4.31 Scientific Notation
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Very Large Numbers
Scientists use powers all the time to write very large and very small numbers more simply and clearly. Example: An astronomer is analyzing our galaxy. The distance across the entire galaxy turns out to be about 590,000,000,000,000,000 (590 thousand trillion) miles – an incredibly huge number. So huge that it would take the astronomer a while just to write out all the zeros. It might take even longer to figure out exactly how big the number is, because the zeros would have to be counted one by one.
Scientists who work regularly with really large number decided to come up with a shorter method for writing these numbers. Since this is a method for scientists, it is call scientific notation. So we would write 590,000,000,000,000,000 as 5.9 x 1017.
Notice that the number contains an exponent (power). Intead of 5 and 9 followed by lots of zeros, it’s 5.9 multiplied by the power 1017. Putting a number into scientific notation doesn’t change its value. Multiplying 5.9 and 1017 will take us right back to 590,000,000,000,000,000. That’s because 1017 is itself a huge number. Actually, its 1 followed by 17 zeros. So 5.9 x 1017 and the original number are equivalent (equal) numbers. Nevertheless, 5.9 x 1017 is much easier to work with, because its much shorter and we don’t have to count all the zeros. We can see how many zeros are in 1017 just by looking at its exponent.
[em]Scientists use powers all the time to write very large and very small numbers more simply and clearly.[/em] Example: An astronomer is analyzing our galaxy. The distance across the entire galaxy turns out to be about 590,000,000,000,000,000 (590 thousand trillion) miles – an incredibly huge number. So huge that it would take the astronomer a while just to write out all the zeros. It might take even longer to figure out exactly how big the number is, because the zeros would have to be counted one by one.
Scientists who work regularly with really large number decided to come up with a shorter method for writing these numbers. Since this is a method for scientists, it is call scientific notation. So we would write 590,000,000,000,000,000 as 5.9 x 10[sup]17[/sup].
Notice that the number contains an exponent (power). Intead of 5 and 9 followed by lots of zeros, it’s 5.9 multiplied by the power 10[sup]17[/sup]. Putting a number into scientific notation doesn’t change its value. Multiplying 5.9 and 10[sup]17[/sup] will take us right back to 590,000,000,000,000,000. That’s because 10[sup]17[/sup] is itself a huge number. Actually, its 1 followed by [sup]17[/sup] zeros. So 5.9 x 10[sup]17[/sup] and the original number are equivalent (equal) numbers. Nevertheless, 5.9 x 10[sup]17[/sup] is much easier to work with, because its much shorter and we don’t have to count all the zeros. [em]We can see how many zeros are in 10[sup]17[/sup] just by looking at its exponent.[/em]
Very Large Numbers
Very Small Numbers
Let’s look at an example of a very small number in scientific notation. Here is the mass of a single hydrogen atom.
0.00000000000000000000000167
This has the same problems as the really large number. All the zeros are hard to write and read. The same number in scientific notation is
1.67 x 10-24
This is obviously much shorter and easier to work with. But notice something. Unlike the really large number (5.9x1017), which had a positive exponent on the 10, this really small number has a negative exponent on the 10. Actually, this makes sense, because 10-24 is the same as 1/1024, which is an incredibly small fraction. So when 1.67 is multiplies by -24, that shrinks 1.67 all the way back down to the value of the original number, 0.000000000000000000000001.67.
Let’s look at an example of a very small number in scientific notation. Here is the mass of a single hydrogen atom.
0.00000000000000000000000167
This has the same problems as the really large number. All the zeros are hard to write and read. [em]The same number in scientific notation is[/em]
1.67 x 10[sup]-24[/sup]
This is obviously much shorter and easier to work with. But notice something. Unlike the really large number (5.9x10[sup]17[/sup]), which had a positive exponent on the 10, [em]this really small number has a negative exponent[/em] on the 10. Actually, this makes sense, because 10[sup]-24[/sup] is the same as 1/10[sup]24[/sup], which is an incredibly small fraction. So when 1.67 is multiplies by [sup]-24[/sup], that shrinks 1.67 all the way back down to the value of the original number, 0.000000000000000000000001.67.
Very Small Numbers
Putting a Number in Scientific Notation
Let’s look at the actual process of taking a number written in the normal way, and converting it into scientific notation.
First, the large number – 590,000,000,000,000,000
First, determine where the decimal point is. Every number has a decimal point even if it is not showing. In a whole number, the decimal point is to the right of the last digit. Our large number becomes
590,000,000,000,000,000.
Next, move the dicemal point to the left until this huge number becomes a number between 1 and 10. That means we need to move the decimal point between the 5 and 9. And, importantly, we’re going to count the number of places we move it. That’s 17 places. Now, instead of a huge number, we’ve got 5.9, which is between 1 and 10. The next step is to multiply 5.9 by a power of 10 that’s just big enough to make the result equal to the original number. And here’s why it was important to count the decimal point moves. The exponent on the 10 should be 17, which is the number of places we moved the decimal point.
5.9 x 1017
Now, the small number – 0.00000000000000000000000167
As before, we need to move the decimal point to a position that will make the new number between 1 and 10. This time, the decimal point has to be moved to the right instead of the left, and it has to go between the 1 and 6. This makes the number equal 1.67, which is between 1 and 10. The 10 needs a negative exponent, though, to shrink 1.67 back down to 0.00000000000000000000000167. Since we moved the decimal point 24 places, the exponent should be negative 24.
1.67 x 10-24
Let’s look at the actual process of taking a number written in the normal way, and converting it into scientific notation.
[tt]First, the large number – 590,000,000,000,000,000[/tt]
First, [em]determine where the decimal point is[/em]. Every number has a decimal point even if it is not showing. In a whole number, the decimal point is to the right of the last digit. Our large number becomes
590,000,000,000,000,000[b].[/b]
Next, [em]move the dicemal point [u]to the left[/u] until this huge number becomes a number between 1 and 10.[/em] That means we need to move the decimal point between the 5 and 9. And, [b]importantly[/b], we’re going to count the number of places we move it. That’s 17 places. Now, instead of a huge number, we’ve got 5.9, which is between 1 and 10. [em]The next step is to multiply 5.9 by a power of 10[/em] that’s just big enough to make the result equal to the original number. And here’s why it was important to count the decimal point moves. [em]The exponent on the 10 should be 17, which is the number of places we moved the decimal point.[/em]
5.9 x 10[sup]17[/sup]
[tt]Now, the small number – 0.00000000000000000000000167[/tt]
As before, [em]we need to move the decimal point to a position that will make the new number between 1 and 10[/em]. This time, the decimal point has to be [em]moved to the right[/em] instead of the left, and it has to go between the 1 and 6. This makes the number equal 1.67, which is between 1 and 10. The 10 needs a negative exponent, though, to shrink 1.67 back down to 0.00000000000000000000000167. [em]Since we moved the decimal point 24 places, the exponent should be [i]negative[/i] 24.[/em]
1.67 x 10[sup]-24[/sup]
Putting a Number in Scientific Notation
Steps for converting a number into Scientific Notation
1. Move the decimal point to a place that gives the number a value somewhere between 1 and 10.
2. Multiply that number by a power of 10, and make the exponent equal the number of places you move the decimal point.
3. If the decimal point was moved to the left, make the exponent positive. If it was moved to the right, make the exponent negative.
Another way to state step 3 is that the exponent is always positive for large numbers and always negative for small numbers.
[tt]1. Move the decimal point to a place that gives the number a value somewhere between 1 and 10.[/tt]
[tt]2. Multiply that number by a power of 10, and make the exponent equal the number of places you move the decimal point.[/tt]
[tt]3. If the decimal point was moved to the left, make the exponent [i]positive[/i]. If it was moved to the right, make the exponent [i]negative[/i].[/tt]
[em]Another way to state step 3 is that the exponent is always positive for large numbers and always negative for small numbers.[/em]
Steps for converting a number into Scientific Notation
Multiplying in Scienctific Notation
Not only are numbers easier to read and write when they are in scientific notation, they are easier to multiply and divide. What if our astronomer needed to multiply the two numbers below?
(3.4x1018)(2.4 x 10[sup}-7)
Imagine how hard this calculation would be if the numbers were written in the normal way, with all the zeros. But since the numbers are in scientific notation, they can be multiplied with no trouble. First, let’s rearrange everything, so the numbers and powers are together.
(3.4 x 2.4 x 1018 x 10-7)
Now we multiply the two number parts.
8.16 x 1018 x 10-7)··········multiplying the numbers
Next, we multiply the powers. And this is where having the numbers in scientific notation becomes a huge time saver. Instead of having to multiply with all those zeros, we can just use the adding exponents shortcut.
8.16 x 1018+(-7)··········multiplying powers
which equals
8.16 x 1011
Since our answer is a really large number, it too should be left in scientific notation. That’s all there is to it. The main point is that to multiply two numbers in scientific notation, we multiply the numbers and powers separately, and on the powers we use the adding exponents shortcut.
[em]Not only are numbers easier to read and write when they are in scientific notation, they are easier to multiply and divide.[/em] What if our astronomer needed to multiply the two numbers below?
(3.4x10[sup]18[/sup])(2.4 x 10[sup}-7[/sup])
Imagine how hard this calculation would be if the numbers were written in the normal way, with all the zeros. But since the numbers are in scientific notation, they can be multiplied with no trouble. First, let’s rearrange everything, so the numbers and powers are together.
(3.4 x 2.4 x 10[sup]18[/sup] x 10[sup]-7[/sup])
Now we multiply the two number parts.
8.16 x 10[sup]18[/sup] x 10[sup]-7[/sup])··········multiplying the numbers
Next, we multiply the powers. And this is where having the numbers in scientific notation becomes a huge time saver. Instead of having to multiply with all those zeros, [em]we can just use the adding exponents shortcut[/em].
8.16 x 10[sup]18+(-7)[/sup]··········multiplying powers
which equals
8.16 x 10[sup]11[/sup]
Since our answer is a really large number, it too should be left in scientific notation. That’s all there is to it. The main point is that [em]to multiply two numbers in scientific notation, we multiply the numbers and powers separately, and on the powers we use the adding exponents shortcut.[/em]
Multiplying in Scienctific Notation
Dividing in Scientific Notation
(1.5x101012) / [(2.5x1013)(2x10-9)]
Before carrying out the division, we need to simplify the divisor by multiplying the numbers first. 2.5 x 2 = 5, which gives us
(1.5x1012) / (5.0x1013x10-9)
New we can multiply the powers by adding their exponents to get 104.
(1.5x1012) / (5.0x104)
With the divisor simplified, we are ready to divide. Once again, we handle the numbers and powers separately. First, let’s divide 1.5 by 5.0 to get 0.3.
[(0.3)x(1012] / (104)
Next, we can divide the powers. And since everything is in scientific notation, we can use the subtracting exponents shortcut.
0.3x1012+(-4) or 0.3x108
We’re not quite done yet, because our answer should itself be in scientific notation. Technically, for a number to be in scientific notation, it has to be a number between 1 and 10 multiplied by a power of 10. In 0.3x108, the number 0.3 is smaller than 1, so we need to make an adjustment. All we have to do is multiply 0.3 by 10 to change it to 3.0. However, to keep the value of the entire number the same, we also need to divide 108 by 10: 108 / 10 = 107. So dividing by 10 just reduces the exponent by 1. After the adjustment, then, we end up with
3.0 x 107
And that’s our final answer. Remember, dividing numbers in scientific notation is almost exactly like multiplying them. The numbers and powers are divided separately, and we can use the subtracting exponents shortcut to divide the powers.
(1.5x10[sup]10[/sup]12) / [(2.5x10[sup]13[/sup])(2x10[sup]-9[/sup])]
Before carrying out the division, we need to simplify the divisor by multiplying the numbers first. 2.5 x 2 = 5, which gives us
(1.5x10[sup]12[/sup]) / (5.0x10[sup]13[/sup]x10[sup]-9[/sup])
New we can multiply the powers by adding their exponents to get 10[sup]4[/sup].
(1.5x10[sup]12[/sup]) / (5.0x10[sup]4[/sup])
With the divisor simplified, [em]we are ready to divide. Once again, we handle the numbers and powers separately[/em]. First, let’s divide 1.5 by 5.0 to get 0.3.
[(0.3)x(10[sup]12[/sup]] / (10[sup]4[/sup])
Next, we can divide the powers. And since everything is in scientific notation, [em]we can use the subtracting exponents shortcut[/em].
0.3x10[sup]12+(-4)[/sup] or 0.3x10[sup]8[/sup]
We’re not quite done yet, because our answer should itself be in scientific notation. Technically, [em]for a number to be in scientific notation, it has to be a number between 1 and 10 multiplied by a power of 10[/em]. In 0.3x10[sup]8[/sup], the number 0.3 is smaller than 1, so we need to make an adjustment. All we have to do is multiply 0.3 by 10 to change it to 3.0. However, to keep the value of the entire number the same, we also need to divide 10[sup]8[/sup] by 10: 10[sup]8[/sup] / 10 = 10[sup]7[/sup]. So dividing by 10 just reduces the exponent by 1. After the adjustment, then, we end up with
3.0 x 10[sup]7[/sup]
And that’s our final answer. Remember, [em]dividing numbers in scientific notation is almost exactly like multiplying them. The numbers and powers are divided separately, and we can use the subtracting exponents shortcut to divide the powers.[/em]
Dividing in Scientific Notation
Calculators and Scientific Notation
Using a calculator is one way to save time when working with large or small numbers. Putting a number into a calculator in scientific notation turns out to be fairly easy. Let’s say we want to put in the number 3.4x1018. It’s usually best to enter the power first and the number between 1 and 10 second. To enter 1018, we press the base(10 first and then press the exponent button, which has the symbol yx (or xy) on it. After that, we enter the exponent (18), and then we just multiply the ower by 3.4. Here are the steps
<10> <yx> <18> <x> <3.4> <=>
One thing can trip up beginners, though, is the way the calculator actually shows the number. Instead of showing 3.4x1018 completely, it usually shows the number part on the far left and then just the exponent on the fart right of the display. Sometimes the letter “E” appears next to the exponent.
Calculators will automatically display a number in scientific notation if it’s too large or small to fit into the display. For example, if we used a calculator to multiply two large numbers whose product was 3,400,000,000,000,000,000, since the calculator wouldn’t have room for all those zeros, it would automatically display the answer in scientific notation.
Using a calculator is one way to save time when working with large or small numbers. Putting a number into a calculator in scientific notation turns out to be fairly easy. Let’s say we want to put in the number 3.4x10[sup]18[/sup]. It’s usually best to enter the power first and the number between 1 and 10 second. [em]To enter 10[sup]18[/sup], we press the base(10 first and then press the exponent button, which has the symbol [i]y[sup]x[/sup][/i] (or [i]x[sup]y[/sup][/i]) on it. After that, we enter the exponent (18), and then we just multiply the ower by 3.4[/em]. Here are the steps
<10> <y[sup]x[/sup]> <18> <x> <3.4> <=>
One thing can trip up beginners, though, is the way the calculator actually shows the number. Instead of showing 3.4x10[sup]18[/sup] completely, it usually shows the number part on the far left and then just the exponent on the fart right of the display. Sometimes the letter “E” appears next to the exponent.
Calculators will automatically display a number in scientific notation if it’s too large or small to fit into the display. For example, if we used a calculator to multiply two large numbers whose product was 3,400,000,000,000,000,000, since the calculator wouldn’t have room for all those zeros, it would automatically display the answer in scientific notation.
Calculators and Scientific Notation
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