Time: 30 hours
Our cannoli story has a fascinating answer that speaks to some deep mathematics first explored in the late 1800s and re-explored in the 1980s (thanks to some chicken nuggets from Mcdonald's). We use math in our daily activities, and it plays an important role in nearly every career you can imagine: from business to cooking, farming, medicine, and beyond. It is no surprise that many call math a universal language: people across the globe use the same numbers, formulas, and equations to help them navigate the world.
In this course, we study essential math concepts that will enrich your understanding of the world and illuminate a larger, mathematically-rich universe. The three courses in the Real World Math series discuss basic algebra and geometry topics and show you how to apply these concepts to everyday life.
The material focuses on how math relates to common real-world situations, transactions, and phenomena, such as personal finance, business, and the sciences. This real-world focus will help you grasp the importance of the mathematical concepts you encounter in these courses and understand why you need quantitative and algebraic skills to succeed in college and your day-to-day life.
For example, fractions allow us to tell interesting and useful stories that involve measurement, ratios, and proportions. Decimals and percentages are merely fractions in disguise. They help us make financial decisions and measure or compare various types of data. This course will help clarify the different ways we represent data visually, such as with a bar or line graph.
We also examine how to interpret data – no matter how it is presented. This skill will help you read a chart that outlines the current mortgage interest rate or make sense of the latest statistics for your fantasy football league. Let's not forget our coffee-infused chocolate cannolis. You will use addition, subtraction, and multiplication to answer her question. However, we will not help you deliver the bad news to your professor-customer.
A note on numbers. Before we begin, let's clarify what we mean by the word numbers. We usually refer to a quantity, such as five, seven, or 10. But mathematicians have created four different categories or types of numbers.
Here is a summary of four types of numbers (with the fancy symbols mathematicians use to refer to them):
- Natural Numbers: all positive whole numbers:
- Integers: all positive and negative whole numbers (and zero, too!):
- Rational Numbers: all positive and negative fractions (including integers):
- Real Numbers: all possible positive and negative numbers (including ):
Since presenting the entire set of real numbers as a collection of quantities is difficult, we usually visualize this set as a solid line of points that stretches on forever in both directions. This set includes all of the four number types in our list above, in addition to some more interesting expressions we won't discuss here. In Unit 1, we discuss the mathematical operations we can perform using all of these numbers: our number properties apply to all real numbers. However, most of our examples will focus on positive whole numbers (integers). In later units, we will explore negative whole numbers (integers) and fractions (also called rationals). You do not need to memorize these symbols, but know that integers can refer to positive and negative whole numbers. Fractions can also be positive or negative.
First, read the course syllabus. Then, enroll in the course by clicking "Enroll me". Click Unit 1 to read its introduction and learning outcomes. You will then see the learning materials and instructions on how to use them.
Unit 1: Number Properties
In this unit, we discuss properties and patterns for adding, subtracting, multiplying, or dividing numbers. You are probably so familiar with these facts that it may seem strange to point them out at all. But there are benefits to naming and using these properties. Some are purely mathematical, but plenty are common, everyday ones too.
First, let's talk about the commutative properties of addition and multiplication. These tell us that it does not matter if you rearrange your numbers when you add or multiply them. Soand both equal five and and both equal 60.
But we cannot swap all of our actions around like this. For example, what about when we bathe our dog, Harry, first and then dry him? Subtracting provides a mathematical example. Subtracting is not commutative:is not the same as . Division is not commutative, either. You get a different result if you divide a 60-dollar dinner bill among 10 diners than if you divide a 10-dollar dinner bill among 60 diners! The order is important.
In other words, some of our basic operations enjoy certain properties (like commutativity), while others do not. You need to keep track of these rules to avoid confusion, incorrect dinner bills, and wet dogs. These properties will not only help you calculate more complicated equations later on, but they will also help you as you consider less intuitive or obvious properties, such as What happens if I divide 60 by 0? Paying careful attention to all of these rules will help you on your mathematical journey and in the real world.
Completing this unit should take you approximately 2 hours.
1.1: Commutative Law of Addition and Multiplication
The commutative law of addition and multiplication tells us that the order you use to add or multiply numbers does not matter. In other words,, and . The same is true of multiplication: , and . You obtain the same result regardless of the order you use to add or multiply numbers.
Note that the commutative property only holds for addition and multiplication. It does not work for subtraction or division. For example,, but . In subtraction, the results are not the same when you change the order of the numbers. Likewise in division: , but . Again, the results are not the same when you change the order of the numbers in the division.
But familiar, old, real numbers are not the only kinds of entities we add and multiply in mathematics. For example, you will learn that matrix multiplication is not commutative when you study matrices in linear algebra and vector calculus. Similarly, we use a more-advanced, exotic kind of number called a quaternion for all sorts of tasks, such as robotics, 3-D animation, and video game design. Quaternion multiplication is noncommutative.
While quaternions and matrices are beyond the scope of this course, they provide an important lesson: rearrangement can affect useful kinds of multiplication. Sometimes order matters! The fact that real number addition and multiplication are commutative warrants an appreciative and careful understanding of the property.
1.2: Associative Law of Addition and Multiplication
Just after closing, you accidentally knock over the cash register at your aunt's coffee shop, and all of the day's money spills out onto the floor. You carefully put the five-dollar bills into one pile, all of the one-dollar bills into another pile, and all of the quarters, dimes, nickels, and pennies into their own piles. Then you add up the total. Thankfully, you check your math, and it matches the day's receipt: $1,500.00 exactly. Would you have tallied up this same amount if you had placed both five- and one-dollar bills into a single pile and grouped all of the coins into another separate pile? Would having a different grouping alter their count?
Here is another puzzling story: Average Joe loves math. He is known far and wide for his love of averaging. Whenever you hand him two numbers (say 2 and 10), Average Joe will add them and then divide by 2 (to give 6). If you were to hand him the numbers 5 and 15, he would happily average them to produce the number 10. But what happens if one day Joe's friend Above-Average Sharon hands him three numbers, say 2, 10, and 14? Will Average Joe first compute the average of 2 and 10, yielding 6, and then compute the average of 6 and 8? Or will he first average 10 and 14, yielding 12, and then compute the average of 12 and 2? Does it matter which grouping Average Joe uses to compute his averages?
In our first story, the change in grouping will not affect the final result, but it does in our second Average Joe story! Certain mathematical combinations or operations depend on how the parts or numbers are grouped, and sometimes they do not.
The associative property is the adjective we use to describe the operations that are unaffected by grouping. More specifically, the associative law of addition and multiplication tells us that no matter how we group or "associate" the numbers we add or multiply, the outcome remains the same.
This may sound awfully similar to the commutative property, but it expresses a different (although similarly useful) property. In arithmetic calculations, we often place parentheses around a set of numbers to indicate we are associating or grouping them together. We always carry out the calculation in parentheses first. For example,. When we calculate this, we first calculate . Then, we add 5 to get 8. We can also write . Here, we first calculate and then add 1 to get a sum of 8. We get the same result regardless of how we group the numbers.
This same law works for multiplication. When we can compute . The placement of the grouping does not change the answer.. We can change the grouping and write
1.3: Identity Property of Addition
In math, the additive identity is the name for a special number that does nothing when it is added to any other number. For real numbers, the number 0 plays this special role; this means that 0 is the additive identity for real numbers. In other words, the identity property of addition simply states that there is an additive identity called 0. This is probably a familiar fact for most of you. For example,or .
It may seem strange to pay so much attention to such a simple or lazy number like 0. What is the big deal? It does nothing in addition. Why make such a fuss or give it a technical name?
As it turns out, paying careful attention to 0's special status early on will help you understand more complicated, unfamiliar numbers and objects. How about other operations? Do they feature a do nothing identity element? If we had not paid enough attention to 0 early on, this question would not have even occurred to us! But now we can ask, "does Average Joe's averaging operation have an identity" element?
1.4: Inverse Property of Addition
The inverse property of addition states that every real number has a special companion that we call its additive inverse. We define this additive inverse in relation to our additive identity, 0. Basically, the sum of a number and its additive equals 0, or the additive inverse of a number and its negative is 0. The fact thattells us that the number is the additive inverse of the number .
Note that there are no additive inverses if we only use positive numbers (the natural numbers)! We must use the larger, richer collection of integers to use our new property.
You can think about this inverse property as mathematical cancel culture, where we regard our additive identity, 0, as a do-nothing or neutral number. The existence of additive inverses simply tells us we can undo, neutralize, or cancel every number. We can undo or cancelby adding ; we can cancel by adding . Here is a fun question to ponder: What is the additive inverse of ?
1.5: Identity Property of Multiplication
Much like the identity property of addition (see section 1.3), the identity property of multiplication states that there is a number that serves as the multiplicative identity that does nothing when it is multiplied against any other number. What special number behaves in this way? Two will not work since, for example,and was not left alone in this multiplication. It multiplied to become ! Can you also see why and fail to be multiplicative identities? Take a minute to explore and practice multiplying various numbers before reading on, and you will likely stumble across that one special, lonely number that works.
As you may have figured out, the numberhas this magical, do nothing property. For example, and . In short, the multiplicative identity property states that if you multiply any number by , the answer is simply the number you started with.
And as you may have also guessed, the reason for emphasizing's special status as a do-nothing multiplicative identity is the same as that for zero's special status as a do-nothing additive identity: it can be useful to pay attention to these special rules and objects, particularly when using other, more abstract or new mathematical operations.
1.6: Inverse Property of Multiplication
The inverse property of multiplication tells us that almost every real number has a multiplicative inverse. Since we treat the multiplicative identity,, as a neutral element, we can cancel numbers (multiplicatively). For example, the multiplicative inverse of is the number ; this follows since .
Note that we would not be able to access multiplicative inverses likeif we only use integers (fractions like are not whole numbers)! We need fractions or rational numbers to be able to cancel numbers multiplicatively.
To cancel a numbermultiplicatively, we always multiply by . While it is correct to call the multiplicative inverse of , we also call it the reciprocal of (just like how we call the additive inverse of a number its negative). Unlike additive inverses, not every real number has a multiplicative inverse: zero is the one special number we cannot cancel (multiplicatively). The reason we cannot invert (multiplicatively) involves the familiar rule you cannot divide by (see section 1.8 below).
1.7: Multiplication by Zero
Photo by Scott Beale
Although zero serves as the additive identity for our numbers – literally leaving them all alone – it has a more destructive impact when used to multiply. Multiplying any number byalways results in . This is why some mathematicians call the annihilator: all other numbers or objects are annihilated when multiplied by . For example, both and . While mathematicians accept this aspect of the additive identity as fact, there are good, everyday, real-world reasons for this rule, too.
For example, let's say your coffee shop manager tells an employee, Every time a customer orders a large coffee, we make. When 45 customers order large coffees, multiplication tells us the shop earned dollars. However, if zero customers come in that day, clearly, the shop has earned : in large coffee sales. We can generalize and adjust this common experience to see that .
1.8: Dividing by Zero Is Undefined
Because division does not enjoy the commutative property, dividing intoand dividing by are two different questions. One of these computations is easy to carry out, while the other is impossible. Can you guess which is which before you read more?
Let's begin by considering division by zero. Many mathematics educators model the operation of division using shared pizzas or pies. If two students show up to a party where there are 10 pizzas, then they can each receive an equal number of pizzas, namelypizzas each; if 80 students show up to that same party, each student will receive of a single pizza, or one (small) slice each.
Thinking about our pizza party example suggests why the answer to this division by zero question is so strange: imagine we have 10 pizzas to share among the students at a party, only now zero students show up! How much pizza does each student receive? A reasonable reply is that the question does not make any sense since no students are present. There is no answer or number we can offer in response to this question!
The mathematical rule for division byreads like this:
Whenever we divide a number by, the answer is undefined.
For example,= undefined. The technical response is, We cannot assign a meaningful value or answer to the expression .
1.9: Distributive Property
Our picture of arithmetic is almost finished. We have our real numbers and operations for combining them (a, , , ), and we have learned about some properties these operations enjoy. But our last puzzle piece asks: How do these operations interact with one another?
Addition and subtraction interact in understandable ways: these are inverse operations. The same is true for multiplication and division. But how do addition and multiplication interact?
Our answer is encoded in the distributive law for numbers. This property tells us how to distribute a multiplication across a sum (we write the sum in parentheses). For example, we can use the distributive property to rewriteas . The answer is the same, and writing it this way makes it easier to simplify large calculations and figure out the answer without having to write it down (mental math).
Here is an abstract statement of the distributive law:
As its name suggests, we are distributing the multiplied numberto each number in the sum.
Unit 1 Assessment
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Your aunt has an excellent recipe for a breakfast sandwich that her customers love, but the cafe is all out of eggs and buns. She needs you to run to the grocery store for supplies. Exactly one egg and one bun are required to make each sandwich. Unfortunately, you can only purchase eggs in packages of 12 and buns in packages of eight. If you return with one dozen eggs and one eight-package of buns, four eggs will go unused. If you return with one dozen eggs and two packages of buns, four buns will be left over. What is the smallest number of eggs and buns you can purchase so there are no leftovers of eggs or buns? (Feel free to take out a sheet of paper and work the numbers! It will not take long to see that the answer is 24).
In this brief unit, we explore the concepts of least common multiple (LCM) and greatest common factor (GCF). Our run-of-the-mill questions, such as our egg-and-bun problem, indicate how these concepts are relevant to our everyday lives, but they also show up in lots of other mathematical questions. Here we are talking about sets of whole numbers: in Unit 1, our operations applied to whole numbers, fractions, and every other type of real number. Consequently, notions of least common multiple and greatest common factor belong to the part of mathematics that studies patterns among whole numbers, an area called number theory.
Completing this unit should take you approximately 1 hour.
We can use our four basic operations to solve so many types of mathematical and real-world problems, but we need to be careful when we combine or repeat them. The commutative and associative laws apply to certain operations but not others. We use the distributive law to guide calculations when we combine different operations. But we need to follow an order of operations, which we apply to any combination of operations we need.
In this unit, we learn how to use exponents and exponential notation to represent repeated multiplication. For example, repeatedly multiplying two against itself five times results in . We will pay careful attention to how negative numbers interact with our operations and clarify how to group different symbols (subjected to different operations) in the correct and clear order.
Completing this unit should take you approximately 4 hours.
Fractions are amazingly helpful; we can use them to tell so many different kinds of stories, but this can also make them seem overwhelming or confusing. How likely will a coin flip result in heads? We use a fraction to express the answer to this probability: our outcome (a heads up) should occur 1 out of 2 times or one-half of the time.
If you serve 10 pizzas to 40 students, how much pizza should each student receive? We use fractions to determine that each student should receiveor one-fourth of a single pizza. We also use fractions to discuss ratios and rates, such as this car gets 40 miles per gallon.
Combining fractions using addition, subtraction, multiplication, or division is a bit more complicated than combining integers, but these combinations are useful in many situations and problems.
For example, if your friend Andy has one-half of a chocolate bar and Tai has one-third of a bar, how much do they have together? To answer this question, we need to add the rational numbers(see Section 4.8). In this unit, we explore these kinds of topics, beginning with a basic review of how to notate a fraction and ending with a discussion on how to use them in multi-step problems.
Completing this unit should take you approximately 8 hours.
In this unit, we study decimals, which are simply another way to express fractions. We use this way of notating fractions (including improper ones and mixed numbers), most notably when dealing with money or currencies.
For example, if the price of your aunt's amazing vanilla latte is $3.50, we understand this to mean it costs three whole dollar bills and fifty cents. The period or decimal point, as mathematicians call it, separates whole numbers from the fractions in this expression – it serves this purpose in this price and in all other decimal expressions. Our whole numbers (in this case, 3) sit to the left of the decimal point, and fractional quantities sit to the right of the decimal point.
Since the digit 5 appears immediately to the right of the decimal, it really means five-tenths. The zero means zero hundredths. Of course, we can reduce the fraction five tenths as
Altogether this means we can translate $3.50 literally or mathematically to three whole dollars and one-half of a dollar. That is
Because decimals are just a different way to express factions, they are just as useful and omnipresent. In this unit, we explore how to convert decimals into explicit fractions and how to add, subtract, multiply, and divide them.
Completing this unit should take you approximately 6 hours.
In this unit, we study another application of fractions, ratios, and proportions. These are mathematical concepts we use all the time, probably without even realizing it. Have you ever compared unit prices for different packages of the same type of food at the grocery store? That is a ratio.
When driving 65 mph (miles per hour) on the highway, have you ever determined how long it will take you to get to your destination? That is a proportion. In sports, statisticians use proportions to predict an athlete's performance based on what they have accomplished in the past. In this unit, we explore how to write ratios, set up and solve proportions, and apply these skills to real-world experiences.
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Cooking is probably where we use ratios most.
Imagine your aunt's coffee shop is hosting a party. She and her staff are being handsomely rewarded for providing the venue and all of the food. She knows her special chocolate raspberry cake feeds 10 people and uses the following ingredients: four cups of flour, two cups of sugar, three sticks of butter, six eggs, and lots of chocolate. However, 25 people will be attending the gathering, and she needs to make a larger cake! Exactly how much flour and sugar does she need to pull this off?
Working through the next few sections will help you use rates and ratios to answer these questions so your aunt can complete this delicious task (and others, too!).
Completing this unit should take you approximately 2 hours.
We use percentages every day. For example, how about when your aunt's coffee shop advertised pumpkin spice lattes for 50% off – you and your aunt knew this meant half off, and so did the individuals in the ensuing rush of new customers. In this unit, we explore how to compute percentages. What does this mean, and how do we think about percentages more generally?
In this unit, we answer this question and lots of others that involve percentage computations and applications. For example, we will convert between percents and fractions or decimals and learn about percentage increases and decreases. We will also explore how to calculate some real-world uses, such as restaurant tips and sale prices.
Completing this unit should take you approximately 2 hours.
After you perform a series of calculations or gather data, you may need or want to visualize your information so it is easier to make sense of. It is usually easier to make sense of a graph or chart than study a long list of numbers. Visual presentations and summaries can make certain trends or patterns apparent and clear. Long lists of numbers can hide these data trends.
For example, we call this image a bar graph. This one shows the results of a survey your aunt asked the customers at her coffee shop to complete. She wanted to know what flavors she should offer next month to increase her sales, so she asked them to choose their favorite latte flavor. The image is pretty clear, even from a distance: you do not need to sort through a long list of numbers to see that pumpkin spice is the most popular flavor.
We use graphs and charts in almost every field, not just with coffee shop customers. Politicians use them to explain demographics and voting trends in campaigns and elections. Businesses use graphs and charts to show growth trends and areas where they are not doing as well.
It is worth noting that unscrupulous politicians and businesses can, and often do, misuse or distort their performance charts so they display information in a way that supports their interests or goals. For example, suppose a chai tea salesman conducted your aunt's coffee shop survey and used this image to present the data results:
Their presentation significantly reduces the visual gap between pumpkin spice and chai tea lattes. They are trying to convince your aunt to buy more chai tea ingredients, leaving her with less money to buy more pumpkin spice flavoring.
Because charts are so ubiquitous and because they are so easy to manipulate, it is important to know how to read and interpret graphs and charts. Learning this skill will also help you create your own visual presentations. In this unit, we discuss how to use different types of graphs and charts in mathematics. We will interpret the results for each type of graph or chart and learn to create them. We will also explore topics related to long lists of numbers (or data sets), including the measures of central tendency.
Completing this unit should take you approximately 3 hours.
This study guide will help you get ready for the final exam. It discusses the key topics in each unit, walks through the learning outcomes, and lists important vocabulary. It is not meant to replace the course materials!
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Certificate Final Exam
Take this exam if you want to earn a free Course Completion Certificate.
To receive a free Course Completion Certificate, you will need to earn a grade of 70% or higher on this final exam. Your grade for the exam will be calculated as soon as you complete it. If you do not pass the exam on your first try, you can take it again as many times as you want, with a 7-day waiting period between each attempt.
Once you pass this final exam, you will be awarded a free Course Completion Certificate.
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