Mathematics
Scientific Notation
Scientific notation is a way of expressing very large or very small numbers in a more concise and manageable form. It involves writing a number as a product of a decimal and a power of 10. This notation is useful in scientific and mathematical calculations, as it simplifies the representation of extremely large or small values.
Written by Perlego with AI-assistance
Related key terms
1 of 5
11 Key excerpts on "Scientific Notation"
eBook - PDF- Linda Almgren Kime, Judith Clark, Beverly K. Michael(Authors)
- 2018(Publication Date)
- Wiley(Publisher)
The one you choose depends on the context. Scientific Notation A number is in Scientific Notation if it is in the form N ⋅ 10 n where N is called the coefficient and 1 ≤ | N | < 10 n is an integer In the following examples, you learn how to write numbers in Scientific Notation. Later we use Scientific Notation to simplify operations with very large and very small numbers. EXAMPLE 3 Using Scientific Notation for Very Large Numbers The distance to Andromeda, our nearest neighboring galaxy, is 15,000,000,000,000,000,000,000 meters. Express this number in Scientific Notation. Solution The coefficient needs to be a number between 1 and 10. We start by identifying the first nonzero digit and then placing a decimal point right after it to create the coefficient of 1.5. The original number written in Scientific Notation will be of the form 1.5 10 ? ⋅ What power of 10 will convert this expression back to the original number? The original number is larger than 1.5, so the exponent will be positive. If we move the decimal place 22 places to the right, we will get back 1.5,000,000,000,000,000,000,000. This is equivalent to multiplying 1.5 by 10 22 . So, in Scientific Notation, 15,000,000,000,000,000,000, 000 is written as 1.5 ⋅ 10 22 EXAMPLE 4 Using Scientific Notation for Very Small Numbers The radius of a hydrogen atom is 0.000 000 000 052 9 meter. Express this number in Scientific Notation. Solution The coefficient is 5.29. The original number written in Scientific Notation will be of the form 5.29 10 ? ⋅ What power of 10 will convert this expression back to the original number? The original number is smaller than 5.29, so the exponent will be negative. If we move the decimal place 11 places to the 210 CHAPTER 4 The Laws of Exponents and Logarithms: Measuring the Universe left, we will get back 0.000 000 000 05.2 9.
eBook - PDF- Morris Hein, Susan Arena, Cary Willard(Authors)
- 2016(Publication Date)
- Wiley(Publisher)
4.5 × 10 9 CHECK YOUR UNDERSTANDING 2.1 Scientific Notation for Numbers Larger Than 10 2.2 Scientific Notation for Numbers Between 0 and 1 ➥ Scientific Notation is a useful way to write very large numbers, such as the distance between the Earth and the moon, or very small numbers, such as the length of these E. coli bacteria (shown here as a colored scanning electron micrograph × 14,000). ONLINE LEARNING MODULE Writing Numbers in Scientific Notation ➥ NASA Dr. Gopal Murti/Science Source Images 16 CHAPTER 2 • Standards for Measurement ENHANCED EXAMPLE 2.3 Write 0.000123 in Scientific Notation. SOLUTION 0.000123 4 Place the decimal between the 1 and the 2. Since the decimal was moved four places to the right, the power of 10 will be −4 and the number 1.23 is multiplied by 10 −4 . 1.23 × 10 −4 P R A C T I C E 2 . 1 Write the following numbers in Scientific Notation: (a) 1200 (four digits) (c) 0.0468 (b) 6,600,000 (two digits) (d) 0.00003 2.2 Measurement and Uncertainty Explain the significance of uncertainty in measurement in chemistry and how significant figures are used to indicate a measurement’s certainty. To understand chemistry, it is necessary to set up and solve problems. Problem solving requires an understanding of the mathematical operations used to manipulate numbers. Measurements are made in an experiment, and chemists use these data to calculate the extent of the physical and chemical changes occurring in the substances that are being studied. By appropriate calculations, an experiment’s results can be compared with those of other experiments and summarized in ways that are meaningful. A measurement is expressed by a numerical value together with a unit of that mea- surement. For example, 70.0 kilograms=154 pounds unit numerical value Whenever a measurement is made with an instrument such as a thermometer or ruler, an estimate is required.
eBook - ePubScience and Technical Writing
A Manual of Style
- Philip Rubens(Author)
- 2002(Publication Date)
- Routledge(Publisher)
- Place a decimal point to the right of the first nonzero digit in the number,
- Delete all nonsignificant zeros (see 7.38 ), and
- Multiply the resulting number by the power of ten required to make the product equal to the original number.
Thus:247,000,000 becomes 2.47 108 in Scientific Notation, and0.000014647 becomes 1.4647 10–5 .7.29 The exponent (the power of ten) can be negative or positive depending on whether the number being represented is smaller or larger than 1 (the number 1 is equal to ten to the zero power). A factor of 100 is customarily omitted. Various examples of small or large numbers and their equivalents in Scientific Notation include 27.5 2.75 × 10 (not 2.75 × 101 ) 1.375 1.375 (not 1.375 × 100 ) 37,040,000 3.704 × 107 (assuming the trailing zeros are not significant) 2,000,000 2 × 106 (not 2.0 × 106 )(assuming the trailing zeros are not significant) 0.000715 7.15 × 10-4 0.09004 9.004 × 10-2 0.000000000145 1.45 × 10-10 10.07 1.007 × 10 7.30 The factor preceding the power of ten, the “mantissa,” is normally between one and ten, as in the preceding examples. However, there is an alternative convention of representing numbers in Scientific Notation so that the power of ten is a multiple of three. This practice fits the structure of the prefixes used in the International System of Units (see 7.41 ). When this convention is used, the mantissa will be between 1 and 999:149.598 × 106 km (not 1.495 98 108 
eBook - PDF- Michael A. Calter, Paul A. Calter, Paul Wraight, Sarah White(Authors)
- 2016(Publication Date)
- Wiley(Publisher)
20 Chapter 1 ◆ Numerical Computation 1–4 Scientific and Engineering Notation Evaluating Powers of 10 We did some work with powers (exponents) in Section 1–2 and saw, for example, that 2 3 means 2 3 = 2 × 2 × 2 = 8 Here, the power 3 tells us how many 2s are to be multiplied to give the product. For powers of 10, the power tells us how many 10s are to be multiplied to give the product. ◆◆◆ Example 39: (a) 10 2 = 10 × 10 = 100 (b) 10 3 = 10 × 10 × 10 = 1000 Negative powers are calculated by x x 1 a a = - (Eq. 35 in Appendix A). ◆◆◆ Example 40: (a) 10 1 10 1 100 0.001 2 2 = = = - (b) 10 1 10 1 100 000 0.000 01 5 5 = = = - Some powers of 10 are summarized in the following table: TABLE 1-8 Powers of Ten Positive Powers Negative Powers 1 000 000 = 10 6 0.1 = 10 -1 100 000 = 10 5 0.01 = 10 -2 10 000 = 10 4 0.001 = 10 -3 1000 = 10 2 0.0001 = 10 -4 100 = 10 3 0.000 01 = 10 -5 10 = 10 1 0.000 001 = 10 -6 1 = 10 0 0.000 000 1 = 10 -7 Scientific and Engineering Notation When we multiply two large numbers 500 000 × 300 000 on our calculator, we get a display like this: 1.5 × 10 11 500000 × 300000 or 1.5E11 500000 × 300000 This is because our answer (150 000 000 000) is too large to fit on the calculator screen, so the calculator automatically defaults to Scientific Notation. The calculator screen here shows two numbers: a decimal number, 1.5, and an integer, 11. Our answer is equal to the decimal number multiplied by 10 raised to the power of the integer. 1.5 × 10 11 500000 × 300000 Decimal number Power of 10 ◆◆◆ ◆◆◆ 21 Section 1–4 ◆ Scientific and Engineering Notation Scientific Notation refers to a number with an absolute value between 1 and 10 which is multi- plied by a power of 10. 1. There is always only a single nonzero digit to the left of the decimal point. 2. The exponent can be any integer. A good way to remember this is “one digit dot.” Engineering notation is similar to Scientific Notation, but with two differences.
eBook - PDF- Dale Ewen(Author)
- 2018(Publication Date)
- Cengage Learning EMEA(Publisher)
Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 118 CHAPTER 2 ◆ Signed Numbers and Powers of 10 Scientific Notation Scientific Notation is a method that is especially useful for writing very large or very small numbers. To write a number in Scientific Notation, write it as a product of a number between 1 and 10 and a power of 10. Write 226 in Scientific Notation. 226 5 2.26 3 10 2 Remember that 10 2 is a short way of writing 10 3 10 5 100. Note that multiplying 2.26 by 100 gives 226. ◆ Write 52,800 in Scientific Notation. 52,800 5 5.28 3 10,000 5 5.28 3 (10 3 10 3 10 3 10) 5 5.28 3 10 4 ◆ Writing a Decimal Number in Scientific Notation To write a decimal number in Scientific Notation, 1. Reading from left to right, place a decimal point after the first nonzero digit. 2. Place a caret ( ^ ) at the position of the original decimal point. 3. If the decimal point is to the left of the caret, the exponent of the power of 10 is the same as the number of decimal places from the caret to the decimal point. 26,638 5 2.6638. 3 10 d 5 2.6638 3 10 4 4. If the decimal point is to the right of the caret, the exponent of the power of 10 is the same as the negative of the number of places from the caret to the decimal point. 0.00986 5 0.009.86 3 10 2 5 9.86 3 10 2 3 5. If the decimal point is already after the first nonzero digit, the exponent of 10 is zero. 2.15 5 2.15 3 10 0 — d ^ ¡ ^ Write 2738 in Scientific Notation. 2738 5 2.738 3 10 5 2.738 3 10 3 ◆ Write 0.0000003842 in Scientific Notation. 0.0000003842 5 0.0000003.842 3 10 2 5 3.842 3 10 2 7 ◆ Example 1 Scientific Notation 2.6 Example 2 Example 3 — ^ Example 4 4 ^ Copyright 2019 Cengage Learning.
eBook - PDF- Leo J. Malone, Theodore O. Dolter(Authors)
- 2012(Publication Date)
- Wiley(Publisher)
The magnitude of that number is almost impossible to determine without tediously counting zeros. A much more convenient and readable way to express this number is as follows 6.022 10 23 When measurements require the use of many nonsignificant zeros, we express these numbers in a form of exponential notation known as Scientific Notation. In Scientific Notation, a given value is expressed as a number written with one nonzero digit to the left of the decimal point and all other significant digits to the right of it (known as the coefficient). The coefficient of 6.022 shown above indicates a precision of four significant figures. This number is then multiplied by 10 raised to a given power, called the exponent. The exponent indicates the magnitude of the number. Following are some powers of 10 and their equivalent numbers. 10 0 1 10 1 1 10 1 0.1 10 1 10 10 2 1 10 2 0.01 10 2 10 10 100 10 3 1 10 3 0.001 10 3 10 * 10 10 1000 10 4 1 10 4 0.0001 10 4 10 * 10 * 10 10 10,000 etc. etc. 1-3.2 Mathematical Manipulation of Scientific Notation In the following exercises, we will give examples of how numbers are expressed in Scientific Notation and how Scientific Notation is handled in multiplication and divi- sion. These examples can serve as a brief review, but if further practice is needed, see Appendix C for additional discussion on adding, squaring, and taking square roots of numbers in Scientific Notation. Before we consider the examples, we can see how sci- entific notation can remove the ambiguity of numbers such as 12,000, where the zeros may or may not be significant. Notice that by expressing the number in scientific nota- tion, we can make it clear whether one or more of the zeros are actually significant. 1.2 10 4 has two significant figures 1.20 10 4 has three significant figures 1.200 10 4 has four significant figures C C OBJECTIVE FOR SECTION 1-3 Perform arithmetic operations involv- ing Scientific Notation.
- Mark Zegarelli(Author)
- 2017(Publication Date)
- For Dummies(Publisher)
In this chapter, you discover Scientific Notation as a handy alternative way of writing very large numbers and very small decimals. Not surprisingly, Scientific Notation is most commonly used in the sciences, where big numbers and precise decimals show up all the time. On the Count of Zero: Understanding Powers of Ten As you discover in Chapter 2, raising a number to a power multiplies the number in the base (the bottom number) by itself as many times as indicated by the exponent (the top number). For example,. Powers often take a long time to calculate because the numbers grow so quickly. For example, 7 6 may look small, but it equals 117,649. But the easiest powers to calculate are powers with a base of 10 — called, naturally, the powers of ten. You can write every power of ten in two ways: Standard notation: As a number, such as 100,000,000 Exponential notation: As the number 10 raised to a power, such as 10 8 Powers of ten are easy to spot, because in standard notation, every power of ten is simply the digit 1 followed by all 0s. To raise 10 to the power of any number, just write a 1 with that number of 0s after it. For example, 1 with no 0s 1 with one 0 1 with two 0s 1 with three 0s To switch from standard to exponential notation, you simply count the zeros and use that as the exponent on the number 10. You can also raise 10 to the power of a negative number. The result of this operation is always a decimal, with the 0s coming before the 1. For example, 1 with one 0 1 with two 0s 1 with three 0s 1 with four 0s When expressing a negative power of ten in standard form, always count the leading 0 — that is, the 0 to the left of the decimal point. For example, has three 0s, counting the leading 0. Q. Write 10 6 in standard notation. A. 1,000,000. The exponent is 6, so the standard notation is a 1 with six 0s after it. Q. Write 100,000 in exponential notation. A
eBook - PDF- John Peterson, Robert Smith(Authors)
- 2018(Publication Date)
- Cengage Learning EMEA(Publisher)
5 Ï 3 e 1 s ef 2 d d 2 s d d 3 ; d 5 2 10.55, e 5 8.26, f 5 2 7.09 26. Ï 4 s mpt 1 pt 1 19 d t 2 1 2 p 2 7 ; m 5 2, p 5 2 2.93, t 5 2 5.86 6–11 Scientific Notation In scientific applications and certain technical fields, computations with very large and very small numbers are required. The numbers in their regular or standard form are inconvenient to read, write, and use in computations. For example, a coulomb (unit of electrical charge) equals 6,241,960,000,000,000,000 electrical charges. Copper expands 0.00000900 per unit of length per degree Fahrenheit. Scientific Notation simplifies read-ing, writing, and computing with large and small numbers. In Scientific Notation, a number is written as a whole number or decimal between 1 and 10 multiplied by 10 with a suitable exponent. For example, a value of 325,000 is written in Scientific Notation as 3.25 3 10 5 . The effect of multiplying a number by 10 is to shift the position of the decimal point. Changing a number from the standard decimal form to Scientific Notation involves counting the number of decimal places the decimal point must be shifted. Expressing Decimal (Standard Form) Numbers in Scientific Notation A positive or negative number whose absolute value is 10 or greater has a positive expo-nent when expressed in Scientific Notation. Copyright 2019 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. Unit 6 SIGNED NUMBERS 245 EXAMPLES Express the following values in Scientific Notation. 1. 146,000 a. Write the number as a value between 1 and 10: 1.46 b.
eBook - PDF- Morris Hein, Susan Arena, Cary Willard(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
Solution 5283. 3 Place the decimal between the 5 and the 2. Since the decimal was moved three places to the left, the power of 10 will be 3 and the number 5.283 is multiplied by 10 3 . 5.283 × 10 3 Try a similar problem in 2.1 Scientific Notation LEARNING OBJECTIVE: Write decimal numbers in Scientific Notation. Scientists often use numbers that are very large or very small in measurements. For example, the Earth’s age is estimated to be about 4,500,000 (4.5 billion) years. Numbers like these are bulky to write, so to make them more compact scientists use powers of 10. Writing a number as the product of a number between 1 and 10 multiplied by 10 raised to some power is called Scientific Notation (see photos). Scientific Notation is a useful way to write very large numbers, such as the distance between the Earth and the moon, or very small numbers, such as the length of these E. coli bacteria (shown here as a colored scanning electron micrograph × 14,000). pockygallery/123 RF Photo Quest Ltd/Science Photo Library/Getty Images To learn how to write a number in Scientific Notation, let’s consider the number 2468. To write this number in Scientific Notation: 1. Move the decimal point in the original number so that it is located after the first nonzero digit. 2468 2.468 (decimal moves three places to the left) 2. Multiply this new number by 10 raised to the proper exponent (power). The proper expo- nent is equal to the number of places that the decimal point was moved. 2.468 × 10 3 3. The sign on the exponent indicates the direction the decimal was moved. moved right negative exponent moved left positive exponent Examples show you problem-solving techniques in a step-by-step form. Study each one and then try the Practice Exercises. CHECK YOUR UNDERSTANDING 2.1 Scientific Notation for Numbers Larger than 10 2.2 Scientific Notation for Numbers between 0 and 1
eBook - PDFGeneral Chemistry I as a Second Language
Mastering the Fundamental Skills
- David R. Klein(Author)
- 2015(Publication Date)
- Wiley(Publisher)
you are counting people at a baseball game and you find that there are exactly 8500? How do you indicate that this is not an estimate, and that the zeros are significant figures? If we use Scientific Notation, this issue goes away. If we want to say that 8500 is just an estimate, rounded to the nearest hundred, then we would say 8.5 10 3 . Notice that there are only two significant figures here. But if we want to show that 8500 is an exact number and the zeros are significant, then we would say 8.500 10 3 . Notice that this number shows that all four digits are significant. In Scientific Notation, counting significant figures is easy. You just count every digit. If you go back to the rules outlined in the previous section, you will see why. In Scientific Notation, the first digit will always be the first significant digit (by defi- nition), and Scientific Notation always has a decimal place (so you always count all digits until you get to the end of the number). For each problem below, assume that the zeros at the end are not significant, and express the number in Scientific Notation: 1.53. 4890 1.54. 240 1.55. 3700 So far, we have seen how to express a large number in Scientific Notation. Now, we turn our attention to expressing small numbers. Consider the number 0.000043. In order to move the decimal place so that there is one non-zero digit to the left of the decimal, you will have to move the decimal five times to the right. 10 CHAPTER 1 NUMBERS AND UNITS 0.0 0 0 0 4 3 We express this by using a negative exponent: 4.3 10 5 . Let’s see why. First we need to understand what a negative exponent means. A negative exponent is used for small numbers. We already saw that 1000 10 3 . So 1 / 1000 would just be 1 / 10 3 . That is where negative exponents come in to the picture: 1 1 0 3 10 3 So, when we use a negative power of 10, we are just saying that number is a small number. In the example above, 4.3 10 5 is a small number.
eBook - PDF- Paul A. Calter, Michael A. Calter(Authors)
- 2011(Publication Date)
- Wiley(Publisher)
This equation is one of the laws of exponents that we will study later. x a # x b x ab We are using a law of exponents for quotients that we will study in Chapter 2. Section 7 ◆ Scientific Notation and Engineering Notation 37 ◆◆◆ Example 93: Here are some quotients of powers of 10. (a) (b) As with multiplication, we divide the decimal parts and the powers of 10 separately. ◆◆◆ ◆◆◆ Example 94: These examples show how to divide numbers in Scientific Notation. (a) (b) (c) ◆◆◆ ◆◆◆ Example 95: An Application. A truck with a capacity of con- tains a load of gravel which weighs Find the density of the sand by dividing the weight by the volume. Solution: = 116 lb/ft 3 ◆◆◆ Scientific and Engineering Notation on the Calculator Displaying Numbers: We can choose the way a number is displayed on a calcula- tor, either in decimal, scientific, or engineering notation. This choice is usually made from a menu. 1.16 10 2 density weight volume 3.77 10 4 3.24 10 2 3.77 10 4 lb. 3.24 10 2 ft 3 7.82 10 2 (1.97 10 3 ) (2.52 10 4 ) 0.782 10 1 12 10 3 4 10 5 3 10 3 5 3 10 2 8 10 5 4 10 2 8 4 10 5 10 2 2 10 52 2 10 3 10 4 10 2 10 4 (2) 10 2 10 5 10 3 10 5 3 10 2 screen for the TI-83/84, showing the Normal, Sci, and Eng modes. MODE screen for the TI-89 showing the NORMAL, SCIENTIFIC, and ENGINEERING modes. MODE Entering Numbers: You can enter a number in any of these notations, regardless of the mode the calculator is in. However, the entered number will be displayed according to the chosen mode. The power of ten is entered using the enter exponent key, usually marked , , or . EEX EXP EE 38 Chapter 1 ◆ Review of Numerical Computation ◆◆◆ Example 96: The keystrokes to enter the number are shown.
Index pages curate the most relevant extracts from our library of academic textbooks. They’ve been created using an in-house natural language model (NLM), each adding context and meaning to key research topics.
