Scientific Notation - Fast Review!
Summary
TLDRThis 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
🔢 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.
🔄 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.
🔁 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.
Mindmap
Keywords
💡Scientific Notation
💡Decimal Point
💡Exponent
💡Coefficient
💡Positive Exponent
💡Negative Exponent
💡Standard Notation
💡Conversion
💡Significant Figures
💡Power of 10
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
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