Why your wrist (or keyboard) will say thanks to you for not creating all those zeros

Note: over there are several numbers top top this web page that might be easier to read if the record is printed out.

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

Why Use clinical Notation?

Scientific Notation was emerged in bespeak to quickly represent numbers that are either very big or very small. Here are two instances of big and little numbers. They are expressed in decimal type instead of clinical notation to assist illustrate the problem:


The Andromeda Galaxy (the closestly one come our Milky method galaxy) includes at the very least 200,000,000,000 stars.


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

As you deserve to see, it can get tedious creating out those number repeatedly. So, a mechanism was emerged to help represent this numbers in a way that was basic 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 created as:

2.0 x 100,000,000,000

It is that huge number, 100,000,000,000 which reasons the problem. However that is simply a multiple of ten. In fact it is ten times 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 much more convenient method of creating 100,000,000,000 is 1011. The little number come the appropriate of the ten is dubbed the “exponent,” or the “power of ten.” It to represent the variety of zeros the follow the 1.

Though we think of zero as having no value, zeroes have the right to make a number much bigger or smaller. Think about the difference in between 10 dollars and also 100 dollars. Any type of one that has well balanced a checkbook knows that one zero deserve to make a large difference in the worth of the number. In the same way, 0.1 (one-tenth) of the us military budget plan is much an ext than 0.01 (one-hundredth) that the budget. (Though one of two people one is probably an ext money than many of united state will ever see in our checkbooks!)

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

2.0 x 1011

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

How Does clinical Notation Work?

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

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

Similarly, 100 = 1, due to the fact that the zero exponent method that no zeros follow the 1.

Negative index number indicate an adverse powers that 10, which are expressed as fractions with 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 also so on.

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

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

Every number can be express in scientific Notation. In our first example, 200,000,000,000 must be created as 2.0 x 1011. In theory, it have the right to be composed as 20 x 1010, yet by convention the number is typically written as 2.0 x 1011 so that the lead number is less than 10, complied with by as plenty of decimal areas as necessary.

It is basic to check out that every the variations above are just various ways to represent the exact 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 method to think around Scientific Notation: the exponent will tell you just how the decimal suggest moves; a positive exponent move the decimal allude to the right, and a an adverse one move it to the left. So because that example:

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

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

Note that clinical Notation is likewise sometimes expressed as E (for exponent), together 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 method of expression makes it much easier to kind in scientific notation.

Sometimes the benefit of clinical notation is not immediately obvious. A number favor 340 is much less complicated to check out in decimal form than in clinical notation:

340 or 3.4 x 102?

What about the number 380,000? in ~ the finish of 1994 the room of Energy’s (DOE) perform of high level radioactive waste was roughly 378,400 cubic meters.


378,400 or 3.784 x 105?

The number 378,400 is also tiny enough to be readable. There are two factors for expressing 378,400 in clinical notation fairly than decimal form:

Computation: clinical Notation provides adding, subtracting, multiplying and dividing numbers much simpler.Creating and reading tables: Let’s look at the table the this number came from:

Historical and also projected accumulation volume that HLW save in tanks, bins, and capsules, by site (Table 2.1 in the department of Energy’s incorporated Data basic Report-1994, DOE/RW-006, Rev. 11)End that Calendar YearVolume, 103 m3

Look at obelisk 6. The brand for the column has units the 103 m3 (1,000 cubic meters). By do the devices 103 m3 quite than m3 the is feasible to use the number 378.4 rather of 378,400 in the table. This renders the graph much easier to read.

Addition and subtraction

The vital to adding or subtracting numbers in scientific Notation is come make certain the exponents space the same. For example,

(2.0 x 102) + (3.0 x 103)

can it is in rewritten as:(0.2 x 103) + (3.0 x 103)

Now girlfriend just add 0.2 + 3 and also keep the 103 intact. Her answer is 3.2 x 103, or 3,200. We can inspect this by convert the numbers first to the much more familiar form.


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

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

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

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

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


When multiplying numbers expressed in scientific notation, the exponents have the right to simply be included together. This is because the exponent to represent the variety of zeros complying with 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 when we inspect we view that: 10 x 1/1000 = 1/100 = 0.01

Look at an additional example:(4.0 x 105) x (3.0 x 10-1).

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The 4 and the 3 are multiplied, offering 12, but the index number 5 and -1 are added, therefore the answer 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 method to see that 100 = 1 is as follows:101 x 10-1 = 101-1 = 100

It is also: 10 x 1/10 = 1

So 100 = 1


Let’s look at 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 acquire 2. The exponent in the denominator is then relocated to the numerator, reversing that is sign. (Remember that little trick from her old math classes?) for this reason we relocate the 105 come the numerator with a an adverse exponent, which climate looks like this:2 x 108 x 10-5

All that’s left currently is to settle this together a multiplication problem, remembering the all you must do for the “108 x 10-5” part is to include the exponents. Therefore the answer is:2.0 x 103 or 2,000

Easy, huh? Well, also Dr. Egghead can’t learn new concepts simply by analysis them over. It takes a small practice. Happy for you, we’ve put together a ScientificNotationWorksheet+Answers>worksheet on scientific notation! You’ll be rattling off substantial numbers choose a pro in no time! great luck!