# Math: Fundamentals 100

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1 Math: Fundamentas 100 Wecome to the Tooing University. This course is designed to be used in conjunction with the onine version of this cass. The onine version can be found at We offer high quaity web -based e -earning that focuses on today's industria manufacturing training needs. We deiver superior training content over the Internet using text, photos, video, audio, and iustrations. Our courses contain "ro -up -your -seeves" content that offers rea -word soutions on subjects such as Meta Cutting, Workhoding, Materias, and CNC with much more to foow. Today's businesses face the chaenge of maintaining a trained workforce. Companies must ocate apprenticeship programs, cover trave and odging expenses, and disrupt operations to cover training needs. Our web -based training offers ow -cost, a -access courses and services to maximize your training initiatives. Copyright 2015 Tooing U, LLC. A Rights Reserved. Cass Outine

2 Cass Outine Objectives The Purpose of Math Addition and Subtraction Integers Adding and Subtracting Integers Mutipication Division Powers Roots The Order of Operations Grouping Symbos Order of Operations: An Exampe Using a Cacuator Summary Lesson: 1/14 Objectives Describe the importance of mathematics for shop empoyees. Sove a basic addition probem. Sove a basic subtraction probem. List integers in order from east to greatest. Sove a basic addition probem containing integers. Sove a basic subtraction probem containing integers. Sove a basic mutipication probem containing integers. Sove a basic division probem containing integers. Sove a basic math probem containing exponents. Sove a basic math probem containing roots. List the correct order of mathematica operations. Sove a basic math probem containing grouping symbos. Sove a math probem requiring a sequence of different operations. Identify types of math probems requiring the use of a cacuator. Figure 1. A cacuator is a very usefu too for soving compex math probems. Copyright 2015 Tooing U, LLC. A Rights Reserved. Figure 2. You must foow specific rues when mutipying positive and negative integers.

3 Lesson: 1/14 Objectives Describe the importance of mathematics for shop empoyees. Sove a basic addition probem. Sove a basic subtraction probem. List integers in order from east to greatest. Sove a basic addition probem containing integers. Sove a basic subtraction probem containing integers. Sove a basic mutipication probem containing integers. Sove a basic division probem containing integers. Sove a basic math probem containing exponents. Sove a basic math probem containing roots. List the correct order of mathematica operations. Sove a basic math probem containing grouping symbos. Sove a math probem requiring a sequence of different operations. Identify types of math probems requiring the use of a cacuator. Figure 1. A cacuator is a very usefu too for soving compex math probems. Figure 2. You must foow specific rues when mutipying positive and negative integers. Lesson: 2/14 The Purpose of Math Without a doubt, one of the most important subjects you can earn for the shop is mathematics. Whie some jobs may require very itte knowedge of math, the various jobs and routine tasks you perform in a manufacturing shop require you to sove numerous math probems. 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 bueprint. For exampe, Figure 1 shows someone totaing parts on an inventory sheet. The subject of mathematics is extensive. Math addresses a wide range of topics, incuding geometry, agebra, trigonometry, and compex cacuus probems. However, a of these subjects rey on the basic rues for probem soving. This cass provides a brief overview of fundamenta math operations such as addition, subtraction, mutipication, and division. You wi aso earn the correct order for using these operations so that you can sove common math probems. Copyright 2015 Tooing U, LLC. A Rights Reserved.

9 Lesson: 7/14 Division Just as subtraction is the reversed operation of addition, division is the reverse operation of mutipication. When you divide a number, you find out how many equa quantities add up to that number. If you know that 5 stacks of 6 parts (5 x 6 = 30) equas 30 parts, you aso know that you can divide a big stack of 30 parts into 5 equa stacks containing 6 parts each (30 5 = 6). Different symbos can be used to indicate division, as shown in Figure 1. The most common symbo is the sign. You may aso see a forward sash ( / ) or the numbers written as a fraction. Unike mutipication, you must divide numbers in a specific order. Reversing the order gives you a wrong answer. When using positive and negative numbers, division foows the same rues as mutipication. Figure 2 summarizes these rues. 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 resut is aways zero. However, it is impossibe to divide any number by zero because you cannot divide a quantity into "zero" smaer, equa parts. Figure 1. Different symbos indicate that division is taking pace. Figure 2. Dividing positive and negative numbers foows the same rues as mutipication. Lesson: 8/14 Powers The ast pair of common math operations is powers and roots. Power operations are often referred to as exponents. Just as mutipication is a simper way to show the same number added to itsef mutipe times, a power indicates how many times a number is mutipied by itsef. This power or exponent is shown as a smaer number paced above and to the right of a number. Consider the operation 3 5 shown in Figure 1. The number 5 is the exponent teing you to mutipy the number 3 by itsef a tota of 5 times. This is read as "3 to the fifth power." Whie 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," respectivey, as shown in Figure 2. As you can see in Figure 3, it is possibe to have a number with 0 or 1 as its exponent. Any number "to the zero power" equas 1. Any number "to the first power" equas that same number. Copyright 2015 Tooing U, LLC. A Rights Reserved. Figure 1. A power is a shortcut for mutipying the same number by itsef mutipe times.

10 Lesson: 8/14 Powers The ast pair of common math operations is powers and roots. Power operations are often referred to as exponents. Just as mutipication is a simper way to show the same number added to itsef mutipe times, a power indicates how many times a number is mutipied by itsef. This power or exponent is shown as a smaer number paced above and to the right of a number. Consider the operation 3 5 shown in Figure 1. The number 5 is the exponent teing you to mutipy the number 3 by itsef a tota of 5 times. This is read as "3 to the fifth power." Whie 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," respectivey, as shown in Figure 2. Figure 1. A power is a shortcut for mutipying the same number by itsef mutipe times. As you can see in Figure 3, it is possibe to have a number with 0 or 1 as its exponent. Any number "to the zero power" equas 1. Any number "to the first power" equas that same number. Figure 2. The most common powers are numbers "squared" or numbers "cubed." Figure 3. A number "to the first power" equas the same number, whie any number "to the zero power" equas 1. Lesson: 9/14 Roots By now, you have earned 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 mutipication. The reverse operation of a power is a root. A root tes you which unknown number is mutipied by itsef 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 equas 5. This is a shorthand way to express that 5 x 5 x 5 x 5 = 625. The symbo for a root is a checkmark sign attached to a horizonta ine, with the tota vaue inside the symbo and the root number outside to the eft. Whie 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 symbo, you assume thattooing it is a square root. Copyright 2015 U, LLC. A 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 resut is the origina number. This rue is shown in Figure 4. Roots are much more Figure 1. Each math operation has a matching reverse operation.

11 Lesson: 9/14 Roots By now, you have earned 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 mutipication. The reverse operation of a power is a root. A root tes you which unknown number is mutipied by itsef 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 equas 5. This is a shorthand way to express that 5 x 5 x 5 x 5 = 625. The symbo for a root is a checkmark sign attached to a horizonta ine, with the tota vaue inside the symbo and the root number outside to the eft. Whie 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 symbo, you assume that it is a square root. Keep in mind that a root is the reverse of a power. If you take the "square root" of any number "squared," the resut is the origina number. This rue is shown in Figure 4. Roots are much more difficut to cacuate than the other math operations. In fact, finding a square root amost aways requires a cacuator. Figure 1. Each math operation has a matching reverse operation. Figure 2. A root finds which unknown number, mutipied by itsef a specific number of times, equas the number contained in the root symbo. Figure 3. A sampe of common square roots and their answers. Figure 4. The square root of any number squared is the origina number. Lesson: Copyright 10/ Tooing U, LLC. A Rights Reserved.

12 Lesson: 10/14 The Order of Operations Whie some probems ony may require you to perform one type of math operation, most probems wi invove a sequence of steps. Math probems are not soved simpy by working from eft to right. Instead, you must cacuate each operation in a specific order, which is isted in Figure 1. This is caed the order of operations. For any math probem, you must sove each operation in this order: 1. Sove each exponent and root probem, working from eft to right. 2. Then, sove each mutipication and division probem, working from eft to right. 3. Finay, sove each addition and subtraction probem, working from eft 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. Foowing the wrong order wi amost aways give you the wrong answer, especiay when soving more compex math probems. Figure 2. Soving a probem with the wrong order of operations amost aways gives you a wrong answer. Lesson: 11/14 Grouping Symbos The order of operations is true for any math probem. However, certain math probems require that you foow these operations "out of sequence." When this is the case, math probems use grouping symbos to indicate the proper order. As you can see in Figure 1, grouping symbos must be soved before moving on to any other operation. There are three types of grouping symbos: parentheses ( ), brackets [ ], and braces { }. As you can see in Figure 2, each pair of grouping symbos fits inside a arger pair. Most math probems use parentheses, with ony a few probems using brackets and braces. If there are mutipe symbos, parentheses are soved first, foowed by brackets, and then braces. In some math probems, you may see ony parentheses of different sizes. Grouping symbos change the order of operations. Whenever you see grouping symbos, you must perform the operations inside them before moving on to other areas. If there are mutipe grouping symbos, you must work "inside out," soving the innermost operations first before moving onto others. In Figure 3, you see that is added inside the parentheses before moving on to the other operations. Within each pair of grouping symbos, you must sove math probems using the norma order of operations. Figure 1. If a probem contains grouping symbos, you must sove numbers contained in the symbos first. Copyright 2015 Tooing U, LLC. A Rights Reserved. Figure 2. Grouping symbos paced inside other

13 Lesson: 11/14 Grouping Symbos The order of operations is true for any math probem. However, certain math probems require that you foow these operations "out of sequence." When this is the case, math probems use grouping symbos to indicate the proper order. As you can see in Figure 1, grouping symbos must be soved before moving on to any other operation. There are three types of grouping symbos: parentheses ( ), brackets [ ], and braces { }. As you can see in Figure 2, each pair of grouping symbos fits inside a arger pair. Most math probems use parentheses, with ony a few probems using brackets and braces. If there are mutipe symbos, parentheses are soved first, foowed by brackets, and then braces. In some math probems, you may see ony parentheses of different sizes. Grouping symbos change the order of operations. Whenever you see grouping symbos, you must perform the operations inside them before moving on to other areas. If there are mutipe grouping symbos, you must work "inside out," soving the innermost operations first before moving onto others. In Figure 3, you see that is added inside the parentheses before moving on to the other operations. Within each pair of grouping symbos, you must sove math probems using the norma order of operations. Figure 1. If a probem contains grouping symbos, you must sove numbers contained in the symbos first. Figure 2. Grouping symbos paced inside other grouping symbos may be shown two different ways. Figure 3. The numbers are first soved inside the parentheses, foowed by numbers contained in the brackets. Lesson: 12/14 Order of Operations: An Exampe The sampe probem in Figure 1 contains a the various math operations. Pus, this probem uses grouping symbos to indicate a specific order for soving each operation. How do you find the correct answer? Your first step is to simpify the operations in each pair of grouping symbos. First, add 3 + 2, which eaves you 5 2. Then, sove the second pair of parentheses foowing the order of operations. By dividing 8 by 4 (8 4 = 2), and then adding 2 and 1 (2 + 1 = 3), you remove a the parentheses from the probem. Copyright 2015 Tooing U, LLC. A Rights Reserved. The next step is to sove roots and powers. The square root of 9 is 3, and the number 5 squared is 25. This eaves you with a probem that appears as x 4 + 2(3).

15 Lesson: 13/14 Using a Cacuator There is more than one way to sove math probems. Most shop empoyees have memorized the basic addition, subtraction, mutipication, and division operations. Math probems containing arger numbers with mutipe digits can often be soved if they are written out by hand. But, as math probems increase in difficuty, you must use a cacuator. Many years ago, most shop empoyees had to sove math probems by hand. But today, a simpe cacuator heps shorten the time it takes to sove a probem and reduces the chance of making a mistake. For exampe, you can use the cacuator in Figure 1 to find the square root by entering the number, foowed by the square root key. Keep in mind that many probems, especiay those containing powers or roots, practicay demand that you rey on the aid of a cacuator. The most important thing to remember is the rues that te you how numbers reate to one another in the math probems you encounter. Figure 1. To find a square root, enter the number, foowed by the square root key. Lesson: 14/14 Summary The most basic units in math are whoe numbers. The first whoe number is zero, foowed by 1, 2, 3, 4, 5, and so on. Together, whoe numbers and their matching negative vaues are caed 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 vaue. Subtracting a negative number is the same as adding its positive vaue. The next pair of math operations is mutipication and division. If you mutipy or divide either two positive numbers or two negative numbers, the resut is aways positive. If you mutipy or divide one positive and one negative number, the resut is aways negative. The ast pair of math operations is powers and roots. A power indicates how many times a specific number is mutipied by itsef, whie a root indicates which unknown number is mutipied by itsef a specific number of times to give you the number contained in the root sign. Each pair of math operations incudes reverse operations. Subtraction is the reverse of addition, division is the reverse of mutipication, and a root is the reverse of a power. When soving a probem, you must foow a specific order of operations. Sove powers and roots first, foowed by mutipication and division, foowed by addition and subtraction. If there are grouping symbos, you must sove the operations incuded in the symbos before moving on to others. Figure 1. Addition provides you with a tota number. Copyright 2015 Tooing U, LLC. A Rights Reserved. Figure 2. You must foow the order of operations when soving math probems.

17 Cass Vocabuary Term Definition Addition Bueprint Division Exponent Fraction Grouping Symbos Integer A mathematica operation that unites two separate quantities into one sum = 4 is an exampe of addition. The instructions and drawings that are used to manufacture a part. A mathematica operation that indicates how many equa quantities add up to a specific number. 8 4 = 2 is an exampe of division. Another term for a power. The exponent is the smaer number above and to the right of the number being mutipied by itsef. A math expression with two numbers paced above and beow a division ine indicating the number of divisions or portions and the size of each division or portion. Mathematica symbos indicating that operations contained within the symbos must be soved before moving on to other operations. Any number incuded in either the set of whoe numbers or their matching negative vaues. The numbers -3, -2, -1, 0, 1, 2, and 3 are a integers. Mathematics The study of numbers and quantities and their reationships. Mathematics requires an understanding of the ogic and rues used to sove numerica probems. Mutipication A mathematica operation that indicates how many times a number is added to itsef. 2 x 4 = 8 is an exampe of mutipication. Order Of Operations Power Root Square Root Subtraction Whoe Number Zero The mathematica rues that determine the correct order for soving any sequence of math operations. Powers and roots are soved before mutipication and division, which in turn are soved before addition and subtraction. A mathematica operation indicating how many times a number is mutipied by itsef. 2 3 = 8 is an exampe of a power. A mathematica operation indicating which unknown number, mutipied by itsef a specific number of times, equas the number incuded inside the root sign. The "square root" of 81 equas 9 is an exampe of a root. The most common root, indicating which unknown number mutipied by itsef equas the number inside the square root sign. A mathematica operation that takes away a quantity from a arger whoe. 4 2 = 2 is an exampe of subtraction. Any number contained in the sequence 0, 1, 2, 3, and so on. The symbo indicating the absence of a quantity or amount. On a number ine, zero indicates the point where negative numbers change into positive numbers. Copyright 2015 Tooing U, LLC. A Rights Reserved.

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