Scientific Notation
When working with very large or very small numbers, scientists, mathematicians, and engineers use scientific notationA convention for writing very large and very small numbers in which a number is expressed as the product of a power of `10` and a number that is greater than or equal to `1` and less than `10` as in `3.2` x `10^4`. to express those quantities. Scientific notation is mathematical shorthand, based on the idea that it is easier to read one exponentThe value that indicates the number of times another value is multiplied by itself in exponential notation. The exponent, also called the power, is written in superscript. In the term `5^3`, `5` is the base and `3` is the exponent. than it is to count many zeros in a number. Very small or very large numbers use less space when written in scientific notation because place values are expressed as a power of `10`. Computation with very small or very large numbers is made easier with scientific notation.
A human red blood cell is very small and is estimated to have a diameter of `0.0065` millimeters. On the other hand, a light year is a very large unit of distance that measures just under `10,000,000,000,000,000` meters. Those figures are both awkward to write, and it would be easy to add or lose a zero or two along the way. But in scientific notation, a red blood cell diameter is written as `6.5 " x "10^-3` millimeters, and a light year as roughly `1" x "10^16` meters. Those figures are easier to use than longhand versions.
Notice that it is the exponent that tells us if the term is a very large or a very small number. If a number is `>=1` in standard decimal notation, the exponent will be `>=0` in scientific notation. In other words, large numbers require positive powers of `10`.
If a number is between `0` and `1` in standard notation, the exponent will be `<0` in scientific notation. Small numbers are described by negative powers of `10`.
Since it’s so useful, let’s look more closely at the details of scientific notation format.
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Scientific Notation Format
The general form for a number in scientific notation is `atext( x )10^n` where `1<=a<10` and `n` is an integer.
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We must pay close attention to these conventions in order to write scientific notation correctly. Let’s look at some examples:
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Number |
Scientific Notation? |
Explanation |
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`1.85text( x )10^-2` |
yes |
`1<=1.85<10`
`-2` is an integer |
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`1.083text( x )10^(1/2)` |
no |
`1/2` is not an integer |
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`0.82text( x )10^14` |
no |
`0.82` is not `>=1` |
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`10text( x )10^3` |
no |
`10` is not `<10` |
Only numbers that follow the proper conventions for all parts of the expression are considered to be in true scientific notation.
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Which number below is written in proper scientific notation format?
A) `4.25text( x )10^0.08`
B) `0.425text( x )10^7`
C) `42.5text( x )10^5`
D) `4.25text( x )10^6`
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Now that we understand the format of scientific notation, let’s compare some numbers expressed in both scientific notation and standard decimal notation in order to understand how to convert from one form to the other. Take a look at the tables below. Pay close attention to the exponent in the scientific notation and the position of the decimal point in the decimal notation.
| Large Numbers | |
| Decimal Notation | Scientific Notation |
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`500.0` |
`5" x "10^2` |
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`80,000.0` |
`8" x "10^4` |
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`43,000,000.0` |
`4.3" x "10^7` |
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`62,500,000,000.0` |
`6.25" x "10^10` |
| Small Numbers | |
| Decimal Notation | Scientific Notation |
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`0.05` |
`5" x "10^-2` |
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`0.0008` |
`8" x "10^-4` |
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`0.00000043` |
`4.3" x "10^-7` |
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`0.000000000625` |
`6.25" x "10^-10` |
Let’s start with large numbers. To write a large number in scientific notation, we first have to move the decimal point to a number between `1` and `10`. Since moving the decimal point changes the value, we have to apply multiplication by the power of `10` that will yield an equivalent value to the original. To figure out the exponent, we just count the number of places we moved the decimal sideways. That number is the exponent for the power of `10`.
Let’s look at an example. To rewrite `180,000` in scientific notation, we first move the decimal point to the left until we have a number greater than or equal to `1` and less than `10`. The decimal point is not written in `180,000`, but if it were, it would be after the last zero. If we start moving the decimal sideways one place at a time, we’ll get to `1.8` after `5` shifts:
`180,000`.
`18,000.0`
`1,800.00`
`180.000`
`18.0000`
`1.80000`
So now we know both the number (`1.8`) and the exponent for the power of `10` multiplier that preserves the original value (`5`). In scientific notation, `180,000` is written as `1.8text( x )10^5`.
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The world population is estimated to be about `6,800,000,000` people. Which answer below correctly expresses this number in scientific notation?
A) `7text( x )10^9`
B) `0.68text( x )10^10`
C) `6.8text( x )10^9`
D) `68text( x )10^8`
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The process for moving between decimal and scientific notation is the same for small numbers (between `0` and `1`), but in this case the decimal moves to the right, and the exponent will be negative. Consider the small number `0.0004`:
`0.0004`
`00.004`
`000.04`
`0000.4`
`00004`.
We moved the decimal point sideways until we got the number `4`, which is between `1` and `10` as required. It took `4` moves, but they were moves that made the number bigger than the original. So we’ll have to multiply by a negative power of `10` to bring the new number back down to the equivalent of the original value. In scientific notation `0.0004` is written as `4.0text( x )10^-4`.
We can also go the other way—numbers written in scientific notation can be translated into decimal notation. For example, a hydrogen atom has a diameter of `5text( x )10^-8` mm. To write this number in decimal notation, we turn that power of `10` back into a series of zeros between the number and the decimal point. Because the exponent is negative, all those zeros go to the left of the number `5`:
`5text( x )10^-8`
`5`.
`0.5`
`0.05`
`0.005`
`0.0005`
`0.00005`
`0.000005`
`0.0000005`
`0.00000005`
For each power of `10`, we move the decimal point one place to the left. Be careful here and don’t get carried away with the zeros—the number of zeros after the decimal point will always be `1` less than the exponent. It takes one power of `10` to shift the decimal point to the left of that first number.
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Rewrite `1.57text( x )10^-10` in decimal notation.
A) `15,700,000,000`
B) `0.000000000157`
C) `0.0000000000157`
D) `157text( x )10^-12`
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Numbers that are written in scientific notation can be multiplied and divided rather simply by taking advantage of a few properties and rules. To multiply numbers in scientific notation, first multiply the numbers that aren’t powers of `10` (the `a` in `atext( x )10^n`). Then multiply the powers of `10` by adding the exponents.
This will produce a new number times a different power of `10`. All we have to do is check to make sure this new value is in scientific notation. If it isn’t, we convert it.
Let’s look at an example:
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Example |
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Problem |
`(3text( x )10^8)(6.8text( x )10^-13)` |
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`(3*6.8)(10^8text( x )10^-13)` |
Regroup using the Commutative and Associative Properties |
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`(20.4)(10^8text( x )10^-13)`
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Multiply the numbers |
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`20.4text( x )10^-5` |
Add the exponents following the rule of exponents |
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`2.04text( x )10^1text( x )10^-5` |
Convert `20.4` into scientific notation |
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`2.04text( x )10^(1+(-5))` |
Add the exponents following the rule of exponents |
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Answer |
`2.04text( x )10^-4`
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In order to divide numbers in scientific notation, we once again apply the properties of numbers and the rules of exponents. We begin by dividing the numbers that aren’t powers of `10` (the `a` in `atext( x )10^n` ). Then we divide the powers of `10` by subtracting the exponents.
This will produce a new number times a different power of `10`. If it isn’t already in scientific notation, we convert it, and then we’re done.
Let’s look at an example:
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Example |
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Problem |
`(2.829text( x )10^-9)/(3.45text( x )10^-3)` |
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`(2.829/3.45)((10^-9)/(10^-3))` |
Regroup using the Associative Property |
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`(0.82)((10^-9)/(10^-3))` |
Divide the numbers |
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`0.82text( x )10^(-9-(-3))`
`0.82text( x )10^-6` |
Subtract the exponents |
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`(8.2text( x )10^-1)text( x )10^-6` |
Convert `0.82` into scientific notation |
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`8.2text( x )10^(-1+(-6))` |
Add the exponents |
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Answer |
`8.2text( x )10^-7` |
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Notice that when we divide exponential terms, we subtract the exponent in the denominator from the exponent in the numerator.
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Evaluate `(4text( x )10^-10)(3text( x )10^5)` and express the result in scientific notation.
A) `1.2text( x )10^-4`
B) `12text( x )10^-5`
C) `7text( x )10^-5`
D) `1.2text( x )10^-50`
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Scientific notation was developed to assist mathematicians, scientists and others when expressing and working with very large and very small numbers. Scientific notation follows a very specific format in which a number is expressed as the product of a number greater than or equal to `1` and less than `10` and a power of `10`. The format is written `atext( x )10^n`, where `1<=a<10` and `n` is an integer.
To multiply numbers in scientific notation, add the exponents. To divide, subtract the exponents.