How do you put scientific notation into a calculator

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How do you put scientific notation into a calculator

Astronomers estimate there are at least 120 sextillion stars in the observable universe. By most accounts, that's a seriously impressive number. One sextillion is written out as a "1" followed by 21 zeros. And when we commit 120 sextillion to paper numerically, it looks like this:120,000,000,000,000,000,000,000But Houston, we have a problem. Long strings of zeros and commas aren't exactly great reading material. Taken in context, this particular sum should make our jaws drop. Just think about its implications: There are more stars in the universe than there are grains of sand in all of Earth's beaches and deserts -- or cells in the human body. Truly, 120 sextillion is a mind-blowing number.Yet comprehension is the key to communication. The plain fact is, one sextillion -- or 1,000,000,000,000,000,000,000 -- isn't a sum most of us think about or interact with every day. So its significance is hard to grasp. Besides, all those lined-up zeros look rather dull, and writing them out by hand or keyboard is a tedious, error-prone chore.Now wouldn't it be great if there was some kind of useful shorthand? Well fortunately, there is. Ladies and gents, let's talk about scientific notation.The Basics of Scientific NotationAs any bank teller should know, 100 is equal to 10 x 10. But instead of writing "10 x 10" out, we could save ourselves some ink and write 102 instead.What's that itty-bitty "2" next to the number 10? We're glad you asked. That's what's called an exponent. And the full-sized number (i.e., 10) to its immediate left is known as the base. The exponent tells you how many times you need to multiply the base by itself.So 102 is just another way of writing 10 x 10. Similarly, 103 means 10 x 10 x 10, which equals 1,000.(By the way, when solving math problems on a computer or calculator, the caret symbol -- or ^ -- is sometimes used to denote exponents. Hence, 102 can also be written as 10^2, but we'll save that conversation for another day.)Scientific notation relies on exponents. Consider the number 2,000. If you wanted to express this sum in scientific notation, you'd write 2.0 x 103.Here's how we made that conversion. When you use scientific notation, what you're really doing is taking a small number (i.e., 2.0) and multiplying it by a specific exponent of 10 (i.e., 103).To get the former, put a decimal point behind the first non-zero digit in the original number. Doing so in this example leaves us with "2.000." Mathematically that can also be written as just "2.0."Obviously, 2.0 is much smaller than the 2,000 we started out with. But a careful count reveals there are three other digits (all zeros) behind the first digit in "2,000." That gives us our exponent value. So, what happens when we multiply 2.0 by 103 -- or 10 x 10 x 10? Lo and behold, we end up with the same sum we started out with: 2,000. Hallelujah.A Sextillion by Another NameAll right, time to have some fun. Through the steps we outlined above, we can use scientific notation to express 4,000 as 4.0 x 103. Likewise, 27,000 becomes 2.7 x 104 and 525,000,000 turns into 5.25 x 108.Ah but dare we convert 120 sextillion, that giant, unwieldy number from our opening sentence? Indeed, we do. Take a good, hard look at 120,000,000,000,000,000,000,000.Altogether, there are 23 digits behind the "1." (Go ahead and count 'em up. We'll wait.) Ergo, in scientific notation, 120,000,000,000,000,000,000,000 is expressed as 1.2 x 1023.But admit it, the latter is way easier on the eyes. Besides, the exponent gives you an immediate sense of how ginormous the total number really is. And it does so in a way that tallying up the zeros never could. Such is the simplifying beauty of scientific notation.Going NegativeYou'll be happy to know this process can be applied to numbers that are smaller than one.Suppose you've only got one-tenth of an apple. Mathematically that means you have 0.10 apples at your disposal. Likewise, if there's only one-millionth of an apple on your lunch tray, you're dealing with a paltry 0.000001 apples. Tough break.There's a way to write this sum down using scientific notation -- and it's not all that different from the technique we've been practicing.Here (again) we'll need to take the existing decimal point and put it to the right of the number's first non-zero digit. Do that and you'll wind up with a plain old "1." In the name of mathematical clarity, we'll write this as "1.0."OK, so in order to get 0.000001, we'll need to multiply our 1.0 by another exponent of 10. But here's the twist: The exponent will be a negative number.Take another gander at 0.000001. See how there are six digits behind the decimal point? That forces us to multiply our 1.0 by 10-6. So in summary, 1.0 x 10-6 is how we express one-millionth, or 0.000001, in scientific notation.By the same token, 6.0 x 10-3 means 0.006. Accordingly, 0.00086 would be written as 8.6 x 10-4. And so on. Happy calculating. Apple's iPhone, with its wide assortment of "apps," has taken the world by storm. Virtually any virtual task you can conceive of, and probably many you couldn't, can be performed through one of the multitude of apps. One basic app is the included calculator. But the calculator can do more than simply add and subtract. Apple allows users to transform the standard calculator into a scientific one, complete with logs, roots, trigonometric formulas and more. Launch the calculator app by touching its icon on the touch screen. Hold the iPhone vertically to perform basic calculator tasks. Rotate the iPhone horizontally, or into landscape mode, to make the calculator automatically transform into a scientific calculator. The Calculator app is one of those iPhone apps that didn't make it's way onto the iPad. It's a shame, because a lot of people, including students, travelers, and home book keepers would get a lot of use out of it. The most popular type of calculator is a scientific calculator -- it can do more than the basic operations (adding, subtracting, multiplying, and dividing), but isn't as different in approach as an RPN (reverse Polish notation) or as advanced as a graphing calculator or CAS (computer algebra system). So which iPad app is the best scientific calculator? Calcbot by Tapbots. One of the great features of Calcbot is that it has large buttons, making it very easy to make entries. In portrait mode, the calculator takes up the whole screen. Along the top is the output screen of the calculator. Under that, you'll find the positive/negative, parentheses, and EE buttons. These are the elements that remain static. The area of the calculator that contains all the numbers and operations is scrollable. Swiping to the left will reveal all the scientific functions. The up-arrow button on the scientific page will replace the trigonometic functions with their inverses, and the logarithm and exponential functions that are base e change to base 2. Visually, these are all the buttons that are light blue in color. As you press buttons on the calculator, their symbols will appear in small, light blue letters at the bottom of the calculator's screen. This is extremely helpful for keeping track of which button you've pressed last. The tracking will reset every time you press enter, when it then gets transferred to the tape. To see the tape, rotate your iPad to landscape. The calculator will become slightly smaller and the tape will slide in from the right. The tape keeps track of all your calculations, including all the buttons that were pressed to get the results. Holding your finger down on an entry pops up a menu to either Use Result, Use Expression, Copy, Send as Email, or Delete. Using the result or expression will automatically add your choice into the calculator. You can also paste into the screen of the calculator or into another app. The Good Large buttons Keeps track of buttons pressed Beautiful UI Landscape orientation reveals the tape Every button emits a subtle sound -- true to Tapbots style (can be disabled) Option to round for currency Universal for iPhone and iPad The Bad Must sacrifice having all the buttons on one screen for the larger buttons The bottom line Calcbot by Tapbots has been my favorite calculator on both the iPhone and iPad for quite some time. It does what I need and does it well. Unfortunately for students, Calcbot likely will not be able to replace a physical calculator because most teachers will not allow the use of iPads during exams. $1.99 - Download Now We may earn a commission for purchases using our links. Learn more. When making a measurement, a scientist can only reach a certain level of precision, limited either by the tools being used or the physical nature of the situation. The most obvious example is measuring distance. Consider what happens when measuring the distance an object moved using a tape measure (in metric units). The tape measure is likely broken down into the smallest units of millimeters. Therefore, there's no way that you can measure with a precision greater than a millimeter. If the object moves 57.215493 millimeters, therefore, we can only tell for sure that it moved 57 millimeters (or 5.7 centimeters or 0.057 meters, depending on the preference in that situation). In general, this level of rounding is fine. Getting the precise movement of a normal-sized object down to a millimeter would be a pretty impressive achievement, actually. Imagine trying to measure the motion of a car to the millimeter, and you'll see that, in general, this isn't necessary. In the cases where such precision is necessary, you'll be using tools that are much more sophisticated than a tape measure. The number of meaningful numbers in a measurement is called the number of significant figures of the number. In the earlier example, the 57-millimeter answer would provide us with 2 significant figures in our measurement. Consider the number 5,200. Unless told otherwise, it is generally the common practice to assume that only the two non-zero digits are significant. In other words, it is assumed that this number was rounded to the nearest hundred. However, if the number is written as 5,200.0, then it would have five significant figures. The decimal point and following zero is only added if the measurement is precise to that level. Similarly, the number 2.30 would have three significant figures, because the zero at the end is an indication that the scientist doing the measurement did so at that level of precision. Some textbooks have also introduced the convention that a decimal point at the end of a whole number indicates significant figures as well. So 800. would have three significant figures while 800 has only one significant figure. Again, this is somewhat variable depending on the textbook. Following are some examples of different numbers of significant figures, to help solidify the concept: One significant figure 4 900 0.00002 Two significant figures 3.7 0.0059 68,000 5.0 Three significant figures 9.64 0.00360 99,900 8.00 900. (in some textbooks) Scientific figures provide some different rules for mathematics than what you are introduced to in your mathematics class. The key in using significant figures is to be sure that you are maintaining the same level of precision throughout the calculation. In mathematics, you keep all of the numbers from your result, while in scientific work you frequently round based on the significant figures involved. When adding or subtracting scientific data, it is only last digit (the digit the furthest to the right) which matters. For example, let's assume that we're adding three different distances: 5.324 + 6.8459834 + 3.1 The first term in the addition problem has four significant figures, the second has eight, and the third has only two. The precision, in this case, is determined by the shortest decimal point. So you will perform your calculation, but instead of 15.2699834 the result will be 15.3, because you will round to the tenths place (the first place after the decimal point), because while two of your measurements are more precise the third can't tell you anything more than the tenths place, so the result of this addition problem can only be that precise as well. Note that your final answer, in this case, has three significant figures, while none of your starting numbers did. This can be very confusing to beginners, and it's important to pay attention to that property of addition and subtraction. When multiplying or dividing scientific data, on the other hand, the number of significant figures do matter. Multiplying significant figures will always result in a solution that has the same significant figures as the smallest significant figures you started with. So, on to the example: 5.638 x 3.1 The first factor has four significant figures and the second factor has two significant figures. Your solution will, therefore, end up with two significant figures. In this case, it will be 17 instead of 17.4778. You perform the calculation then round your solution to the correct number of significant figures. The extra precision in the multiplication won't hurt, you just don't want to give a false level of precision in your final solution. Physics deals with realms of space from the size of less than a proton to the size of the universe. As such, you end up dealing with some very large and very small numbers. Generally, only the first few of these numbers are significant. No one is going to (or able to) measure the width of the universe to the nearest millimeter. This portion of the article deals with manipulating exponential numbers (i.e. 105, 10-8, etc.) and it is assumed that the reader has a grasp of these mathematical concepts. Though the topic can be tricky for many students, it is beyond the scope of this article to address. In order to manipulate these numbers easily, scientists use scientific notation. The significant figures are listed, then multiplied by ten to the necessary power. The speed of light is written as: [blackquote shade=no]2.997925 x 108 m/s There are 7 significant figures and this is much better than writing 299,792,500 m/s. The speed of light is frequently written as 3.00 x 108 m/s, in which case there are only three significant figures. Again, this is a matter of what level of precision is necessary. This notation is very handy for multiplication. You follow the rules described earlier for multiplying the significant numbers, keeping the smallest number of significant figures, and then you multiply the magnitudes, which follows the additive rule of exponents. The following example should help you visualize it: 2.3 x 103 x 3.19 x 104 = 7.3 x 107 The product has only two significant figures and the order of magnitude is 107 because 103 x 104 = 107 Adding scientific notation can be very easy or very tricky, depending on the situation. If the terms are of the same order of magnitude (i.e. 4.3005 x 105 and 13.5 x 105), then you follow the addition rules discussed earlier, keeping the highest place value as your rounding location and keeping the magnitude the same, as in the following example: 4.3005 x 105 + 13.5 x 105 = 17.8 x 105 If the order of magnitude is different, however, you have to work a bit to get the magnitudes the same, as in the following example, where one term is on the magnitude of 105 and the other term is on the magnitude of 106: 4.8 x 105 + 9.2 x 106 = 4.8 x 105 + 92 x 105 = 97 x 105 or 4.8 x 105 + 9.2 x 106 = 0.48 x 106 + 9.2 x 106 = 9.7 x 106 Both of these solutions are the same, resulting in 9,700,000 as the answer. Similarly, very small numbers are frequently written in scientific notation as well, though with a negative exponent on the magnitude instead of the positive exponent. The mass of an electron is: 9.10939 x 10-31 kg This would be a zero, followed by a decimal point, followed by 30 zeroes, then the series of 6 significant figures. No one wants to write that out, so scientific notation is our friend. All the rules outlined above are the same, regardless of whether the exponent is positive or negative. Significant figures are a basic means that scientists use to provide a measure of precision to the numbers they are using. The rounding process involved still introduces a measure of error into the numbers, however, and in very high-level computations there are other statistical methods that get used. For virtually all of the physics that will be done in the high school and college-level classrooms, however, correct use of significant figures will be sufficient to maintain the required level of precision. Significant figures can be a significant stumbling block when first introduced to students because it alters some of the basic mathematical rules that they have been taught for years. With significant figures, 4 x 12 = 50, for example. Similarly, the introduction of scientific notation to students who may not be fully comfortable with exponents or exponential rules can also create problems. Keep in mind that these are tools which everyone who studies science had to learn at some point, and the rules are actually very basic. The trouble is almost entirely remembering which rule is applied at which time. When do I add exponents and when do I subtract them? When do I move the decimal point to the left and when to the right? If you keep practicing these tasks, you'll get better at them until they become second nature. Finally, maintaining proper units can be tricky. Remember that you can't directly add centimeters and meters, for example, but must first convert them into the same scale. This is a common mistake for beginners but, like the rest, it is something that can very easily be overcome by slowing down, being careful, and thinking about what you're doing.

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