Math, the Way it Should Be (Unit 1.3: Numbers, Part 3)

Not related to math, but today is my five year anniversary with my fiancée!  I'm so excited to spend the day with her!


As always, the start of the series is here and the start of the unit is here.  When we left off last, we had just talked about all of the integers.  We had learned to solve algebra problems involving addition, subtraction, multiplication, and some instances of division.  But we ran into a problem with division: sometimes we can’t split an integer number of objects into an integer number of groups, each with the same integer number of objects.  In the example last time, we had twelve objects we wanted to put into 5 groups and we couldn't find an integer that worked.

What’s the solution?  Come up with a new set of numbers.  In this case, we use the rational numbers.  The rational numbers are characterized by the ability to be written as the ratio of two integers.  That is, the answer to our 12 objects in 5 groups is 12/5 or x, the number of objects in each group is 12/5:

This is also the solution to the equation:

5x = 12

This set of numbers looks like fractions, but please don’t start hyperventilating until the end of the lesson.  Some things to take note of about the rational numbers:  

• All of the integers are already rational numbers (think of them being divided by one: 5/1 = 5)  
• The rational numbers are an ordered set (5/3 < 5) 
• The rational numbers are the first group of numbers that are said to be “dense”.

Density is a pretty cool property for a group of numbers to have.  In essence, it means between any two rational numbers, there are an infinite number of rational numbers.  Why am I just bringing this awesome property up now?  I haven’t mentioned it yet because the counting numbers and the integers are not dense.  Let’s look at an example.  If we think about two integers, say 3 and 7, we can see exactly how many integers are between them (we can do this because the integers are an ordered set).  We have 4, 5, and 6.  There are a finite (three) number of integers between 3 and 7.  Now let’s look at the rational numbers between 3 and 7.  We still have 4, 5, and 6, but we also have 7/2, 9/2, 11/2, 13/2, 13/3, 14/3, 16/3, 17/3, 19/3, 20/3… and the list goes on forever.

Hold off on the hyperventilation for a bit longer while we delve into decimals.  Decimals are a special way of representing numbers and involve the addition of fractions.  Say for instance, we have the number 3.7.  This number in the ones’ place indicates how many whole objects we have, in this case, three.  The next number after the decimal place indicates how many tenths we have (seven).  So this number is the same as 3 + (7/10).  Now we can change the way the number 3 looks by multiplying by a funny-looking version of the number one.  You would agree that if you put 10 objects evenly into 10 groups, you would have one object in each group, right?  So 10/10 is equal to 1.  Now to multiply two fractions, you multiply their top numbers and multiply their bottom numbers like so:

This is a great result!  We have changed the way the number 3 looks without changing its value!  Now we can add 3 and 0.7.  To add fractions, you make sure their bottom number is the same (in this case, 10) and then add their top numbers:

And all of a sudden, decimals are demystified.  We can see from the above result that because 3.7 can be written as the ratio of two integers, it is a rational number.

I personally prefer fractions and like to avoid decimals because sometimes decimals can only reasonably approximate an actual number.  Take 1/3 for example.  There it is in a simple fraction, for all the world to see.  We can imagine exactly what it is: one-third.  But try writing it as a decimal and you get 0.333333333333… and the threes just keep coming!  You can write it, but only with an infinite number of threes.  This is why I typically don’t deal in decimals.  I also avoid percentages because they are stupid: simply take the decimal and multiply by 100.  So I have good news for you, if fractions, decimals, and percentages give you pain: I won’t be dealing in either of the last two things.  Just remember that to add fractions you make the bottoms the same (by multiplying by 1) and then add the tops and to multiply two fractions you multiply the tops and the bottoms.

Now to solve a problem requiring division:

5x = 12

(5x)/5 = 12/5

1x = 12/5

x = 12/5

Using rational numbers, we can now solve any multiplication or division problem.  The next problem we need to solve looks like this: x² = 5.  That requires two new types of numbers, but after that, we’re done and will be moving on!  One other thing to remember about multiplying and dividing with negative numbers: if you combine two negative numbers or two positive numbers, you get a positive number.  If you combine one of each, you get a negative number.

Homework:

• Practice adding and multiplying fractions.         
• I should have said this before, but throw away your calculator.  It is a crutch that makes you rely on decimals.     
• Forget about degrees.  The better you do this, the better future lessons will go.  Start now, though, because it will be harder than you think.

-Lane