Without a question, you know should’ve heard about and used division throughout your life. But maybe you just learned the basics to push your way through school and found the topic boring but necessary at the time. However, you’re at college or even started working, and for some reason, you’d like to catch up to at least sound smart and competent in your job interview. And that’s something you’ll need for sure if you go into programming – something more and more people re-qualify and sign up for. Let’s not dwell on it and get to the bottom of the matter – what is division?
Here are some of the most basic things that you should know about the division function in mathematics. If you are into maths, you should be aware of these things.
How division works
It’s actually a basic operating of the arithmetic field of mathematics, next to subtraction, addition, and multiplication. When you’re dividing two numbers, you’re essentially calculating how many times is the second number contained within the first one. Here’s a simple example – you have 15 oranges split into three groups of five. That means the number 15 divided by 3 (groups) gives 5 (oranges). If we had to write it, it-d be either 15 / 3 = 5, 15 ÷ 3 = 5, or by using a line to divide them and placing the bigger number on top.
You can look at a division in two ways – either as a partition or as a quotition. Don’t worry, we’ll explain. If we look at it as a quotition, we’re looking for a number of times 3 has to be added to get to 15. On the other hand, if we’re looking from a partition side, we’re looking at the size of each part that 15 has been divided to if it’s divided into 3 groups. Since division is the opposite of multiplication, the following is correct: a ⋅ b = c means a = c / b, but only if b isn’t 0. In most cases, dividing by zero is undefined because you can’t deduce a from b and c if b=0.
The main terms in use are these: the dividend is divided by the divisor in order to get a quotient. Using the example we’ve given, it means 15 is the dividend, 3 is the divisor, and 5 is the quotient. That brings us to a topic of a remainder because not every division is perfect. Take this as an example, dividing 15 by 2 will give you a result of 7, but with a remainder of 1, because 7 x 2 isn’t 15. We usually add the remainder in notation as a fractional part, either as 7 and 1/2 or 7.5.
Division in other branches of mathematics
The application goes further than numbers, and abstract and physical objects can be included – we can divide real and complex numbers, and even fields and vector spaces. It’s commonly considered division is the hardest and most mentally challenging out of all basic operations. That’s because the set of integers isn’t closed by division. In simple words, multiplying two numbers can’t have a remainder, while dividing two does. That’s exactly why fractions had to be included. However, that’s exactly what gives students ability and confidence to navigate their way through arithmetic and algebra division of variables, matrices, and polynomials. Don’t worry, we’ll remind you of everything in a minute.
Before you start dividing and figuring out the answers, you must know how to recognize this operation. That’s because there are multiple ways division is written in textbooks and in the end, your notebook. Some teachers use one way or the other, but the most common one is with a big line that separates two numbers you’re diving.
This is the most common way – used in your school textbooks and is read as “a divided by b”, “an over b” or even “a by b”.
is the way computer programming languages recognizes division because a slash can be typed without any additional characters. There are other ways to write and are enabled by default in the mathematical software you use on your PC – eitheror
Finally, the form many professionals agree shouldn’t be used today, but important to mention if you encounter it – . The sign is called obelus and has dots to represent numbers. It was introduced way back in 1659, but the one most of us use on our computers when having to write a note in a class is a : b. However, that can bring problems, since it was invented by Leibniz, and mostly used in non-English-speaking countries. The character “:” which is a colon is used to express a ratio, in our case “ratio a to b”.
First of all, the division is right-distributive over addition and subtraction. If you’re not sure what that means – it’s simple:
(a + b) / c = a / c + b / c.
The division is also left-associative, which means if there are multiple division, you should perform the operation from left to right.
a / b / c = (a / b) / c = a / (b ⋅ c) = a ⋅ b^-1 ⋅ c^-1
Division of integers
Like we mentioned, the division isn’t closed, which means the quotient, or the result isn’t an integer unless you can multiply the divisor to get a dividend. Simply put, 25 cannot be divided by 4. However, that can be written in a few ways.
- Just say it cannot be divided. That’s called a partial function.
- Leave the end result as a fraction – 25/4.
- Write an approximation as a decimal fraction 6.25 or 6 and 1/4.
- Write the answer with a remainder. 25/4 = 6 remainder 1
- Leave the integer (whole number) as the result. 25/4 = 6. It’s known as integer division.
Division of rational numbers
Dividing two rational numbers will give you another rational number as a result unless the divisor is 0. two rational numbers, p/q and r/s can be written and calculated like this.
All numbers are integers, and only p can be zero. The proof of this division is by doing a multiplication to confirm it.
Division of real numbers
Dividing two real numbers will give you another real number as a quotient unless divisor is 0. a / b = c iff(if and only if) a = c x b (b =/= 0)
Division by zero
We’ve already explained this, but in case you need a quick reminder, here it is. Division of any chosen number by zero is undefined because any finite result multiplied by zero will always be zero, and not the dividend. If you try this with a calculator, you’ll usually get an error message.
You know the remainders we mentioned above? This is a mathematical formulation that can compute the outcome of the process of division when it comes to integers. If we have two integers, a as the dividend and b as the divisor and b isn’t 0, we will get integers q as the quotient and r as the remainder, and it’s true that a = bq + r and 0 ≤ r < |b| (absolute value of b, always a positive number)
The process of division, both when it comes to multiple branches of mathematics like algebra, arithmetic, and calculus, is irreplaceable. It was, and it still is of utmost importance to us and the world around us, helping us understand the changes in nature, and giving us a hand in many areas of day-to-day life. Just think about it, how would you split students for an art project? Or split the term paper into multiple days and multiple small projects in order to get a result. Finally, cities depend on land division, bands have to split obligations and roles – the examples are endless.