Why her wrist (or keyboard) will thank you for not writing all those zeros

Note: there are many numbers top top this web page that might be simpler to check out if the paper is printed out.

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A ScientificNotationWorksheet+Answers>scientific notation worksheet accompanies this lesson. Be certain to inspect it out!

Why Use scientific Notation?

Scientific Notation was occurred in bespeak to conveniently represent number that room either very big or an extremely small. Right here are two instances of big and small numbers. They space expressed in decimal type instead of clinical notation to aid illustrate the problem:

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The Andromeda Galaxy (the the next one to our Milky way galaxy) contains at the very least 200,000,000,000 stars.

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On the various other hand, the weight of an alpha particle, which is emitted in the radioactive decay of Plutonium-239, is 0.000,000,000,000,000,000,000,000,006,645 kilograms.

As you can see, it can get tedious creating out those numbers repeatedly. So, a system was arisen to assist represent these numbers in a means that was straightforward to read and understand: clinical Notation.

What is scientific Notation?

Using one of the over examples, the number of stars in the Adromeda Galaxy can be composed as:

2.0 x 100,000,000,000

It is that huge number, 100,000,000,000 which reasons the problem. However that is just a multiple of ten. In reality it is ten time itself eleven times:

10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 = 100,000,000,000

A an ext convenient method of composing 100,000,000,000 is 1011. The tiny number to the appropriate of the ten is referred to as the “exponent,” or the “power the ten.” It to represent the variety of zeros the follow the 1.

Though us think of zero as having no value, zeroes deserve to make a number much bigger or smaller. Think about the difference in between 10 dollars and also 100 dollars. Any kind of one that has balanced a checkbook knows the one zero deserve to make a huge difference in the worth of the number. In the very same way, 0.1 (one-tenth) of the united state military budget is much much more than 0.01 (one-hundredth) the the budget. (Though one of two people one is probably much more money than most of united state will ever see in ours checkbooks!)

So we would write 200,000,000,000 in clinical notation as:

2.0 x 1011

This number is review as follows: “two point zero time ten to the eleventh.”

How Does scientific Notation Work?

As we stated above, the exponent refers to the variety of zeros the follow the 1. So:

101 = 10;102 = 100;103 = 1,000, and so on.

Similarly, 100 = 1, because the zero exponent method that no zeros monitor the 1.

Negative index number indicate an unfavorable powers the 10, which are expressed together fractions v 1 in the numerator (on top) and the strength of 10 in the denominator (on the bottom).

So:10-1 = 1/10;10-2 = 1/100;10-3 = 1/1,000, and so on.

This permits us come express other small numbers this way. For example:

2.5 x 10-3 = 2.5 x 1/1,000 = 0.0025

Every number can be expressed in clinical Notation. In our an initial example, 200,000,000,000 should be composed as 2.0 x 1011. In theory, it deserve to be created as 20 x 1010, but by convention the number is normally written together 2.0 x 1011 so the the command number is less than 10, adhered to by as countless decimal areas as necessary.

It is simple to view that all the variations above are just different ways to represent the very same number:

200,000,000,000 =20 x 1010 (20 x 10,000,000,000)2.0 x 1011 (2.0 x 100,000,000,000)0.2 x 1012 (.2 x 1,000,000,000,000)

This illustrates another way to think about Scientific Notation: the exponent will tell you exactly how the decimal suggest moves; a hopeful exponent moves the decimal allude to the right, and also a an adverse one moves it to the left. So because that example:

4.0 x 102 = 400 (2 areas to the ideal of 4);

while4.0 x 10-2 = 0.04 (2 locations to the left of 4).

Note that scientific Notation is also sometimes expressed as E (for exponent), as in 4 E 2 (meaning 4.0 x 10 elevated to 2). Likewise 4 E -2 method 4 times 10 raised to -2, or = 4 x 10-2 = 0.04. This an approach of expression makes it less complicated to form in scientific notation.

Sometimes the benefit of clinical notation is not immediately obvious. A number favor 340 is much much easier to read in decimal form than in scientific notation:

340 or 3.4 x 102?

What about the number 380,000? at the end of 1994 the room of Energy’s (DOE) inventory of high level radioactive waste was approximately 378,400 cubic meters.

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378,400 or 3.784 x 105?

The number 378,400 is also small enough to it is in readable. There are two reasons for express 378,400 in scientific notation rather than decimal form:

Computation: scientific Notation makes adding, subtracting, multiplying and also dividing numbers much simpler.Creating and also reading tables: Let’s look in ~ the table that this number come from:

Historical and also projected cumulative volume that HLW stored in tanks, bins, and also capsules, by website (Table 2.1 in the room of Energy’s combined Data basic Report-1994, DOE/RW-006, Rev. 11)End of Calendar YearVolume, 103 m3
HanfordINELSRSWVDPTotal
1990253.612.0131.71.2398.5
1991256.410.4127.91.7396.5
1992258.711.2126.91.6398.3
1993261.710.5129.32.0403.5
1994238.911.0126.32.2378.4

Look at obelisk 6. The brand for the column has actually units the 103 m3 (1,000 cubic meters). By make the systems 103 m3 rather than m3 the is possible to use the number 378.4 instead of 378,400 in the table. This provides the graph much easier to read.

Addition and also subtraction

The vital to including or subtracting numbers in clinical Notation is come make certain the exponents space the same. Because that example,

(2.0 x 102) + (3.0 x 103)

can be rewritten as:(0.2 x 103) + (3.0 x 103)

Now girlfriend just include 0.2 + 3 and keep the 103 intact. Her answer is 3.2 x 103, or 3,200. We can check this by convert the numbers an initial to the more familiar form.

So:

2 x 102 + 3.0 x 103 = 200 + 3,000 = 3,200 = 3.2 x 103

Let’s shot a subtraction example.(2.0 x 107) – (6.3 x 105)

The trouble needs to it is in rewritten so the the exponents room the same. So we can write(200 x 105) – (6.3 x 105) = 193.7 x 105,

which in clinical Notation would certainly be written 1.937 x 107.

Let’s check by working it an additional way:2 x 107 – 6.3 x 105 = 20,000,000 – 630,000 = 19,370,000 = 1.937 x 107

Multiplication:

When multiplying numbers expressed in clinical notation, the exponents deserve to simply be included together. This is since the exponent to represent the number of zeros following the one. So:

101 x 102 = 10 x 100 = 1,000 = 103

Checking that us see: 101 x 102 = 101+2 = 103

Similarly 101 x 10-3 = 101-3 = 10-2 = 0.01

Again once we examine we watch that: 10 x 1/1000 = 1/100 = 0.01

Look at one more example:(4.0 x 105) x (3.0 x 10-1).

See more: How Much Is A 1881 Silver Dollar Worth, 1881 Morgan Silver Dollar Value

The 4 and also the 3 are multiplied, giving 12, yet the index number 5 and also -1 are added, so the prize is:12 x 104, or 1.2 x 105

Let’s check:(4 x 105) x (3 x 10-1) = 400,00 x 0.3 = 123,000 = 1.2 x 105.

Interesting note: another way to view that 100 = 1 is as follows:101 x 10-1 = 101-1 = 100

It is also: 10 x 1/10 = 1

So 100 = 1

Division:

Let’s look in ~ a basic example:(6.0 x 108) ÷ (3.0 x 105)

To deal with this problem, very first divide the 6 by the 3, to get 2. The exponent in the denominator is then relocated to the numerator, reversing that sign. (Remember that tiny trick from your old mathematics classes?) therefore we relocate the 105 come the numerator through a an unfavorable exponent, which then looks like this:2 x 108 x 10-5

All that’s left now is to fix this as a multiplication problem, remembering that all you should do for the “108 x 10-5” part is to add the exponents. So the prize is:2.0 x 103 or 2,000

Easy, huh? Well, even Dr. Egghead can’t learn brand-new concepts merely by reading them over. It takes a tiny practice. Happy for you, we’ve placed together a ScientificNotationWorksheet+Answers>worksheet on scientific notation! You’ll be rattling off substantial numbers favor a agree in no time! good luck!