Lesson 2: The Properties of Real Numbers

The Properties of Algebra In total there are 10 key properties of real numbers. Nine of them are listed in this video and the tenth (the Distributive Property) will be addressed in a later lesson.These properties are the foundation of what you will need to be successful in Algebra II this coming school year. Please watch the following video and fill in the notes accordingly. Fill in the Blanks - Part 1Think back to the video, replay and pause it if need be, and fill in the blanks below. This first part covers the video from start up until 9:18.The wording may be slightly different, but is close enough to the narration that you should be able to get the answers. The Properties of real numbers do not apply to two operations, subtractionand division (written in order of least importance to greatest). However, the properties do exist for the operations of addition and multiplication.Associative Property (Multiplication)Example 1: a( b * c) is exactly the same as (a * b) * cThe Associative Property means no matter how you group things that are multiplied together, you will get the same product (aka the result of multiplying).Example 2: a(b * c) is the same as a * b * c.This is something that is not shown as often textbooks. The Associative Property allows us to throw away the grouping symbols, the parentheses, when we have ALL multiplication.Associative Property (Addition)Example 3: a + (b + c) is exactly the same as (a + b) + cThe Associative Property states that you should be able to regroup things that are added together and still get the same sum (aka the result of adding). It also means we can just throw away the grouping symbols all together. We can prove this is true with a quick example.Example 4: 5 + (6 + 7) and 5 + 6 + 7, check it out for yourself. The results should be the same!Commutative Property (Multiplication)Example 5: a * b * c = c * a * bExample 6: 3 * 4 * 5 = 4 * 5 * 3, check it out for yourself. The results should be the same!The Commutative Property says you can move around all the factors in a product and still get the same product. We can also rearrange terms that are being added together to get the same sum.I might try and trick you, so I might ask a question like this: Which property is being demonstrated by (3 * 4 ) * 5 = 5 * (4 * 3)?Look, there are parentheses, but notice you are not regrouping the numbers. The only thing that has changed here is that 3 and 4 are in different positions. That means we are using the CommutativeProperty, even though there are parentheses.Commutative Property (Addition)Example 7: a + b + c = b + c + aExample 8: 4 + 7 + 9 = 7 + 4 + 9, check it out for yourself. The results should be the same!The Commutative Property says we can rearrangeterms and get the same sum.In summary:The Associative Property states that you can change the groupings and still get the same results.The Commutative Property states that you can change the order and still get the same results. Fill in the Blanks - Part 2Think back to the video, replay and pause it if need be, and fill in the blanks below. This second part covers the video from 9:18 up until the end.The wording may be slightly different, but is close enough to the narration that you should be able to get the answers. Do keep things in the order that the video has given them to you. Example 1: a * (1 = 11 is known as the inverse of a, with regard to multiplication. We have a fancy way of saying it, the Multiplicative Inverse. The Multiplicative Inverse is used when we are trying to get a variable all by itself. We have another name for it and that other name is the reciprocal. Whenever we multiply a number by its multiplicative Inverse, the result is the number (written as a word) one.Example 2: a + (-a) = 0We call -athe Additive Inverse of a. The Additive Inverse is also known as the opposite of a number and is equally as important when isolating variables. Whenever we add a number and its Additive Inverse (so a + -a) the result will be the number (written as a word) zero.Example 3: 1 * b = bIdentity is what we think of ourselves, and so a is the same as a. A number is the same as itself. We can get back to a if we mulitply a times the number 1. One is know as the Multiplicative Identity. What we are saying is that if we have 1 times x, that's just x! This is the reason why we can just throw away that 1 when solving equations!Example 4: c + 0 = cAdding zero to any number just gets you that number. Thererfore, 0 is known as the Additive Identity. You've used this property countless times (you know it by heart!) because you use it every time you balance an equation by adding or subtracting a value from both sides.Example 5: 3 (-7) (48) (√2) (13.7) (0) (14) = 0The Zero Property states that if we take any number and then multiply by 0, we just get 0! This property will be the foundation of our work in Units 6 and 7, both of which heavily rely on factoring.When we have something like (x + 3)*(x - 2) = 0, we can use this property to figure out what x is. We simply say that either x + 3 = 0 or x - 2 = 0, which gives us two possible solutions. What does it look like? Match the property to its algebraic equivalent. Associative Property of Addition 4*5 + 2 -7 = 4*5 + (2 - 7) Commutative Property of Addition 5(10b + 12b) + 7b = 5(12b + 10b) + 7b Associative Property of Multiplication 1 * (6 * 2) = (1 * 6) * 2 Commutative Property of Multiplication (3 * 4) * 5 = (4 * 3) * 5 Additive Identity 9 + x = 9 Additive Inverse (-8 + 8) + 15 = 0 + 15 Multiplicative Identity 15x = 15 Multiplicative Inverse 7 * (1 = 1 The Zero Property (x - 2) * (x - 3) = 0 Inverses, What's the Deal With That? For each of the 3 values, state both the additive and multiplicative inverses.For the sake of simplicity, write all NON-WHOLE answers as fractions. Do not use spaces for negative values- Ex. Negative 1 should be -1 not - 1 Number Additive Inverse Multiplicative Inverse 15 -15 1 -4 4 -5 Say My Name, Say My Name State the property that best describes each step in the problem below. The first one is an example (don't try to do it, it's already done!) Steps Properties -(3g + 3h) + 5g - 10h Original -3g - 3h + 5g - 10h Distributive Property -3g + 5g - 3h - 10h Commutative Property of Addition (-3g + 5g) + (- 3h - 10h) Associative Property of Addition 2g - 13h Answer (via Substitution Like Terms) ----------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------- 1 (15d + 3c) - 1 (8c + 10d) Original (5d + 1c) + (-4c - 5d) Distributive Property Inverse (5d + c) + (-4c - 5d) Multiplicative Identity 5d + c + -4c - 5d Associative Property of Addition 5d - 5d + c - 4c Commutative Property of Addition 0 + c - 4c Additive Inverse c - 4c Additive Identity -3c Answer (via Substitution Like Terms)