01.1 Algebra Concept Overview
ALGEBRA | CONCEPT OVERVIEW The TOPIC of ALGEBRA is not directly provided to us in the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. However, it is of upmost importance that we understand the fundamental concepts and applications revolving around this subject independent of the NCEES Supplied Reference Handbook.
CONCEPT INTRO: The study of ALGEBRA must start at the most foundational level with a study on NUMBERS. Here we can begin with INTEGERS, which can also be referred to as COUNTING NUMBERS, WHOLE NUMBERS or NATURAL NUMBERS. Some will say that each of these categories are distinct, while others will say that are equivalents. Whether itβs COUNTING NUMBERS, WHOLE NUMBERS or NATURAL NUMBERS, an INTERGER is a number that is whole and terminating. Here is an example of a string of INTERGERS: {-5 , -4, -3, -2, -1, 0, 1, 2, 3, 4, 5}
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A RATIONAL NUMBER is one that can be written in terms of INTERGERS, or rather WHOLE NUMBERS. A rational number can be expressed as a ratio that has integers in both the numerator and denominator. INTEGERS are rational numbers because they can be expressed as a ratio using whole numbers. For example, 9 is an INTEGER that is a RATIONAL NUMBER because it can be written as the ratio 9/1. An IRRATIONAL NUMBER on the other hand is one that cannot be expressed in the form of a ratio, or a fractionβ¦or in other terms, the numerator or denominator of the fraction is not an integer. Examples of IRRATIONAL NUMBERS include any number with a radical (i.e. square root), numbers that are non-repeating and numbers that are non-terminating. A PRIME NUMBER is an integer that has no factors except itself and 1. For example, the number 8 is not a PRIME NUMBER because it can be divided by itself, 8, and 1β¦as well as 2 and 4.However, 7 is a PRIME NUMBER because it can only be divided by itself, 7, and 1. There are a few fundamental ALGEBRAIC rules defined that allow us to rearrange, expand, and simplify mathematical relationships as we work through our engineering problems. Made with
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They include the Commutative Law for Addition, Commutative Law for Multiplication, Associate Law for Addition, Associate Law for Multiplication, and the Distributive Law. Assuming that the symbols "a", "b", and "c" represent any real numbers, each law can be represented as follows: Commutative Law for Addition: π+π = π+π Commutative Law for Multiplication: πβπ = πβπ Associate Law for Addition: π + (π + π) = (π + π) + π Associate Law for Multiplication: π β (π β π) = (π β π) β π Distributive Law: π β (π + π) = π β π + π β π
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ALGEBRA | CONCEPT EXAMPLE The following problem introduces the concept reviewed within this module. Use this content as a primer for the subsequent material and practice problems to come. Given the following expression: 2 β (4 + 3) The βLawβ represented as well as itβs is: A. Associate Law for Multiplication and 12 B. Distributive Law and 14 C. Commutative Law for Addition and 14 D. None of the above
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SOLUTION: The TOPIC of ALGEBRA is not directly provided to us in the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. However, it is of upmost importance that we understand the fundamental concepts and applications revolving around this subject independent of the NCEES Supplied Reference Handbook. Although these concepts present themselves as pretty straight forward, we must make sure that we know them by heart. These are giveaway problems that many will trip up on simply because they failed to solidify them as second nature. Revisiting the problem statement, we are given the expression: 2 β (4 + 3) Reviewing the laws from above, and by order of operations, we see that this expression lines up perfectly with the Distributive Law which states that: π β (π + π) = π β π + π β π In our case, each variable is defined as: β’ a=2 β’ b=4 β’ c=3
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With this information defined, we simply just need to plug and play, either by hand, using our calculator, or crunching the numbers in our head. Doing so we get: Simplifying, we find our answer is: 2 β (4 + 3) = 2 β 4 + 2 β 3 = 14 Therefore, the correct answer choice is B. The Distributive Law and 14
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CONCEPT INTRO: The study of ALGEBRA must start at the most foundational level with a study on NUMBERS. Here we can begin with INTEGERS, which can also be referred to as COUNTING NUMBERS, WHOLE NUMBERS or NATURAL NUMBERS. Some will say that each of these categories are distinct, while others will say that are equivalents. Whether itβs COUNTING NUMBERS, WHOLE NUMBERS or NATURAL NUMBERS, an INTERGER is a number that is whole and terminating. Here is an example of a string of INTERGERS: {-5 , -4, -3, -2, -1, 0, 1, 2, 3, 4, 5}
Made with
by Prepineer | Prepineer.com
A RATIONAL NUMBER is one that can be written in terms of INTERGERS, or rather WHOLE NUMBERS. A rational number can be expressed as a ratio that has integers in both the numerator and denominator. INTEGERS are rational numbers because they can be expressed as a ratio using whole numbers. For example, 9 is an INTEGER that is a RATIONAL NUMBER because it can be written as the ratio 9/1. An IRRATIONAL NUMBER on the other hand is one that cannot be expressed in the form of a ratio, or a fractionβ¦or in other terms, the numerator or denominator of the fraction is not an integer. Examples of IRRATIONAL NUMBERS include any number with a radical (i.e. square root), numbers that are non-repeating and numbers that are non-terminating. A PRIME NUMBER is an integer that has no factors except itself and 1. For example, the number 8 is not a PRIME NUMBER because it can be divided by itself, 8, and 1β¦as well as 2 and 4.However, 7 is a PRIME NUMBER because it can only be divided by itself, 7, and 1. There are a few fundamental ALGEBRAIC rules defined that allow us to rearrange, expand, and simplify mathematical relationships as we work through our engineering problems. Made with
by Prepineer | Prepineer.com
They include the Commutative Law for Addition, Commutative Law for Multiplication, Associate Law for Addition, Associate Law for Multiplication, and the Distributive Law. Assuming that the symbols "a", "b", and "c" represent any real numbers, each law can be represented as follows: Commutative Law for Addition: π+π = π+π Commutative Law for Multiplication: πβπ = πβπ Associate Law for Addition: π + (π + π) = (π + π) + π Associate Law for Multiplication: π β (π β π) = (π β π) β π Distributive Law: π β (π + π) = π β π + π β π
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ALGEBRA | CONCEPT EXAMPLE The following problem introduces the concept reviewed within this module. Use this content as a primer for the subsequent material and practice problems to come. Given the following expression: 2 β (4 + 3) The βLawβ represented as well as itβs is: A. Associate Law for Multiplication and 12 B. Distributive Law and 14 C. Commutative Law for Addition and 14 D. None of the above
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by Prepineer | Prepineer.com
SOLUTION: The TOPIC of ALGEBRA is not directly provided to us in the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. However, it is of upmost importance that we understand the fundamental concepts and applications revolving around this subject independent of the NCEES Supplied Reference Handbook. Although these concepts present themselves as pretty straight forward, we must make sure that we know them by heart. These are giveaway problems that many will trip up on simply because they failed to solidify them as second nature. Revisiting the problem statement, we are given the expression: 2 β (4 + 3) Reviewing the laws from above, and by order of operations, we see that this expression lines up perfectly with the Distributive Law which states that: π β (π + π) = π β π + π β π In our case, each variable is defined as: β’ a=2 β’ b=4 β’ c=3
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by Prepineer | Prepineer.com
With this information defined, we simply just need to plug and play, either by hand, using our calculator, or crunching the numbers in our head. Doing so we get: Simplifying, we find our answer is: 2 β (4 + 3) = 2 β 4 + 2 β 3 = 14 Therefore, the correct answer choice is B. The Distributive Law and 14
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