Math: Fundamentals 100 - Tooling U-sme

Transcript

Math: Fundamentals 100 Welcome to the Tooling University. This course is designed to be used in conjunction with the online version of this class. The online version can be found at http://www.toolingu.com. We offer high quality web -based e -learning that focuses on today's industrial manufacturing training needs. We deliver superior training content over the Internet using text, photos, video, audio, and illustrations. Our courses contain "roll -up -your -sleeves" content that offers real -world solutions on subjects such as Metal Cutting, Workholding, Materials, and CNC with much more to follow. Today's businesses face the challenge of maintaining a trained workforce. Companies must locate apprenticeship programs, cover travel and lodging expenses, and disrupt operations to cover training needs. Our web -based training offers low -cost, all -access courses and services to maximize your training initiatives.   Copyright © 2015 Tooling U, LLC. All Rights Reserved. Class Outline   Class Outline Objectives The Purpose of Math Addition and Subtraction Integers Adding and Subtracting Integers Multiplication Division Powers Roots The Order of Operations Grouping Symbols Order of Operations: An Example Using a Calculator Summary   Lesson: 1/14 Objectives l Describe the importance of mathematics for shop employees. l Solve a basic addition problem. l Solve a basic subtraction problem. l List integers in order from least to greatest. l Solve a basic addition problem containing integers. l Solve a basic subtraction problem containing integers. l Solve a basic multiplication problem containing integers. l Solve a basic division problem containing integers. l Solve a basic math problem containing exponents. l Solve a basic math problem containing roots. l List the correct order of mathematical operations. l Solve a basic math problem containing grouping symbols. l Solve a math problem requiring a sequence of different operations. l Identify types of math problems requiring the use of a calculator.      Figure 1. A calculator is a very useful tool for solving complex math problems. Copyright © 2015 Tooling U, LLC. All Rights Reserved.   Figure 2. You must follow specific rules when multiplying positive and negative integers.   Lesson: 1/14 Objectives l Describe the importance of mathematics for shop employees. l Solve a basic addition problem. l Solve a basic subtraction problem. l List integers in order from least to greatest. l Solve a basic addition problem containing integers. l Solve a basic subtraction problem containing integers. l Solve a basic multiplication problem containing integers. l Solve a basic division problem containing integers. l Solve a basic math problem containing exponents. l Solve a basic math problem containing roots. l List the correct order of mathematical operations. l Solve a basic math problem containing grouping symbols. l Solve a math problem requiring a sequence of different operations. l Identify types of math problems requiring the use of a calculator.      Figure 1. A calculator is a very useful tool for solving complex math problems. Figure 2. You must follow specific rules when multiplying positive and negative integers.   Lesson: 2/14 The Purpose of Math Without a doubt, one of the most important subjects you can learn for the shop is mathematics. While some jobs may require very little knowledge of math, the various jobs and routine tasks you perform in a manufacturing shop require you to solve numerous math problems. You use math in the shop every time you determine the number of parts to make, measure the features of a part, or compare a part to its blueprint. For example, Figure 1 shows someone totaling parts on an inventory sheet. The subject of mathematics is extensive. Math addresses a wide range of topics, including geometry, algebra, trigonometry, and complex calculus problems. However, all of these subjects rely on the basic rules for problem solving. This class provides a brief overview of fundamental math     operations such as addition, subtraction, multiplication, and division. You will also learn the correct order for using these operations so that you can solve common math problems. Copyright © 2015 Tooling U, LLC. All Rights Reserved.   Lesson: 2/14 The Purpose of Math Without a doubt, one of the most important subjects you can learn for the shop is mathematics. While some jobs may require very little knowledge of math, the various jobs and routine tasks you perform in a manufacturing shop require you to solve numerous math problems. You use math in the shop every time you determine the number of parts to make, measure the features of a part, or compare a part to its blueprint. For example, Figure 1 shows someone totaling parts on an inventory sheet. The subject of mathematics is extensive. Math addresses a wide range of topics, including geometry, algebra, trigonometry, and complex calculus problems. However, all of these subjects rely on the basic rules for problem solving. This class provides a brief overview of fundamental math     operations such as addition, subtraction, multiplication, and division. You will also learn the correct order for using these operations so that you can solve common math problems. Figure 1. Finding part measurements, working with inventory, and reading blueprints all require a solid grasp of math.   Lesson: 3/14 Addition and Subtraction The most basic units in math are whole numbers. The first whole number is zero, which is followed by 1, 2, 3, 4, 5, and so on. These are the numbers you use for counting. A whole number tells you the number of "units" that are present. Two basic operations are addition and subtraction. You use these operations when you are counting the number of units. For example, consider the two bins in Figure 1. The first bin has 7 parts, and the second bin has 12 parts. To find the total number of parts, you add 7 and 12 (7 + 12 = 19) to find that there are 19 total parts. The order that you add the numbers does not affect the answer. As you can see in Figure 2, the result is always the same. Subtraction is simply the opposite of addition. Imagine that a bin has 19 parts in it. How many parts remain if you take out 8 parts? If you subtract 8 from 19 (19 – 8 = 11), you find that 11 parts remain in the bin. Keep in mind that, when you are subtracting, placing the numbers in the wrong order gives you the wrong answer, as you can see in Figure 3. Simple addition and subtraction problems are most often done by memory or written out by hand. More complicated problems with very large numbers may require you to use a calculator. Figure 1. A simple addition problem provides you with a total.      Figure 2. The order that you add numbers does not affect the result. Copyright © 2015 Tooling U, LLC. All Rights Reserved.   Lesson: 3/14 Addition and Subtraction The most basic units in math are whole numbers. The first whole number is zero, which is followed by 1, 2, 3, 4, 5, and so on. These are the numbers you use for counting. A whole number tells you the number of "units" that are present. Two basic operations are addition and subtraction. You use these operations when you are counting the number of units. For example, consider the two bins in Figure 1. The first bin has 7 parts, and the second bin has 12 parts. To find the total number of parts, you add 7 and 12 (7 + 12 = 19) to find that there are 19 total parts. The order that you add the numbers does not affect the answer. As you can see in Figure 2, the result is always the same. Subtraction is simply the opposite of addition. Imagine that a bin has 19 parts in it. How many parts remain if you take out 8 parts? If you subtract 8 from 19 (19 – 8 = 11), you find that 11 parts remain in the bin. Keep in mind that, when you are subtracting, placing the numbers in the wrong order gives you the wrong answer, as you can see in Figure 3. Simple addition and subtraction problems are most often done by memory or written out by hand. More complicated problems with very large numbers may require you to use a calculator. Figure 1. A simple addition problem provides you with a total.      Figure 2. The order that you add numbers does not affect the result. Figure 3. Subtracting numbers in the wrong order gives you a wrong answer.   Lesson: 4/14 Integers An important number for mathematics is zero. Zero represents the absence of any quantity. If you add a number to zero (0 + 5 = 5), you get the same value. Likewise, if you subtract zero from a number (5 – 0 = 5), you get the same value as well. Every whole number has a matching negative value. These whole numbers and their negative equivalents are called integers. While it may seem strange to have a "negative" number, negative values represent a reduction or absence of a quantity. As you can see in Figure 1, integers can be arranged on a number line. Zero is in the center, with Figure 1. On a number line, negative integers are on the left, and positive integers are on negative numbers to the left and positive numbers to the right. If you select any two numbers on the number line, Tooling the number the rightReserved. is larger. That means that 5 is greater than 2, 2 is greater the right. Copyright © 2015 U, LLC.toAll Rights      than -2, and -2 is greater than -5. Keep in mind that "larger" negative numbers are positioned farther to the left on the number line and are therefore less than "smaller" negative numbers. Lastly, if you add any number and its matching negative value, the result is always zero, as shown   Lesson: 4/14 Integers An important number for mathematics is zero. Zero represents the absence of any quantity. If you add a number to zero (0 + 5 = 5), you get the same value. Likewise, if you subtract zero from a number (5 – 0 = 5), you get the same value as well. Every whole number has a matching negative value. These whole numbers and their negative equivalents are called integers. While it may seem strange to have a "negative" number, negative values represent a reduction or absence of a quantity. As you can see in Figure 1, integers can be arranged on a number line. Zero is in the center, with Figure 1. On a number line, negative integers are on the left, and positive integers are on negative numbers to the left and positive numbers to the right. If you select any two numbers on the number line, the number to the right is larger. That means that 5 is greater than 2, 2 is greater the right.      than -2, and -2 is greater than -5. Keep in mind that "larger" negative numbers are positioned farther to the left on the number line and are therefore less than "smaller" negative numbers. Lastly, if you add any number and its matching negative value, the result is always zero, as shown in Figure 2. Figure 2. Any integer plus its matching negative value equals zero.   Lesson: 5/14 Adding and Subtracting Integers Like whole numbers, all integers can be added and subtracted. However, you must follow specific rules when working with negative numbers. These rules are illustrated in Figures 1 and 2: l l l l If you add two positive or two negative numbers, you add the numbers together and keep the same sign. If you add one positive and one negative number, you subtract the numbers and keep the sign of the "larger" number. Subtracting a positive number is the same as adding its negative value. Subtracting a negative number is the same as adding its positive value. Adding and subtracting positive and negative numbers means you must keep careful track of the proper sign for each number. A negative number always has a minus sign (-) in front of it. Positive numbers have either a plus sign (+) or no sign at all. Keep in mind that, whenever you subtract a number, you are essentially changing its sign and adding it to another number. Figure 1. The positive or negative signs of the numbers you add affect the correct answer.      Copyright © 2015 Tooling U, LLC. All Rights Reserved.   Lesson: 5/14 Adding and Subtracting Integers Like whole numbers, all integers can be added and subtracted. However, you must follow specific rules when working with negative numbers. These rules are illustrated in Figures 1 and 2: l l l l If you add two positive or two negative numbers, you add the numbers together and keep the same sign. If you add one positive and one negative number, you subtract the numbers and keep the sign of the "larger" number. Subtracting a positive number is the same as adding its negative value. Subtracting a negative number is the same as adding its positive value. Adding and subtracting positive and negative numbers means you must keep careful track of the proper sign for each number. A negative number always has a minus sign (-) in front of it. Positive numbers have either a plus sign (+) or no sign at all. Keep in mind that, whenever you subtract a number, you are essentially changing its sign and adding it to another number. Figure 1. The positive or negative signs of the numbers you add affect the correct answer.      Figure 2. Subtracting a positive number is the same as adding its negative value; subtracting a negative number is the same as adding its positive value.   Lesson: 6/14 Multiplication The next common pair of math operations is multiplication and division. Multiplication is a simpler way to express how many times a number is added to itself. For example, consider the series of numbers added in Figure 1. The number 3 is added to itself 5 times, which equals 15. However, a more simple way to express this is to use multiplication. If you multiply 3 and 5 (3 x 5 = 15), you get this same value. The number 5 indicates how many times 3 is added to itself. Different symbols can be used to indicate multiplication. As you can see in Figure 2, an "x", a small dot, and parentheses all indicate that you must multiply numbers. The order that you use to multiply the numbers does not matter because the result is always the same. When multiplying positive and negative numbers, you must follow specific rules, which are shown in Figure 3: Multiplying two positive numbers--the result is positive. Multiplying two negative numbers--the result is positive. l Multiplying a positive and a negative number--the result is negative. Copyright © 2015 Tooling U, LLC. All Rights Reserved. l Multiplying any number by zero--the result is zero. l l Like addition, multiplication is simply another way to think of quantities. Imagine that you have 7 Figure 1. Multiplication is a shortcut for adding the same number to itself multiple times.   Lesson: 6/14 Multiplication The next common pair of math operations is multiplication and division. Multiplication is a simpler way to express how many times a number is added to itself. For example, consider the series of numbers added in Figure 1. The number 3 is added to itself 5 times, which equals 15. However, a more simple way to express this is to use multiplication. If you multiply 3 and 5 (3 x 5 = 15), you get this same value. The number 5 indicates how many times 3 is added to itself. Different symbols can be used to indicate multiplication. As you can see in Figure 2, an "x", a small dot, and parentheses all indicate that you must multiply numbers. The order that you use to multiply the numbers does not matter because the result is always the same. When multiplying positive and negative numbers, you must follow specific rules, which are shown in Figure 3: l l l l Multiplying Multiplying Multiplying Multiplying Figure 1. Multiplication is a shortcut for adding the same number to itself multiple times. two positive numbers--the result is positive. two negative numbers--the result is positive. a positive and a negative number--the result is negative. any number by zero--the result is zero. Like addition, multiplication is simply another way to think of quantities. Imagine that you have 7 bins, each containing 50 parts. To find the total number of parts, you can multiply 7 and 50 (7 x 50     Figure 2. Different symbols indicate = 350) to find that all seven bins contain a total of 350 parts. multiplication is taking place. Figure 3. Multiplying two positive or two negative numbers gives you a positive answer; multiplying one positive and one negative number gives you a negative answer.   Lesson: 7/14 Division Just as subtraction is the reversed operation of addition, division is the reverse operation of multiplication. When you divide a number, you find out how many equal quantities add up to that number. If you know that 5 stacks of 6 parts (5 x 6 = 30) equals 30 parts, you also know that you can divide a big stack of 30 parts into 5 equal stacks containing 6 parts each (30 ÷ 5 = 6). Different symbols can be used to indicate division, as shown in Figure 1. The most common symbol is the ÷ sign. You may also see a forward slash ( / ) or the numbers written as a fraction. Unlike Copyright © 2015 Tooling LLC. All Rights Reserved. multiplication, you mustU, divide numbers in a specific order. Reversing the order gives you a wrong answer.   Lesson: 7/14 Division Just as subtraction is the reversed operation of addition, division is the reverse operation of multiplication. When you divide a number, you find out how many equal quantities add up to that number. If you know that 5 stacks of 6 parts (5 x 6 = 30) equals 30 parts, you also know that you can divide a big stack of 30 parts into 5 equal stacks containing 6 parts each (30 ÷ 5 = 6). Different symbols can be used to indicate division, as shown in Figure 1. The most common symbol is the ÷ sign. You may also see a forward slash ( / ) or the numbers written as a fraction. Unlike multiplication, you must divide numbers in a specific order. Reversing the order gives you a wrong answer. When using positive and negative numbers, division follows the same rules as multiplication. Figure 2 summarizes these rules. Dividing two positive or two negative numbers gives you a positive answer. Dividing a positive and negative number gives you a negative number. The number zero is different. If you divide zero by any number, the result is always zero. However, it is impossible to divide any number by zero because you cannot divide a quantity into "zero" smaller, equal parts. Figure 1. Different symbols indicate that division is taking place.      Figure 2. Dividing positive and negative numbers follows the same rules as multiplication.   Lesson: 8/14 Powers The last pair of common math operations is powers and roots. Power operations are often referred to as exponents. Just as multiplication is a simpler way to show the same number added to itself multiple times, a power indicates how many times a number is multiplied by itself. This power or exponent is shown as a smaller number placed above and to the right of a number. Consider the operation 3 5 shown in Figure 1. The number 5 is the exponent telling you to multiply the number 3 by itself a total of 5 times. This is read as "3 to the fifth power." While an exponent can be any number, the most common powers are 2 and 3. These are read as a number "squared" or a number "cubed," respectively, as shown in Figure 2. As you can see in Figure 3, it is possible to have a number with 0 or 1 as its exponent. Any number "to the zero power" equals 1. Any number "to the first power" equals that same number. Copyright © 2015 Tooling U, LLC. All Rights Reserved. Figure 1. A power is a shortcut for multiplying the same number by itself multiple times.   Lesson: 8/14 Powers The last pair of common math operations is powers and roots. Power operations are often referred to as exponents. Just as multiplication is a simpler way to show the same number added to itself multiple times, a power indicates how many times a number is multiplied by itself. This power or exponent is shown as a smaller number placed above and to the right of a number. Consider the operation 3 5 shown in Figure 1. The number 5 is the exponent telling you to multiply the number 3 by itself a total of 5 times. This is read as "3 to the fifth power." While an exponent can be any number, the most common powers are 2 and 3. These are read as a number "squared" or a number "cubed," respectively, as shown in Figure 2. Figure 1. A power is a shortcut for multiplying the same number by itself multiple times. As you can see in Figure 3, it is possible to have a number with 0 or 1 as its exponent. Any number "to the zero power" equals 1. Any number "to the first power" equals that same number.      Figure 2. The most common powers are numbers "squared" or numbers "cubed." Figure 3. A number "to the first power" equals the same number, while any number "to the zero power" equals 1.   Lesson: 9/14 Roots By now, you have learned that math operations have matching reverse operations. As you can see in Figure 1, subtraction is the reverse of addition, and division is the reverse of multiplication. The reverse operation of a power is a root. A root tells you which unknown number is multiplied by itself a specific number of times to give you the number written inside the root sign. As you can see in Figure 2, the fourth root of 625 equals 5. This is a shorthand way to express that 5 x 5 x 5 x 5 = 625. The symbol for a root is a checkmark sign attached to a horizontal line, with the total value inside the symbol and the root number outside to the left. While a root can be any number, the most common root is a square root, as shown in Figure 3. If a root has no number outside the symbol, you assume thatTooling it is a square root. Copyright © 2015 U, LLC. All Rights Reserved. Keep in mind that a root is the reverse of a power. If you take the "square root" of any number "squared," the result is the original number. This rule is shown in Figure 4. Roots are much more Figure 1. Each math operation has a matching reverse operation.   Lesson: 9/14 Roots By now, you have learned that math operations have matching reverse operations. As you can see in Figure 1, subtraction is the reverse of addition, and division is the reverse of multiplication. The reverse operation of a power is a root. A root tells you which unknown number is multiplied by itself a specific number of times to give you the number written inside the root sign. As you can see in Figure 2, the fourth root of 625 equals 5. This is a shorthand way to express that 5 x 5 x 5 x 5 = 625. The symbol for a root is a checkmark sign attached to a horizontal line, with the total value inside the symbol and the root number outside to the left. While a root can be any number, the most common root is a square root, as shown in Figure 3. If a root has no number outside the symbol, you assume that it is a square root. Figure 1. Each math operation has a matching reverse operation. Keep in mind that a root is the reverse of a power. If you take the "square root" of any number "squared," the result is the original number. This rule is shown in Figure 4. Roots are much more difficult to calculate than the other math operations. In fact, finding a square root almost always requires a calculator. Figure 2. A root finds which unknown number, multiplied by itself a specific number of times, equals the number contained in the root symbol.      Figure 3. A sample of common square roots and their answers. Figure 4. The square root of any number squared is the original number.   Lesson: 10/14 Copyright © 2015 Tooling U, LLC. All Rights Reserved.   Lesson: 10/14 The Order of Operations While some problems only may require you to perform one type of math operation, most problems will involve a sequence of steps. Math problems are not solved simply by working from left to right. Instead, you must calculate each operation in a specific order, which is listed in Figure 1. This is called the order of operations. For any math problem, you must solve each operation in this order: 1. Solve each exponent and root problem, working from left to right. 2. Then, solve each multiplication and division problem, working from left to right. 3. Finally, solve each addition and subtraction problem, working from left to right. Figure 1. Each step in the order of operations focuses on a different pair of math operations. As you can see in Figure 2, the order of operations is very important. Following the wrong order will almost always give you the wrong answer, especially when solving more complex math problems.      Figure 2. Solving a problem with the wrong order of operations almost always gives you a wrong answer.   Lesson: 11/14 Grouping Symbols The order of operations is true for any math problem. However, certain math problems require that you follow these operations "out of sequence." When this is the case, math problems use grouping symbols to indicate the proper order. As you can see in Figure 1, grouping symbols must be solved before moving on to any other operation. There are three types of grouping symbols: parentheses ( ), brackets [ ], and braces { }. As you can see in Figure 2, each pair of grouping symbols fits inside a larger pair. Most math problems use parentheses, with only a few problems using brackets and braces. If there are multiple symbols, parentheses are solved first, followed by brackets, and then braces. In some math problems, you may see only parentheses of different sizes. Grouping symbols change the order of operations. Whenever you see grouping symbols, you must perform the operations inside them before moving on to other areas. If there are multiple grouping symbols, you must work "inside out," solving the innermost operations first before moving onto others. In Figure 3, you see that 3 + 4 is added inside the parentheses before moving on to the other operations. Within each pair of grouping symbols, you must solve math problems using the normal order of operations. Figure 1. If a problem contains grouping symbols, you must solve numbers contained in the symbols first.      Copyright © 2015 Tooling U, LLC. All Rights Reserved. Figure 2. Grouping symbols placed inside other   Lesson: 11/14 Grouping Symbols The order of operations is true for any math problem. However, certain math problems require that you follow these operations "out of sequence." When this is the case, math problems use grouping symbols to indicate the proper order. As you can see in Figure 1, grouping symbols must be solved before moving on to any other operation. There are three types of grouping symbols: parentheses ( ), brackets [ ], and braces { }. As you can see in Figure 2, each pair of grouping symbols fits inside a larger pair. Most math problems use parentheses, with only a few problems using brackets and braces. If there are multiple symbols, parentheses are solved first, followed by brackets, and then braces. In some math problems, you may see only parentheses of different sizes. Grouping symbols change the order of operations. Whenever you see grouping symbols, you must perform the operations inside them before moving on to other areas. If there are multiple grouping symbols, you must work "inside out," solving the innermost operations first before moving onto others. In Figure 3, you see that 3 + 4 is added inside the parentheses before moving on to the other operations. Within each pair of grouping symbols, you must solve math problems using the normal order of operations. Figure 1. If a problem contains grouping symbols, you must solve numbers contained in the symbols first.      Figure 2. Grouping symbols placed inside other grouping symbols may be shown two different ways. Figure 3. The numbers are first solved inside the parentheses, followed by numbers contained in the brackets.   Lesson: 12/14 Order of Operations: An Example The sample problem in Figure 1 contains all the various math operations. Plus, this problem uses grouping symbols to indicate a specific order for solving each operation. How do you find the correct answer? Your first step is to simplify the operations in each pair of grouping symbols. First, add 3 + 2, which leaves you 5 2. Then, solve the second pair of parentheses following the order of operations. By dividing 8 by 4 (8 ÷ 4 = 2), and then adding 2 and 1 (2 + 1 = 3), you remove all the  parentheses from the problem. Copyright © 2015 Tooling U, LLC. All Rights Reserved. The next step is to solve roots and powers. The square root of 9 is 3, and the number 5 squared is 25. This leaves you with a problem that appears as 3 + 25 – 7 x 4 + 2(3).   Lesson: 12/14 Order of Operations: An Example The sample problem in Figure 1 contains all the various math operations. Plus, this problem uses grouping symbols to indicate a specific order for solving each operation. How do you find the correct answer? Your first step is to simplify the operations in each pair of grouping symbols. First, add 3 + 2, which leaves you 5 2. Then, solve the second pair of parentheses following the order of operations. By dividing 8 by 4 (8 ÷ 4 = 2), and then adding 2 and 1 (2 + 1 = 3), you remove all the  parentheses from the problem. The next step is to solve roots and powers. The square root of 9 is 3, and the number 5 squared is 25. This leaves you with a problem that appears as 3 + 25 – 7 x 4 + 2(3). After solving roots and powers, you must solve each multiplication and division problem. You find that 7 x 4 equals 28, and 2(3) equals 6. Remember that parentheses are another symbol indicating      multiplication. This leaves you with 3 + 25 – 28 + 6. Now all that is left are addition and subtraction operations. You must solve these from left to right. If you add 3 + 25 (3 + 25 = 28), then subtract 28 from 28 (28 – 28 = 0), and finally add 0 and 6 (0 + 6 = 6), you find that the entire problem equals 6. No matter how complex the problem, correctly following the order of operations provides you with the correct answer. Figure 1. Numbers highlighted in red are solved with the answer in the next line of the problem.   Lesson: 13/14 Using a Calculator There is more than one way to solve math problems. Most shop employees have memorized the basic addition, subtraction, multiplication, and division operations. Math problems containing larger numbers with multiple digits can often be solved if they are written out by hand. But, as math problems increase in difficulty, you must use a calculator. Many years ago, most shop employees had to solve math problems by hand. But today, a simple calculator helps shorten the time it takes to solve a problem and reduces the chance of making a mistake. For example, you can use the calculator in Figure 1 to find the square root by entering the number, followed by the square root key. Keep in mind that many problems, especially those containing powers or roots, practically demand that you rely on the aid of a calculator. The most important thing to remember is the rules that tell you how numbers relate to one another in the      math problems you encounter. Copyright © 2015 Tooling U, LLC. All Rights Reserved.   Figure 1. To find a square root, enter the number, followed by the square root key.   Lesson: 13/14 Using a Calculator There is more than one way to solve math problems. Most shop employees have memorized the basic addition, subtraction, multiplication, and division operations. Math problems containing larger numbers with multiple digits can often be solved if they are written out by hand. But, as math problems increase in difficulty, you must use a calculator. Many years ago, most shop employees had to solve math problems by hand. But today, a simple calculator helps shorten the time it takes to solve a problem and reduces the chance of making a mistake. For example, you can use the calculator in Figure 1 to find the square root by entering the number, followed by the square root key. Keep in mind that many problems, especially those containing powers or roots, practically demand that you rely on the aid of a calculator. The most important thing to remember is the rules that tell you how numbers relate to one another in the      math problems you encounter. Figure 1. To find a square root, enter the number, followed by the square root key.   Lesson: 14/14 Summary The most basic units in math are whole numbers. The first whole number is zero, followed by 1, 2, 3, 4, 5, and so on. Together, whole numbers and their matching negative values are called integers. The most basic math operations are addition and subtraction. The sign of an integer affects how you add and subtract numbers. Subtracting a positive number is the same as adding its negative value. Subtracting a negative number is the same as adding its positive value. The next pair of math operations is multiplication and division. If you multiply or divide either two positive numbers or two negative numbers, the result is always positive. If you multiply or divide one positive and one negative number, the result is always negative. The last pair of math operations is powers and roots. A power indicates how many times a specific number is multiplied by itself, while a root indicates which unknown number is multiplied by itself a specific number of times to give you the number contained in the root sign. Each pair of math operations includes reverse operations. Subtraction is the reverse of addition, division is the reverse of multiplication, and a root is the reverse of a power. When solving a problem, you must follow a specific order of operations. Solve powers and roots first, followed by multiplication and division, followed by addition and subtraction. If there are grouping symbols, you must solve the operations included in the symbols before moving on to others.      Figure 1. Addition provides you with a total number. Copyright © 2015 Tooling U, LLC. All Rights Reserved. Figure 2. You must follow the order of operations when solving math problems.   Lesson: 14/14 Summary The most basic units in math are whole numbers. The first whole number is zero, followed by 1, 2, 3, 4, 5, and so on. Together, whole numbers and their matching negative values are called integers. The most basic math operations are addition and subtraction. The sign of an integer affects how you add and subtract numbers. Subtracting a positive number is the same as adding its negative value. Subtracting a negative number is the same as adding its positive value. The next pair of math operations is multiplication and division. If you multiply or divide either two positive numbers or two negative numbers, the result is always positive. If you multiply or divide one positive and one negative number, the result is always negative. The last pair of math operations is powers and roots. A power indicates how many times a specific number is multiplied by itself, while a root indicates which unknown number is multiplied by itself a specific number of times to give you the number contained in the root sign. Each pair of math operations includes reverse operations. Subtraction is the reverse of addition, division is the reverse of multiplication, and a root is the reverse of a power. When solving a problem, you must follow a specific order of operations. Solve powers and roots first, followed by multiplication and division, followed by addition and subtraction. If there are grouping symbols, you must solve the operations included in the symbols before moving on to others.      Figure 1. Addition provides you with a total number. Figure 2. You must follow the order of operations when solving math problems.   Class Vocabulary Term Addition Blueprint Division Exponent Fraction Grouping Symbols Definition A mathematical operation that unites two separate quantities into one sum. 2 + 2 = 4 is an example of addition. The instructions and drawings that are used to manufacture a part. A mathematical operation that indicates how many equal quantities add up to a specific number. 8 ÷ 4 = 2 is an  example of division. Another term for a power. The exponent is the smaller number above and to the right of the number being multiplied by itself. A math expression with two numbers placed above and below a division line indicating the number of divisions or portions and the size of each division or portion. Mathematical symbols indicating that operations contained within the symbols must be solved before moving on to other operations. Integer Any number included in either the set of whole numbers or their matching negative values. The numbers -3, -2, -1, 0, 1,All 2,Rights and 3Reserved. are all integers. Copyright © 2015 Tooling U, LLC. Mathematics The study of numbers and quantities and their relationships. Mathematics requires an understanding of the logic   Class Vocabulary Term Definition Addition Blueprint Division Exponent Fraction Grouping Symbols Integer A mathematical operation that unites two separate quantities into one sum. 2 + 2 = 4 is an example of addition. The instructions and drawings that are used to manufacture a part. A mathematical operation that indicates how many equal quantities add up to a specific number. 8 ÷ 4 = 2 is an  example of division. Another term for a power. The exponent is the smaller number above and to the right of the number being multiplied by itself. A math expression with two numbers placed above and below a division line indicating the number of divisions or portions and the size of each division or portion. Mathematical symbols indicating that operations contained within the symbols must be solved before moving on to other operations. Any number included in either the set of whole numbers or their matching negative values. The numbers -3, -2, -1, 0, 1, 2, and 3 are all integers. Mathematics The study of numbers and quantities and their relationships. Mathematics requires an understanding of the logic and rules used to solve numerical problems. Multiplication A mathematical operation that indicates how many times a number is added to itself. 2 x 4 = 8 is an example of multiplication. Order Of Operations Power Root Square Root Subtraction Whole Number Zero The mathematical rules that determine the correct order for solving any sequence of math operations. Powers and roots are solved before multiplication and division, which in turn are solved before addition and subtraction. A mathematical operation indicating how many times a number is multiplied by itself. 2 3 = 8 is an example of a power. A mathematical operation indicating which unknown number, multiplied by itself a specific number of times, equals the number included inside the root sign. The "square root" of 81 equals 9 is an example of a root. The most common root, indicating which unknown number multiplied by itself equals the number inside the square root sign. A mathematical operation that takes away a quantity from a larger whole. 4 – 2 = 2 is an example of subtraction. Any number contained in the sequence 0, 1, 2, 3, and so on. The symbol indicating the absence of a quantity or amount. On a number line, zero indicates the point where negative numbers change into positive numbers. Copyright © 2015 Tooling U, LLC. All Rights Reserved.