine six

and we need to move the decimal point

seven units to the right so one two

three four five six seven

and so we need to add

five zeros

so the answer is going to be three nine

six zero zero

zero zero zero

so that's 39 million six hundred

thousand

now let's work on some examples with

negative exponents

so a negative exponent is going to be

associated with a small number so this

time

we need to move to the left

so let's move three spaces to the left

one

two

three

so we need to add two zeros so therefore

this is going to be

point zero zero

three seven

let's try this example four point one

six times ten to the negative five

so we need to move five spaces to the

left

one

two three four five

so this is going to be point zero zero

zero

zero four one six

now let's work on a mixed review

go ahead and convert the following

numbers

into scientific notation

let's see if you remember how to do this

so the first one is a small number so

it's going to be associated with a

negative exponent we need to move the

decimal point between the seven and the

three between the first two

non-zero numbers

so since we move it three units to the

right it's going to be 7.35

times ten to the negative three

now let's move on to the next example we

have a large number

and we need to put the decimal between

the first two non-zero numbers between

the three and six

so we're going to move it three four

five spaces to the left

so this is going to be 3.64

times 10 to the positive 5

since we have a large number

now the next example is a small number

and we only need to move it two spaces

to the left

so this is going to be 1.5

times 10 to the minus 2.

and for the last example

we have a large number and we're going

to move it three spaces to the left so

this is going to be

2.8 times 10 to the 3.

so keep that in mind anytime you have

positive exponents it always will be

associated with large numbers

and small numbers that are between 0 and

1

are associated with negative exponents

that will help you to determine which

direction you need to move the decimal

point

so let's try some more examples

1.8 times 10 to the minus 3

four point one times ten to the two

one point two times ten to the negative

five

and two point seven times ten to the

four

so let's convert this to standard

notation

so let's start with the first example

should we move the decimal point to the

left or to the right

this is particularly useful if you need

to convert it from scientific notation

to standard form

since we have a negative exponent we

need a small number

so we got to move to the left

one two three

so we're going to fill these spaces with

zeros so therefore this is going to be

.0018

now for the next example we have a

positive exponent

so that's associated with a large number

therefore we need to move the decimal

point to the right

two spaces

so we're going to add a zero here

therefore that's going to be

410.

now for the next example we need to move

it to the left one

two three four five

so therefore that's going to be

point zero zero

zero zero one two

and for the last one

we need to move it to the right

so one two three four

and so that's going to be 27 000

so hopefully this video gave you a good

introduction into scientific notation

and how to convert back and forth into

standard notation

so thanks again for watching

you