Math A-100 IUSB Sec.1.3 Week 1 Doing calculations with Signed Numbers Whole Numbers There are 7 categories of numbers that we will work with during this course: 1) Natural Numbers (also called the Counting Numbers) – these are the numbers used in Counting: 1, 2, 3, … NOTE: 0 is not included in the Natural Numbers – if someone does not exist in Nature, why do you need a number for it? 2) Whole numbers – non-negative (so 0 IS included here) numbers that have no fraction or decimal part: 0, 1, 2, 3, …, √4 = 2, √9 = 3 3) Integers – all numbers with no fraction or decimal part: …-3, -2, -1, 0, 1, 2, 3, … 4) Rational Numbers – all numbers that CAN be written as a fraction (but don’t have to be). These include: Integers – since any integer can be written as a fraction: 3 = 3/1 , -7 = -7/1, 0 = 0/1, etc. Proper fractions: 2/3, -1/4, etc. Improper fractions: 6/5, -7/3, etc. Mixed numbers: 3 ½ = 7/2, -4 3/5 = -23/5, Etc. Decimals: 0.75 = ¾, - 0.2 = -1/5, 5.1 = 5 1/10 = 51/10, - 6.35 = - 6 35/100 = -6 7/20 = -127/20, etc. Repeating decimals: 0.66666… = 2/3, - 0.571428571428… = -4/7, 0.4166666 = 5/12, etc. 5) Irrational Numbers – all numbers that CAN NOT be written as fractions, i.e. decimals that never repeat and never end. Examples: π, √3, √2 6) Real Numbers – all numbers which you can find/place on our number line, i.e. all the possibilities listed above (π is on the number line between 3 and 4, √2 is on the number line just to the left of 1.5, etc.) 7) Unreal Number (also called Imaginary Numbers) – all number that we have a concept of, but cannot be put on our number line, such as √−2. NOTE: Fortunately, in this class we are dealing with the first 6 categories only. M-107 will deal with the Unreal numbers. Integers - Signed Numbers w/ no fraction or decimal part “Signed” numbers are simply positive and negative numbers. Positive numbers generally indicate an increase or improvement. For example, depositing $200 in your bank would be +200 or just 200, going up in elevation by 3000 feet would be +3000 or just 3000, etc. Negative numbers generally indicate a decrease or lowering. For example, withdrawing $200 from your bank account would be represented by -200, charging $45 on your credit card would be represented by -45, diving to 75 feet below sea level would be represented by -75. Larger vs smaller How can you tell if one number is bigger than another? With positive numbers, it is pretty obvious. With negative numbers, however, it can be confusing: 100 is bigger than 75, but -100 is NOT bigger than -75. Students often tell me things like: “-75 is bigger than – 100 because it is closer to 0.” There is a problem here, because that works with -75 and -100, but does not work with 75 and 100 (because 75 is closer to 0, but is NOT bigger than 100). So let’s try something the works ALL the time: bigger numbers are farther to the right on the number line. 100 is farther to the right on the number line than 75, so 100 is bigger. -75 is farther to the right on the number line than -100, so -75 is bigger. < vs > Students often get these two symbols confused as well. < means “less than” and > means “greater than”, but who sues those words in real-life? Let’s say instead: < means “smaller” > means “bigger” and If you have another memory trick to use for remembering, like drawing in teeth and “the alligator always eats the bigger meal”, then feel free to use them! Another one is “the small end always points to the smaller number”