The Alphabet of Math: The rational numbers

So far, we've only looked at whole numbers. We've had no way to represent parts of numbers or numbers in between the points on a numberline. Rational numbers give us the ability to represent points in between the whole numbers. To understand the points between whole numbers we need to look at the concewpt of infinity.

Infinity is a number that is so large or so small that it can't be represented, but we know it exists. Something can be infinitly large meaning it's so big there isn't a number that can be written to represent it. There are an infinite number of whole numbers because for any whole number you can add 1 to it and come up with another whole number.

Everyone has heard the joke routine of a person counting down to an ultimatum. I'm going to count to 5 and then I'm going to ... Then the person counts; one, two ,three, four, four and one half, four and three quarters... This joke works because there are an infinite number of parts bewteen 4 and 5. Have you ever wondered why the football never crosses the goalline when a team is third and goal and the defense gets charged with a penalty that results in half the distance to the goal? That's because you can always cut the distacne to the goal in half and get closer to the goal, but you never actually cross the goal.

Cutting something in half can;t be represented by whole numbers because half isn't a whole anymore. We use something called rational numbers to represent halves. Rational numbers are also called fractions because they represent fractions of a whole. Rational numbers are ratios of a number to a whole. One half is a ratio of 1 to 2. It is also a fraction of 1 part of 2 parts of a whole.

A rational number is made up of two parts. The number is represented as a number over another number. The number on top, called the numerator, is the part of the whole that we're talking about. The number on the bottom, called the denominator, is the number of parts that the whole is divided into. In the number 1/2, we are talking about 1 part of 2. Two halves make a whole.

The number 1/2 is between 0 and 1 on the numberline.
<——0——½——1——>

If we divide the distance between 0 and 1 into fourths we get this numberline
<——0—¼—½—¾—1——>

Notice how we count up the fourths from 1 to 3. At 1, we would be at 4 fourths. When the numerator equals the denominator, we are at 1 or a whole. If we continued to count fourths passed 1 we would get 5/4, 6/4, 7/4, 8/4.

Notice that 8 is a multiple of 4. 8/4 is the same as 2. Also notice that the gcd of 6 and 4 is 2. If we divide 6 and 4 by 2 we get 3 and 2. 6/4 is the same as 3/2. If the numerator and denominator of a rational number have a gcd, we have to use this gcd to divide the rational number to its reduced form. We never represent rational numbers with a gcd between the numerator and denominator. Rational numbers must always be represented in their reduced form.

Anytime the denominator is 1, the reduced number is a whole number. 4/1 would be 4 wholes or just 4. Anytime you see a rational number where the denomintaor is 1, simply drop the denominator.

The numberline divided into fourths would look like this
<——0—1/4—1/2—3/4—1—5/4—3/2—7/4—2—9/4—5/2—11/4—3——>

Multiplying ratinoal numbers is easy. You simply multiply the numerators and multiply the denominators. Then reduce the rational number by removing the gcd of the numeratoar and denominator, if it exists.

3/2×4/3 = (3×4)/(2×3) = 12/6

The gcd of 12 and 6 is 6, so divide the numerator by 6 and divide the denominator by 6
(12÷6)/(6÷6) = 2/1 = 2

Adding rational numbers requires us to add numbers with common denominators. To get a common denominators, we have to have the lcm of the denominators. To refresh the concept of the lcm look at the blog Division Revisited.Then we change each rational number so it looks different, but isn't really changed. We can do this by multiplying each number by 1. Remember that in multiplication, 1 is the identity meaning that when we multiply by 1 we don't change the value of the number. From above, if the numerator and denominator are the same, the rational number is actually 1. So we are going to multiply each of the rational numbers by a rational number where the numerator and the denominator are the same. This changes the way the rational numbers look but not their actual value. This is confusing, so let's work one out. Let's add 1/2 to 1/3.

First, we need the lcm of 2 and 3. The lcm of 2 and 3 is 6. So we need to convert 1/2 and 1/3 to sixths. If we multiply 2 by 3 we get 6, so we multiply 1/2 by 3/3 to get 3/6. If we multiply 3 by 2 we get 6, so we multiply 1/3 by 2/2 to get 2/6. Now that we have a common denominator, we simply add the numerators.

1/2+1/3 = 3/6+2/6 = (3+2)/6 = 5/6

So to add two rational numbers, we first have to find the lcm of the denominators, multiply both numbers by 1 to get a common denominator, then add the numerators and reduce if needed.

Subtraction works like addition. Get a common denominator then subtract the second numerator from the first.

5/6-1/3 = 5/6 - 2/6 = (5-2)/6 = 3/6 = 1/2

Division works similar to multiplication. But we flip the divisor then we multiply the numerators and the denominators and reduce if necessary.

Rational numbers give us the ability to represent the inverse for multiplication. Remember from addition that the inverse is the number that we add to a number to get the identity which is 0 in addition. In multiplication, the identity is 1, so we want the number that when we multiply we get 1. In multiplication, the inverse is called the reciprocal and we get it by flipping the rational number. The reciprocal of 1/2 is 2/1, and if we multiply 1/2 by 2/1 we get (1×2)/(2×1) = 2/2 = 1. When we multiplied by the reciprocal, we got the identity. So the reciprocal is the inverse.

Rational numbers actually represent a division problem and we'll investigate this further in the next blog.