What are rational numbers and what methods can we use to divide integers?


We can now introduce rational numbers which are definted as the ratio of two integers. Whenwever you hear "ratio" think of division. When you have a ratio you are comparing two thing with each other. That is what the division operator does.

Any rational number (lets call it $r$) is a ratio fo two integers which we will call $a$, and $b$. When we give these numbers names, we have power over them, so we can write, \begin{equation} r = \frac{a}{b}, \quad b \neq 0 \end{equation} Why do we have to specify that $b$ cannot be zero? This is done because dividing by zero is not possible mathematically, and if we donot state it clearly it can cause some confusion.

The integet $b$ can be any integer besides 0. So lets try to set it to the next most simple number, 1. If $b=1$ then $r = \frac{a}{1}$. Any number divided by 1 is itself, so $\frac{a}{1} = a$. This means that $a = r$; moreover, since $a$ is an integer and $r$ is a rational number, this implies that any integer is also a rational number. This is becuase any number is a ratio of itself and 1.

The division operator has a lot of interesting properties. For one it is the opposite of multiplication. For example if I multiply 2 by 3, I get 6. If I want my original number back I can divide 6 by 2 to get 3 or I can divide 6 by 3 to get 2.

The Divison Algorithm

Lets try to divide 11 by 4. Les look at what it fundamentally means to divide two numbers. We can ask, how many groups of fours are made so that it can equal 11? If we try that we get, \begin{align*} 11 - 4 = 7, \quad \text{1 group of four} \\ 7 - 4 = 3, \quad \text{2 groups of four} \\ \end{align*} Three is less than four so the remainder is 3. This style of groping fours gives us our answer, 11/4 = 2 R3. The 3 called the remainder, the 2 is called the quotent, the 4 is the divisor and the 11 is the divident.

We can also divide in the following way, $4 \cdot 2 = 8$, and 8 is less than 11. So 4 groups 11 at least 2 times; however, we do not know if this is the higest groups of four we can get. To check if this is the higest grouping is simple, we can try $4 \cdot 3 = 12$, and 12 is definitely greater than 11, so we know that 2 is the higest grouping of 4. Now we have to add 3 to 8 ($2\cdot 4$) to get 11. We write it like this, \begin{align*} 11 = 2 \cdot 4 + 3 \end{align*}

Generalized Division

We can generalize this division with some symbols. Lets call our divident, $n$; our quotient, $q$; our remainder, $r$; and our divisor, $d$. \begin{equation} n = d \cdot q + r, \quad 0 \leq r < d \end{equation} Practice: Why is it that $r$ is greater than or equal to $0$ and less than the divisor, $d$.

Practice: Solve for the quotent and the remainder,

  1. $n = 54$, $d = 4$
  2. $n = -54$, $d = 4$
  3. $n = 54$, $d = 70$

Practice: Lets say today is tuesday. What day will it be 1 year from now? We will say that Monday = 0, Tuesday = 1, Wednesday = 2, ..., Sunday = 6.

Answer: There are 365 days in 1 year and 7 days in 1 week. Our $n = 365$, $d = 7$, begin{align*} 365 = 7 \cdot 52 + 1 \end{align*} We know that 365-1 = 364 days from now, we will have Tuesday as our day, so in 365 days we will have wednesday (which is our answer).

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