Math A-100 IUSB Sec.1.6 Notes – Week 1 Properties in Math “Properties” in math are simply “things that are always true”. You are probably already familiar with some of them. Our textbook wants to focus on of them: Associative Property of Addition Associative Property of Multiplication Commutative Property of Addition Commutative Property of Multiplication The Identity Property of Addition The Identity Property of Multiplication The Additive Inverse Property The Multiplicative Inverse Property The Zero-factor Property Fortunately, in my class, you don’t need to know these names (and certainly not memorize them, i.e. I will not put them on quizzes or tests) but you will have to recognize them. So let’s take a look at each. Commutative and Associative Properties There are several “properties” in math. A “property” is simply something that is always true. For example, you can always: 1) Add two numbers together before adding a third number, such as (2 + 3) + 4 2) multiply two numbers together before multiplying by a third number (2 x 3) x 4 3) Rearrange the order of the numbers when you add 2+3+4=2+4+3 4) Rearrange the order of the numbers when you multiply 2x3x4=2x4x3 When you put two numbers together, you are grouping them, i.e. they are being “associated”. As we see above, you can re-group (re-associate it) if you wish. This is called the Associative Property of Addition and the Associative Property of Multiplication. When you rearrange the numbers, you are move them around, i.e. the numbers are commuting to a new location. As we see above, you can rearrange the numbers, i.e. move them, however if you wish. This is called the Commutative Property of Addition and the Commutative Property of Multiplication.’ NOTE: you cannot re-group or rearrange the numbers when subtracting or dividing. Subtracting: (12 – 4) – 3 is not the same as 12 – (4 – 3).  (12 – 4) – 3 = 5 but 12 – (4 – 3) = 11. and 12 - 4 -3 is not the same as 4 - 12 – 3.  12 – 4 – 3 = 5 but 4 – 12 – 3 = - 11. Dividing: (12/4) /3 is not the same as 12 /(4/3)  (12/4) /3 = 1 but 12 /(4/3) = 9 and 12/4/3 is not the same as 4/12/3  12/4/3 = 1 but 4/12/3 = 1/9 Example 1: use the Associative Property to re-write (a + 3) + b To associate means to “group together” so re-group them (a + 3) + b becomes a + (3 + b) Example 2: use the Associative Property to re-write (a * 3) * b To associate means to “group together” so re-group them (a * 3) * b becomes a * (3 * b) Example 3: use the Commutative Property to re-write (a + 3) + b To commute means to go or move, so move them around (a + 3) + b becomes b + (a + 3) or a + b + 3 or b+3+a etc. b*3*a etc. Example 4: use the Commutative Property to re-write (a * 3) * b To commute means to go or move, so move them around (a * 3) * b becomes b * (a * 3) or a * b * 3 or The Identity Property of Addition The Identity Property of Addition simply says “what can you add to a number and not change its identity? 0 Yes, this property basically says adding 0 to any number does not change it, i.e. x + 0 = x The Identity Property of Multiplication The Identity Property of Multiplication simply says “what can you multiply to a number and not change its identity? 1 Yes, this property basically says multiplying any number by 1 does not change it, i.e. (x)(1) = x The Inverse Properties reverses the order… The Additive Inverse Property The “Additive Inverse Property” simply says “what can you add to a number and get an answer of 0? The number’s opposite  a.k.a. its additive inverse Yes, this property basically says adding the opposite to any number gets an answer of 0, i.e. 1x + -1x = 0 The Multiplicative Inverse Property (i.e. the reciprocal) The “Multiplicative Inverse Property” simply says “what can you multiply a number by to get an answer of 1? The number’s reciprocal  a.k.a. its multiplicative inverse. The reciprocal is simply flipping the number: the reciprocal: of ¾ is 4/3; of 1/5 is 5/1 = 5; of 7  7/1 so it is 1/7 Yes, this property basically says multiplying any number by its reciprocal gets an answer of 1, i.e. (x)(1/x) = 1 The Zero-factor Property “Factors” in math are “numbers multiplied together”. So, the Zero-factor Property simply says that if you multiply any number by 0, you get an answer of 0, i.e. (x)(0) = 0 Word questions  math expressions In the real world, no one has ever been given a piece of paper with a math equation on it and then told to “solve for x”. If you go to the store with $20 in your pocket and are looking to buy hotdogs for $3 per pack and buns for $2 per pack, no one gives you a piece of paper with 2x + 3y < 20. So, we need to be able to convert word statements into math expressions and equations in order to solve them. Look for “key words” that tell you what is happening: Adding:  “sum”, total of, increased by, gain, etc. Subtracting:  decreased by, difference, reduced by, loss, etc. Multiplying:  times, product, etc. Dividing:  quotient, ratio, etc. Special “key words” “a number” = x “twice” = times 2 “thrice” = times 3 “double” = times 2 “triple” = times 3 “squared” = to the 2nd power, i.e. exponent of 2 “cubed” = to the 3rd power, i.e. exponent of 3 “halved” = times ½ or divided by 2 “quartered” = times 1/4 or divided by 4 “__ times the sum of” = __( __ + __) “__ times the difference of” = __(__ - __) “subtracted from” = reverses the order Example 1: a number subtracted from 6  6 – x Example 2: 13 subtracted from a number  x – 13 Example 1: “the sum of a number and 25” “Sum” means adding, so you have x + 25 Example 2: “the product of a number and 25” “product” means multiplying, so you have 25x Example 3: “the product of 12 and the sum of a number and 4” “product” means multiplying, and “the sum of a number and “ means (x + __), so you have 12(x + 4) Example 4: “half the difference of a number and -5” “half” means multiplying by 1/2, and “the difference of a number and ___“ means (x - __), so you have ½(x - -5) NOTE: Some of your HW in MML ask you to only write the expression and do nothing else with it. Example 5: “the sum of twice a number and -5” “twice” means multiplying by 2, so you have (2x + -5) or just 2x + -5