# Figure 1. A typical Laboratory Thermometer graduated in C.

Size: px
Start display at page:

Transcription

2 Celsius degrees. When this thermometer is placed in an ice bath at 0 C, it reads -1 C and when placed in boiling water at 100 C, it reads 99 C. This thermometer has been roughly calibrated over a 100 temperature range and has been found to be 1 in error. As a correction factor, 1 must be added to all temperature readings in this temperature range. It would be better, however, to check the thermometer at several different temperatures within this range to verify that the error is indeed linear before the uniform application of the correction factor at all temperatures. Figure 2 A meter stick graduated in centimeters. The graduations are sufficiently far apart that it is relatively easy to estimate the length of the object shown above it. The length of the object is estimated to be 26.3±0.1 cm. The doubtful or uncertain digit here is the last one recorded. Figure 3 A meter stick graduated in millimeters. The graduations are close enough together that it is difficult to estimate to the nearest tenth of a millimeter. In reporting the length of the object shown above the meter stick, it is sufficient to report it to the nearest millimeter. Thus the length is reported to be 26.3±0.1 cm. The uncertain digit here is the smallest -graduation on the meter stick, ±0.1 cm (or ±1 mm). When using a meter stick, even one that is finely machined, there is enough unevenness in the object being measured, the meter stick itself, and the difficulty in placing it on the item to be measured to make it pointless to attempt to estimate tenths of a millimeter. Even under ideal conditions, estimation of anything less than one-half a millimeter becomes more of a guess than a measurement. Whether the information from a series of measurements is obtained first-hand or second hand through another source, the number of significant figures must be determined in order to keep all the results meaningful. The rules for writing and identifying significant figures are: 1. All nonzero digits (digits from 1 to 9) are significant. 254 contains three significant figures 4.55 contains three significant figures contains six significant figures 2

3 2. Zero digits that occur between nonzero digits are significant. 202 contains three significant figures In these examples, the zeros contains four significant figures are part of a measurement contains six significant figures 3. Zeros at the beginning of a number (i.e., on the left-hand side) are considered to be placeholders and are not significant contains two significant figures In these examples, the zeros contains three significant figures on the left are placeholders contains three significant figures It is common practice to place a zero in front of the decimal point preceding a decimal fraction. It acts as a placeholder only. 4. Zeros that occur at the end of a number (i.e., on the right-hand side) that include an expressed decimal point are significant. The presence of the decimal point is taken as an indication that the measurement is exact to the places indicated contains five significant figures contains four significant figures In these examples, the zeros contains four significant figures on the right express part of a contains five significant figures. measurement contains four significant figures 5. Zeros that occur at the end of a number (i.e., on the right-hand side) without an expressed decimal point are ambiguous (i.e., we have no information on wether they are significant or not) and are not considered to be significant contains three significant figures In these examples, the zeros may only be 2000 contains one significant figure placeholders. Do not count them unless contains four significant figures a decimal point is present. One way of indicating that some or all the zeros are significant is to write the number in scientific notation form (See Section 6). For example, a number such as 2000 would be written as 2 x 10 3 if it contains one significant figure, and as 2.0 x 10 3 if it contains two significant figures. Problems: Significant Figures State the number of significant figures in each of the following measurements. a) 230 cm ans. a) b) 34.0 ml b) c) g c) d) g d) e) km e) f) ml f)

4 g) m g) h) g h) 2. ROUNDING-OFF NUMBERS When dealing with significant figures, it is often necessary to round-off numbers in order to keep the results of calculations significant. To round-off a number such as to three significant figures means to express it as the nearest three digit number. Since is between 64.8 and 64.9, but closer to 64.8, then the result of the round-off is A number such as is equally close to 64.8 and In this case and in similar cases, the rule to observe is to round off to the nearest even number which is This rule assumes that in a series of numbers which are to be rounded off, there will be approximately the same number of times that you would have to roundoff upward to the nearest even number as you would have to round-off downward. 1. Round off to three significant figures. Answer: is between 75.5 and Since is closer to 75.5, then the answer is Round off to two decimal places. Answer: When expressed to two decimal places, falls between 9.08 and Since it is closer to 9.08, then the answer is Round off to a whole number. Answer: is between 1345 and The number, is closer to 1346, thus, the answer is Round off to three significant figures. Answer: To round off , use the first three significant figures. Therefore, is between and , but it is closer to The zeros must be maintained as placeholders. The answer is Round off to two significant figures. Answer: is between and Since it is equally close to both these numbers, then it will be rounded off to the nearest even number. Round off upward to Problems: Rounding-off numbers Round off each of the following numbers to three significant figures. a) ans. a) b) b) c) c)

5 d) d) e) e) f) f) g) g) h) h) 3. NUMERICAL OPERATIONS WITH SIGNIFICANT FIGURES A. Addition and subtraction When adding (or subtracting) approximate numbers, round off the sum (or difference) to the last column in which each number has a significant figure. 1. Add the following numbers: Answer: First arrange the numbers in a column: The sum is Applying the rule for addition of significant figures, it is observed that the last column in which every one of the four numbers has a significant figure is the tenths column (the first decimal place). Thus the sum must be rounded off to one decimal place. The answer would properly be reported as (6 significant figures) 2. Subtract the following numbers: Answer: Set up the subtraction problem: The difference is The last column in which both of these numbers contain a significant figure is the tenths column. Round off the answer to one decimal place. The difference is properly reported as (4 significant figures)

6 B. Multiplication and division When multiplying (or dividing) approximate numbers, round off the product (or -quotient) so that the final answer contains only as many significant figures as the least approximate number involved in the calculation. 1. Multiply the following numbers: 3.01 x 1.4 x Answer: The product of multiplication is: 3.01 x 1.4 x = The process of multiplication can result in many more digits in the answer than in any of the original numbers in the problem. If we examine each of the initial numbers in the problem, we find 3.01 contains 3 significant figures; 1.4 contains 2 significant figures; and contains 4 significant figures. The number of significant figures in the answer will be determined by the least approximate number, the 1.4 which contains 2 significant figures. The final product is rounded off and reported as 3100 (2 significant figures) 2. Divide the following numbers: / 6.01 Answer: The quotient of this division problem is: / 6.01 = Examining the initial numbers, you will observe that contains 5 significant figures and 6.01 contains 3 significant figures. Thus, 6.01 is the least approximate number and the quotient is rounded off to 3 significant figures. The final quotient will be reported as (3 significant figures) Problems: Numerical operations with significant figures Perform the indicated mathematical operations in each of the following. Round off the answers to the proper number of significant figures. a) g g g b) cm cm c) 75.5 m x 8.66 m x 44 m d) g/ 3.45 g e) 2334 cm x cm x 21.2 cm

7 f) 8.6 ml ml ml g) L/ L h) cm mm 4. EXPONENTS Occasionally, in scientific work, we encounter a product in which the same number is used more than once as a factor, as in the following two examples: 2 x 2 x 2 x 2 or 3 x 3 x 3 x 3 x 3 x 3 This method of writing such numbers is cumbersome and in order to simplify this, we use exponential notation. The two examples given above can be written in exponential notation as: The answers to these examples are: 2 4 where the 4 means that there are four 2 s to be multiplied together. 3 6 where the 6 means that six 3 s are to be multiplied together. 2 4 = 16 and 3 6 = 729 In general form an exponential number is given as: a n = a x a x a x a x a... x a (n times) where the symbol a n is read as the n th power of a or a to the n th. The repeated factor, a, is called the base and the number of factors to be multiplied together, n, is called the exponent. Two exponents have common names: and a 2 is usually read as a squared a 3 is usually read as a cubed The types of exponents that you will be encountering in scientific work will be described in the following sections. In most cases, we will be mainly concerned with exponential powers of ten. A. Positive Exponents If the exponent, n, is a positive integer, then a n denotes the product of n factors where each factor is equal to a. This can be written in the general form: a n = a x a x a x a... x a (n times)

8 4 3 can be written as the product of three fours: 4 3 = 4 x 4 x 4 = can be written as the product of three tens: 10 3 = 10 x 10 x 10 = can be written as the product of six tens: 10 6 = 10 x 10 x 10 x 10 x 10 x 10 = A summary of some positive powers of ten are given below: 10 1 = = = = = = = = = Examining the numbers above, you may observe that there is a relationship between the positive exponent of ten and the number of zeros in the product. This can be summarized as: For powers of ten where the exponent is a positive integer, n, the product can be written as a 1 with n zeros following it. Occasionally, one encounters fractions that are raised to a power. An example of this is: x 2 x 2 8 ( ) = = x 3 x 3 27 B. Zero Exponents If the exponent, n, is zero, then by definition, any number raised to the zero power is equal to unity (i.e., 1) = = = 1 (4 x 10 3 ) 0 = 1

9 C. Negative Exponents If the exponent, n, is a negative quantity, then the expression a -n can be written as shown below: 1 1 a -n = ( ) n = a a n Note that when the number with a negative exponent is rewritten in fractional form, the sign of the exponent becomes positive. 3-2 can be evaluated as shown below: = = = 3 3 x can be rewritten and evaluated as shown below: = = can be written and evaluated as shown below: = = = = x 10 x A summary of some negative powers of ten are given below: 10-1 = = = = = = = = = Examining the numbers above, you may observe that there is a relationship between the negative exponent of ten and the number of zeros in the product. This can be summarized as: For powers of ten where the exponent is a negative integer, -n, the product can be written as a decimal fraction with (n-1) zeros between the decimal point and the 1. Occasionally, one encounters fractions that are raised to a negative power. An example of this is: x 3 x 3 27 ( ) -3 = ( ) 3 = = x 2 x 2 8

10 D. Fractional Exponents m If the exponent is a fraction of the form, where m and n are integers, then an n exponential number such as a m/n is read as take the n th root of a and raise it to the m th power. This can be written in the form: _ a m/n = ( n a ) m 10 1/2 can be rewritten and evaluated as shown below: 10 1/2 = 10 = /3 is evaluated as shown below: _ 1/4 8 2/3 = ( 3 8 ) 2 = (2) 2 = ( ) 1/4 is evaluated: 1 ( ) 81 1/ = = = Problems: Exponents Evaluate the following exponential numbers: a) 3 4 b) 5 3 c) 10 2 d) 10 5 e) f) ( ) 2 4 g) 3-2 h) 2-5 i) 10-2

11 j) 10-5 k) l) ( ) -2 4 m) 64 2/3 o) 4-3/2 5. MATHEMATICAL OPERATIONS WITH EXPONENTIAL NUMBERS This tutorial will address multiplication, division, and powers with exponential numbers. Addition and subtraction with exponential numbers is not commonly encountered in chemistry courses. A. Multiplication To multiply two exponential numbers having the same base, add the exponents: a n x a m = a (n+m) Evaluate 3 2 x x 3 3 = 3 (2+3) = 3 5 = 243 Evaluate 3 2 x x 3-3 = 3 2+(-3) = 3-1 = 3 Note: If one or more of the exponents have a negative sign, addition of exponents is with respect to sign. Evaluate 10 2 x 10 3 x x 10 3 x 10 4 = = 10 9 = Evaluate 10 2 x 10-3 x x 10-3 x 10-4 = 10 2+(-3)+(-4) = 10-5 =

12 B. Division To find the quotient of two exponential numbers having the same base, subtract the exponent of the denominator form the exponent of the numerator: a n a m = a n-m Evaluate 5 6 / /5 3 = = = 5 3 = Evaluate 5 3 / /5-2 = = 5 3-(-2) = = 5 5 = Evaluate 10 8 / /10 12 = = = 10-4 = C. Raising to Powers To raise a exponential number to a power, multiply the exponents: (a n ) m = a n m Evaluate (2 3 ) 2 (2 3 ) 2 = = 2 6 = 64 Evaluate (10 4 ) 3 (10 4 ) 3 = = = Evaluate (10-3 ) 2 (10-3 ) 2 = 10 (-3) 2 = 10-6 = Evaluate (10 1/2 ) 8 (10 1/2 ) 8 = 10 (1/2) 8 = 10 4 =

13 Problems: Mathematical operations with exponential numbers Evaluate each of the following: a) 2 3 x 2 2 b) 4 5 x 4-3 c) 2-5 / 2-3 d) 4 3 /4 5 e) (2-2 ) 2 f) (4 2 ) 5 g) 10 2 x 10 6 h) 10 3 x 10 5 x 10-4 i) 10-3 x 10-2 j) 10 4 /10 7 h) 10 2 /10-5 l) 10-4 /10-7 m) (10 3 ) 2 n) (10 3 ) -4 o) (10-3 ) -2 (10-2 x 10 6 ) p) 10 8

14 6. SCIENTIFIC NOTATION In chemistry, we frequently have to deal with very large or very small numbers. For example, one mole of any substance contains approximately particles of that substance. If we consider a substance such as gold, one atom of gold will weigh grams Numbers such as these are difficult to write and are even more difficult to work with, especially in calculations where the number may have to be used a few times. To simplify working with these numbers, we use what is known as scientific notation. In scientific notation we make the assumption that a number such as can be written as the product of two numbers 2.5 and To further simplify this expression, the number can be written in exponential form. Thus, this number, in scientific notation becomes = 2.5 x 10 7 In proper scientific notation form, the significant figures are written so that the decimal point is located between the first and second significant figures (counting from left to right). The power of ten is the indicator of how many spaces the decimal point had to be moved to place it between the first two significant figures. In the first number, the number of particles in a mole, given above, in order to place the decimal point between the first two significant figures (the 6 and the 0) the decimal point must be moved twenty-three spaces form the right to the left. The number of spaces the decimal point is moved will be expressed as a positive integer. In proper scientific notation form, the number of particles in a mole is 6.02 x particles In the second number, the number of grams in one atom of gold, the decimal point must be moved twenty-two spaces form the left to the right. The number of spaces the decimal point was moved will be expressed as a negative exponent. The weight of one atom of gold, expressed in scientific notation x gram As you can observe, these two numbers are easier to manage in scientific notation. Some more examples of numbers written in scientific notation are: 4500 = 4.5 x 10 3 (decimal point moved 3 spaces to the left) = 3.05 x 10 5 (decimal point moved 5 spaces to the left) = 2.50 x 10-3 (decimal point moved 3 spaces to the right)

15 Remember, if the decimal point was moved from the right to the left in order to place it between the first two significant figures, the exponent will be positive. If the decimal point had to be moved from the left to the right, the exponent will be negative. Another advantage of scientific notation is that only the significant figures are kept, all placeholders are contained in the power of ten. This eliminates zeros which are ambiguous. For example, if the number is rounded off to three significant figures, the result would be Only the zero following the 4 should be significant, the other zeros are place holders. It is not apparent from looking at the number that one of the zeros is significant while the others are not. In scientific notation, this number would be written as 2.40 x 10 4 Thus, the significant zero is included, but the placeholders are not. Problems: Scientific notation 1. Write the following numbers in proper scientific notation form: a) ans. a) b) b) c) c) d) 9800 d) e) 2025 x 10 3 e) f) x 10-2 f) 2. Round off the following numbers as indicated and write the answer in scientific notation form: a) Round off to 3 significant figures ans. a) b) Round off to 4 significant figures b) c) Round off to 3 significant figures c) d) Round off to 3 significant figures d) 3. Express the following exponential numbers in non-exponential form: a) 7.90 x 10 5 ans. a) b) 5.70 x 10-4 b) c) x 10-9 c) d) x 10 8 d) e) 9.09 x 10-3 e)

16 7. NUMERICAL OPERATIONS WITH SCIENTIFIC NOTATION NUMBERS A. Multiplication To multiply two (or more) numbers written in scientific notation form, first rearrange the numbers grouping the non-exponential parts of the numbers together and grouping the powers of ten together. The non-exponential numbers are multiplied together and the exponential numbers are added. Multiply: (3 x ) x (2 x 10 6 ) Answer: (3 x ) x (2 x 10 6 ) = 3 x x 2 x 10 6 = 3 x 2 x x 10 6 (rearrange) = 6 x (multiply) = 6 x Multiply: (4.50 x 10-8 ) x (3.00 x 10 4 ) Answer: (4.50 x 10-8 ) x (3.00 x 10 4 ) = 4.50 x 10-8 x 3.00 x 10 4 = 4.50 x 3.00 x 10-8 x 10 4 = 13.5 x = 13.5 x 10-4 = 1.35 x 10-3 Do not forget to put the final answer in proper scientific notation form (i.e., one number in front of the decimal place.) B. Division To divide two numbers written in scientific notation, first rearrange them to group the non-exponential numbers together and the powers of ten together. The non-exponential numbers are divided and the powers of ten are subtracted. Divide: (3.0 x ) / (2.0 x ) Answer: 3.0 x = x (rearrange) 2.0 x = 1.5 x (divide) = 1.5 x 10-4

17 Divide: (6.60 x 10-3 ) / (2.20 x 10 4 ) Answer: 6.60 x = x 2.20 x = 3.00 x 10-3-(4) = 3.00 x 10-7 C. Raising to a Power To raise a number in scientific notation to a power, separate the non-exponential part from the power of ten, raise the non-exponetial part to the required power, and multiply the exponents. Example: Evaluate (3.0 x 10 2 ) 3 Answer: (3.0 x 10 2 ) 3 = (3.0) 3 x (10 2 ) 3 = 27 x = 27 x 10 6 = 2.7 x 10 7 Problems: Numerical operations with scientific notation numbers Evaluate each of the following and express the answers in proper scientific notation form. a) (4.50 x 10 6 ) x (2.1 x 10-2 ) b) (5.50 x 10-3 ) x (3.50 x 10-4 ) c) (1.5 x 10 3 ) x (6.6 x 10 4 ) d) 9000 / (2.5 x 10 2 ) e) (4.48 x 10-3 ) / (6.60 x 10 2 ) f) (8.5 x 10-5 ) / (3.5 x 10-8 ) g) (2.00 x 10-3 ) 4 h) (3.0 x 10 4 ) 3 i) (1.60 x 10-5 ) x x 150 j) x 4200 x k) ( ) 2/3

18 ( ) 3 x (6000) 2 l) ( ) 4 x (400) 1/2 m) ( ) x ( ) n) 4000 / (2.12 x 10-3 ) 0

### Session 29 Scientific Notation and Laws of Exponents. If you have ever taken a Chemistry class, you may have encountered the following numbers:

Session 9 Scientific Notation and Laws of Exponents If you have ever taken a Chemistry class, you may have encountered the following numbers: There are approximately 60,4,79,00,000,000,000,000 molecules

### AP Physics 1 and 2 Lab Investigations

AP Physics 1 and 2 Lab Investigations Student Guide to Data Analysis New York, NY. College Board, Advanced Placement, Advanced Placement Program, AP, AP Central, and the acorn logo are registered trademarks

### EXERCISE # 1.Metric Measurement & Scientific Notation

EXERCISE # 1.Metric Measurement & Scientific Notation Student Learning Outcomes At the completion of this exercise, students will be able to learn: 1. How to use scientific notation 2. Discuss the importance

### This is a square root. The number under the radical is 9. (An asterisk * means multiply.)

Page of Review of Radical Expressions and Equations Skills involving radicals can be divided into the following groups: Evaluate square roots or higher order roots. Simplify radical expressions. Rationalize

### Chapter 1 Lecture Notes: Science and Measurements

Educational Goals Chapter 1 Lecture Notes: Science and Measurements 1. Explain, compare, and contrast the terms scientific method, hypothesis, and experiment. 2. Compare and contrast scientific theory

### Exponents, Radicals, and Scientific Notation

General Exponent Rules: Exponents, Radicals, and Scientific Notation x m x n = x m+n Example 1: x 5 x = x 5+ = x 7 (x m ) n = x mn Example : (x 5 ) = x 5 = x 10 (x m y n ) p = x mp y np Example : (x) =

### Chapter 1: Chemistry: Measurements and Methods

Chapter 1: Chemistry: Measurements and Methods 1.1 The Discovery Process o Chemistry - The study of matter o Matter - Anything that has mass and occupies space, the stuff that things are made of. This

### 2.2 Scientific Notation: Writing Large and Small Numbers

2.2 Scientific Notation: Writing Large and Small Numbers A number written in scientific notation has two parts. A decimal part: a number that is between 1 and 10. An exponential part: 10 raised to an exponent,

### A.2. Exponents and Radicals. Integer Exponents. What you should learn. Exponential Notation. Why you should learn it. Properties of Exponents

Appendix A. Exponents and Radicals A11 A. Exponents and Radicals What you should learn Use properties of exponents. Use scientific notation to represent real numbers. Use properties of radicals. Simplify

### Rational Exponents. Squaring both sides of the equation yields. and to be consistent, we must have

8.6 Rational Exponents 8.6 OBJECTIVES 1. Define rational exponents 2. Simplify expressions containing rational exponents 3. Use a calculator to estimate the value of an expression containing rational exponents

1 Decimals Adding and Subtracting Decimals are a group of digits, which express numbers or measurements in units, tens, and multiples of 10. The digits for units and multiples of 10 are followed by a decimal

### How do you compare numbers? On a number line, larger numbers are to the right and smaller numbers are to the left.

The verbal answers to all of the following questions should be memorized before completion of pre-algebra. Answers that are not memorized will hinder your ability to succeed in algebra 1. Number Basics

### 0.8 Rational Expressions and Equations

96 Prerequisites 0.8 Rational Expressions and Equations We now turn our attention to rational expressions - that is, algebraic fractions - and equations which contain them. The reader is encouraged to

### 3.1. RATIONAL EXPRESSIONS

3.1. RATIONAL EXPRESSIONS RATIONAL NUMBERS In previous courses you have learned how to operate (do addition, subtraction, multiplication, and division) on rational numbers (fractions). Rational numbers

### MULTIPLICATION AND DIVISION OF REAL NUMBERS In this section we will complete the study of the four basic operations with real numbers.

1.4 Multiplication and (1-25) 25 In this section Multiplication of Real Numbers Division by Zero helpful hint The product of two numbers with like signs is positive, but the product of three numbers with

### Chapter 3 Review Math 1030

Section A.1: Three Ways of Using Percentages Using percentages We can use percentages in three different ways: To express a fraction of something. For example, A total of 10, 000 newspaper employees, 2.6%

### A Short Guide to Significant Figures

A Short Guide to Significant Figures Quick Reference Section Here are the basic rules for significant figures - read the full text of this guide to gain a complete understanding of what these rules really

### Chapter 2 Measurement and Problem Solving

Introductory Chemistry, 3 rd Edition Nivaldo Tro Measurement and Problem Solving Graph of global Temperature rise in 20 th Century. Cover page Opposite page 11. Roy Kennedy Massachusetts Bay Community

### Welcome to Physics 40!

Welcome to Physics 40! Physics for Scientists and Engineers Lab 1: Introduction to Measurement SI Quantities & Units In mechanics, three basic quantities are used Length, Mass, Time Will also use derived

### Quick Reference ebook

This file is distributed FREE OF CHARGE by the publisher Quick Reference Handbooks and the author. Quick Reference ebook Click on Contents or Index in the left panel to locate a topic. The math facts listed

### Fractions and Linear Equations

Fractions and Linear Equations Fraction Operations While you can perform operations on fractions using the calculator, for this worksheet you must perform the operations by hand. You must show all steps

### 5.1 Radical Notation and Rational Exponents

Section 5.1 Radical Notation and Rational Exponents 1 5.1 Radical Notation and Rational Exponents We now review how exponents can be used to describe not only powers (such as 5 2 and 2 3 ), but also roots

### PREPARATION FOR MATH TESTING at CityLab Academy

PREPARATION FOR MATH TESTING at CityLab Academy compiled by Gloria Vachino, M.S. Refresh your math skills with a MATH REVIEW and find out if you are ready for the math entrance test by taking a PRE-TEST

### CHAPTER 4 DIMENSIONAL ANALYSIS

CHAPTER 4 DIMENSIONAL ANALYSIS 1. DIMENSIONAL ANALYSIS Dimensional analysis, which is also known as the factor label method or unit conversion method, is an extremely important tool in the field of chemistry.

### Vocabulary Cards and Word Walls Revised: June 29, 2011

Vocabulary Cards and Word Walls Revised: June 29, 2011 Important Notes for Teachers: The vocabulary cards in this file match the Common Core, the math curriculum adopted by the Utah State Board of Education,

### UNIT (1) MEASUREMENTS IN CHEMISTRY

UNIT (1) MEASUREMENTS IN CHEMISTRY Measurements are part of our daily lives. We measure our weights, driving distances, and gallons of gasoline. As a health professional you might measure blood pressure,

### DIMENSIONAL ANALYSIS #2

DIMENSIONAL ANALYSIS #2 Area is measured in square units, such as square feet or square centimeters. These units can be abbreviated as ft 2 (square feet) and cm 2 (square centimeters). For example, we

### Chapter 7 - Roots, Radicals, and Complex Numbers

Math 233 - Spring 2009 Chapter 7 - Roots, Radicals, and Complex Numbers 7.1 Roots and Radicals 7.1.1 Notation and Terminology In the expression x the is called the radical sign. The expression under the

### Multiplying and Dividing Signed Numbers. Finding the Product of Two Signed Numbers. (a) (3)( 4) ( 4) ( 4) ( 4) 12 (b) (4)( 5) ( 5) ( 5) ( 5) ( 5) 20

SECTION.4 Multiplying and Dividing Signed Numbers.4 OBJECTIVES 1. Multiply signed numbers 2. Use the commutative property of multiplication 3. Use the associative property of multiplication 4. Divide signed

### = 800 kg/m 3 (note that old units cancel out) 4.184 J 1000 g = 4184 J/kg o C

Units and Dimensions Basic properties such as length, mass, time and temperature that can be measured are called dimensions. Any quantity that can be measured has a value and a unit associated with it.

### Fractions to decimals

Worksheet.4 Fractions and Decimals Section Fractions to decimals The most common method of converting fractions to decimals is to use a calculator. A fraction represents a division so is another way of

### Basic numerical skills: FRACTIONS, DECIMALS, PROPORTIONS, RATIOS AND PERCENTAGES

Basic numerical skills: FRACTIONS, DECIMALS, PROPORTIONS, RATIOS AND PERCENTAGES. Introduction (simple) This helpsheet is concerned with the ways that we express quantities that are not whole numbers,

### AP Chemistry A. Allan Chapter 1 Notes - Chemical Foundations

AP Chemistry A. Allan Chapter 1 Notes - Chemical Foundations 1.1 Chemistry: An Overview A. Reaction of hydrogen and oxygen 1. Two molecules of hydrogen react with one molecule of oxygen to form two molecules

### Paramedic Program Pre-Admission Mathematics Test Study Guide

Paramedic Program Pre-Admission Mathematics Test Study Guide 05/13 1 Table of Contents Page 1 Page 2 Page 3 Page 4 Page 5 Page 6 Page 7 Page 8 Page 9 Page 10 Page 11 Page 12 Page 13 Page 14 Page 15 Page

### Welcome to Math 19500 Video Lessons. Stanley Ocken. Department of Mathematics The City College of New York Fall 2013

Welcome to Math 19500 Video Lessons Prof. Department of Mathematics The City College of New York Fall 2013 An important feature of the following Beamer slide presentations is that you, the reader, move

### Negative Exponents and Scientific Notation

3.2 Negative Exponents and Scientific Notation 3.2 OBJECTIVES. Evaluate expressions involving zero or a negative exponent 2. Simplify expressions involving zero or a negative exponent 3. Write a decimal

### The gas can has a capacity of 4.17 gallons and weighs 3.4 pounds.

hundred million\$ ten------ million\$ million\$ 00,000,000 0,000,000,000,000 00,000 0,000,000 00 0 0 0 0 0 0 0 0 0 Session 26 Decimal Fractions Explain the meaning of the values stated in the following sentence.

### MEASUREMENT. Historical records indicate that the first units of length were based on people s hands, feet and arms. The measurements were:

MEASUREMENT Introduction: People created systems of measurement to address practical problems such as finding the distance between two places, finding the length, width or height of a building, finding

### Chapter 1 Chemistry: The Study of Change

Chapter 1 Chemistry: The Study of Change This introductory chapter tells the student why he/she should have interest in studying chemistry. Upon completion of this chapter, the student should be able to:

### Exponents. Exponents tell us how many times to multiply a base number by itself.

Exponents Exponents tell us how many times to multiply a base number by itself. Exponential form: 5 4 exponent base number Expanded form: 5 5 5 5 25 5 5 125 5 625 To use a calculator: put in the base number,

### MATH-0910 Review Concepts (Haugen)

Unit 1 Whole Numbers and Fractions MATH-0910 Review Concepts (Haugen) Exam 1 Sections 1.5, 1.6, 1.7, 1.8, 2.1, 2.2, 2.3, 2.4, and 2.5 Dividing Whole Numbers Equivalent ways of expressing division: a b,

### How to Verify Performance Specifications

How to Verify Performance Specifications VERIFICATION OF PERFORMANCE SPECIFICATIONS In 2003, the Centers for Medicare and Medicaid Services (CMS) updated the CLIA 88 regulations. As a result of the updated

### Exponents. Learning Objectives 4-1

Eponents -1 to - Learning Objectives -1 The product rule for eponents The quotient rule for eponents The power rule for eponents Power rules for products and quotient We can simplify by combining the like

### 3.1. Solving linear equations. Introduction. Prerequisites. Learning Outcomes. Learning Style

Solving linear equations 3.1 Introduction Many problems in engineering reduce to the solution of an equation or a set of equations. An equation is a type of mathematical expression which contains one or

### To Evaluate an Algebraic Expression

1.5 Evaluating Algebraic Expressions 1.5 OBJECTIVES 1. Evaluate algebraic expressions given any signed number value for the variables 2. Use a calculator to evaluate algebraic expressions 3. Find the sum

### MBA Jump Start Program

MBA Jump Start Program Module 2: Mathematics Thomas Gilbert Mathematics Module Online Appendix: Basic Mathematical Concepts 2 1 The Number Spectrum Generally we depict numbers increasing from left to right

### SUNY ECC. ACCUPLACER Preparation Workshop. Algebra Skills

SUNY ECC ACCUPLACER Preparation Workshop Algebra Skills Gail A. Butler Ph.D. Evaluating Algebraic Epressions Substitute the value (#) in place of the letter (variable). Follow order of operations!!! E)

### Mathematics. Steps to Success. and. Top Tips. Year 5

Pownall Green Primary School Mathematics and Year 5 1 Contents Page 1. Multiplication and Division 3 2. Positive and Negative Numbers 4 3. Decimal Notation 4. Reading Decimals 5 5. Fractions Linked to

### SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS

(Section 0.6: Polynomial, Rational, and Algebraic Expressions) 0.6.1 SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS LEARNING OBJECTIVES Be able to identify polynomial, rational, and algebraic

### Following are Summaries from Two Chemistry Education Web Sites Concerning Significant Figure Rules

Following are Summaries from Two Chemistry Education Web Sites Concerning Significant Figure Rules From http://dbhs.wvusd.k12.ca.us/sigfigs/sigfigrules.html There are three rules on determining how many

### CHAPTER 2: MEASUREMENT AND PROBLEM SOLVING

CHAPTER 2: MEASUREMENT AND PROBLEM SOLVING Problems: 1-64, 69-88, 91-120, 123-124 2.1 Measuring Global Temperatures measurement: a number with attached units When scientists collect data, it is important

### Section 1 Tools and Measurement

Section 1 Tools and Measurement Key Concept Scientists must select the appropriate tools to make measurements and collect data, to perform tests, and to analyze data. What You Will Learn Scientists use

### Revision Notes Adult Numeracy Level 2

Revision Notes Adult Numeracy Level 2 Place Value The use of place value from earlier levels applies but is extended to all sizes of numbers. The values of columns are: Millions Hundred thousands Ten thousands

### 5.1 Introduction to Decimals, Place Value, and Rounding

5.1 Introduction to Decimals, Place Value, and Rounding 5.1 OBJECTIVES 1. Identify place value in a decimal fraction 2. Write a decimal in words 3. Write a decimal as a fraction or mixed number 4. Compare

### Numerator Denominator

Fractions A fraction is any part of a group, number or whole. Fractions are always written as Numerator Denominator A unitary fraction is one where the numerator is always 1 e.g 1 1 1 1 1...etc... 2 3

### Answer Key for California State Standards: Algebra I

Algebra I: Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences.

### Solutions of Linear Equations in One Variable

2. Solutions of Linear Equations in One Variable 2. OBJECTIVES. Identify a linear equation 2. Combine like terms to solve an equation We begin this chapter by considering one of the most important tools

### Order of Operations More Essential Practice

Order of Operations More Essential Practice We will be simplifying expressions using the order of operations in this section. Automatic Skill: Order of operations needs to become an automatic skill. Failure

### 1.6 The Order of Operations

1.6 The Order of Operations Contents: Operations Grouping Symbols The Order of Operations Exponents and Negative Numbers Negative Square Roots Square Root of a Negative Number Order of Operations and Negative

### 23. RATIONAL EXPONENTS

23. RATIONAL EXPONENTS renaming radicals rational numbers writing radicals with rational exponents When serious work needs to be done with radicals, they are usually changed to a name that uses exponents,

### Experiment #1, Analyze Data using Excel, Calculator and Graphs.

Physics 182 - Fall 2014 - Experiment #1 1 Experiment #1, Analyze Data using Excel, Calculator and Graphs. 1 Purpose (5 Points, Including Title. Points apply to your lab report.) Before we start measuring

### 1 Introduction The Scientific Method (1 of 20) 1 Introduction Observations and Measurements Qualitative, Quantitative, Inferences (2 of 20)

The Scientific Method (1 of 20) This is an attempt to state how scientists do science. It is necessarily artificial. Here are MY five steps: Make observations the leaves on my plant are turning yellow

### Chapter Test B. Chapter: Measurements and Calculations

Assessment Chapter Test B Chapter: Measurements and Calculations PART I In the space provided, write the letter of the term or phrase that best completes each statement or best answers each question. 1.

### Training Manual. Pre-Employment Math. Version 1.1

Training Manual Pre-Employment Math Version 1.1 Created April 2012 1 Table of Contents Item # Training Topic Page # 1. Operations with Whole Numbers... 3 2. Operations with Decimal Numbers... 4 3. Operations

### YOU CAN COUNT ON NUMBER LINES

Key Idea 2 Number and Numeration: Students use number sense and numeration to develop an understanding of multiple uses of numbers in the real world, the use of numbers to communicate mathematically, and

### 6.4 Normal Distribution

Contents 6.4 Normal Distribution....................... 381 6.4.1 Characteristics of the Normal Distribution....... 381 6.4.2 The Standardized Normal Distribution......... 385 6.4.3 Meaning of Areas under

### CS321. Introduction to Numerical Methods

CS3 Introduction to Numerical Methods Lecture Number Representations and Errors Professor Jun Zhang Department of Computer Science University of Kentucky Lexington, KY 40506-0633 August 7, 05 Number in

### Charlesworth School Year Group Maths Targets

Charlesworth School Year Group Maths Targets Year One Maths Target Sheet Key Statement KS1 Maths Targets (Expected) These skills must be secure to move beyond expected. I can compare, describe and solve

### Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any.

Algebra 2 - Chapter Prerequisites Vocabulary Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. P1 p. 1 1. counting(natural) numbers - {1,2,3,4,...}

### Chapter 2 Measurements in Chemistry. Standard measuring device. Standard scale gram (g)

1 Chapter 2 Measurements in Chemistry Standard measuring device Standard scale gram (g) 2 Reliability of Measurements Accuracy closeness to true value Precision reproducibility Example: 98.6 o F 98.5 o

### Equations, Inequalities & Partial Fractions

Contents Equations, Inequalities & Partial Fractions.1 Solving Linear Equations 2.2 Solving Quadratic Equations 1. Solving Polynomial Equations 1.4 Solving Simultaneous Linear Equations 42.5 Solving Inequalities

### Unit 1 Number Sense. In this unit, students will study repeating decimals, percents, fractions, decimals, and proportions.

Unit 1 Number Sense In this unit, students will study repeating decimals, percents, fractions, decimals, and proportions. BLM Three Types of Percent Problems (p L-34) is a summary BLM for the material

### REVIEW SHEETS INTRODUCTORY PHYSICAL SCIENCE MATH 52

REVIEW SHEETS INTRODUCTORY PHYSICAL SCIENCE MATH 52 A Summary of Concepts Needed to be Successful in Mathematics The following sheets list the key concepts which are taught in the specified math course.

### Chapter 5. Decimals. Use the calculator.

Chapter 5. Decimals 5.1 An Introduction to the Decimals 5.2 Adding and Subtracting Decimals 5.3 Multiplying Decimals 5.4 Dividing Decimals 5.5 Fractions and Decimals 5.6 Square Roots 5.7 Solving Equations

### Negative Integral Exponents. If x is nonzero, the reciprocal of x is written as 1 x. For example, the reciprocal of 23 is written as 2

4 (4-) Chapter 4 Polynomials and Eponents P( r) 0 ( r) dollars. Which law of eponents can be used to simplify the last epression? Simplify it. P( r) 7. CD rollover. Ronnie invested P dollars in a -year

### Integers, I, is a set of numbers that include positive and negative numbers and zero.

Grade 9 Math Unit 3: Rational Numbers Section 3.1: What is a Rational Number? Integers, I, is a set of numbers that include positive and negative numbers and zero. Imagine a number line These numbers are

### CHAPTER 5 Round-off errors

CHAPTER 5 Round-off errors In the two previous chapters we have seen how numbers can be represented in the binary numeral system and how this is the basis for representing numbers in computers. Since any

### This unit will lay the groundwork for later units where the students will extend this knowledge to quadratic and exponential functions.

Algebra I Overview View unit yearlong overview here Many of the concepts presented in Algebra I are progressions of concepts that were introduced in grades 6 through 8. The content presented in this course

### 5.1 Simple and Compound Interest

5.1 Simple and Compound Interest Question 1: What is simple interest? Question 2: What is compound interest? Question 3: What is an effective interest rate? Question 4: What is continuous compound interest?

### Review of Scientific Notation and Significant Figures

II-1 Scientific Notation Review of Scientific Notation and Significant Figures Frequently numbers that occur in physics and other sciences are either very large or very small. For example, the speed of

### Math Review. for the Quantitative Reasoning Measure of the GRE revised General Test

Math Review for the Quantitative Reasoning Measure of the GRE revised General Test www.ets.org Overview This Math Review will familiarize you with the mathematical skills and concepts that are important

### Measurement and Calibration

Adapted from: H. A. Neidig and J. N. Spencer Modular Laboratory Program in Chemistry Thompson Learning;, University of Pittsburgh Chemistry 0110 Laboratory Manual, 1998. Purpose To gain an understanding

### Negative Integer Exponents

7.7 Negative Integer Exponents 7.7 OBJECTIVES. Define the zero exponent 2. Use the definition of a negative exponent to simplify an expression 3. Use the properties of exponents to simplify expressions

### Section 4.1 Rules of Exponents

Section 4.1 Rules of Exponents THE MEANING OF THE EXPONENT The exponent is an abbreviation for repeated multiplication. The repeated number is called a factor. x n means n factors of x. The exponent tells

### Math 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers.

Math 0980 Chapter Objectives Chapter 1: Introduction to Algebra: The Integers. 1. Identify the place value of a digit. 2. Write a number in words or digits. 3. Write positive and negative numbers used

### Decimals and other fractions

Chapter 2 Decimals and other fractions How to deal with the bits and pieces When drugs come from the manufacturer they are in doses to suit most adult patients. However, many of your patients will be very

8. Simplification of Radical Expressions 8. OBJECTIVES 1. Simplify a radical expression by using the product property. Simplify a radical expression by using the quotient property NOTE A precise set of

### Multiplying Fractions

. Multiplying Fractions. OBJECTIVES 1. Multiply two fractions. Multiply two mixed numbers. Simplify before multiplying fractions 4. Estimate products by rounding Multiplication is the easiest of the four

### Solving Rational Equations

Lesson M Lesson : Student Outcomes Students solve rational equations, monitoring for the creation of extraneous solutions. Lesson Notes In the preceding lessons, students learned to add, subtract, multiply,

### Continued Fractions and the Euclidean Algorithm

Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction

### Indices and Surds. The Laws on Indices. 1. Multiplication: Mgr. ubomíra Tomková

Indices and Surds The term indices refers to the power to which a number is raised. Thus x is a number with an index of. People prefer the phrase "x to the power of ". Term surds is not often used, instead

### SIMPLIFYING SQUARE ROOTS

40 (8-8) Chapter 8 Powers and Roots 8. SIMPLIFYING SQUARE ROOTS In this section Using the Product Rule Rationalizing the Denominator Simplified Form of a Square Root In Section 8. you learned to simplify

### SQUARES AND SQUARE ROOTS

1. Squares and Square Roots SQUARES AND SQUARE ROOTS In this lesson, students link the geometric concepts of side length and area of a square to the algebra concepts of squares and square roots of numbers.

### Number Conversions Dr. Sarita Agarwal (Acharya Narendra Dev College,University of Delhi)

Conversions Dr. Sarita Agarwal (Acharya Narendra Dev College,University of Delhi) INTRODUCTION System- A number system defines a set of values to represent quantity. We talk about the number of people

### Section 1.5 Exponents, Square Roots, and the Order of Operations

Section 1.5 Exponents, Square Roots, and the Order of Operations Objectives In this section, you will learn to: To successfully complete this section, you need to understand: Identify perfect squares.

### Student Outcomes. Lesson Notes. Classwork. Discussion (10 minutes)

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 5 8 Student Outcomes Students know the definition of a number raised to a negative exponent. Students simplify and write equivalent expressions that contain