# Arithmetic properties

Contents

This tutorial will help us make sure we can go deep on arithmetic. We'll explore various ways to represent whole numbers, place value, order of operations, rounding and various other properties of arithmetic.

In this tutorial we'll learn about place value. This is key to better understanding what we're really doing when we count, carry, regroup, multiply, and divide with mult-digit numbers.

If you're looking to create an army of robot dogs, will it really make a difference if you have 10,300 dogs, 9,997 dogs or 10,005 dogs? Probably not. All you really care about is how many dogs you have to, say, the nearest thousand (10,000 dogs).
In this tutorial, we'll learn about conventions for rounding whole numbers. Very useful when we might not need to (or cannot) be completely precise.

Regrouping involves taking value from one place and giving it to another. It is a great way to make sure you understand place value. It is also super useful when subtracting multi-digit numbers (the process is often called "borrowing" even though you never really "pay back" the value taken from one place and given to another).

If you have the expression "3 + 4 x 5", do you add the 3 to the 4 first or multiply the 4 and 5 first? To clear up confusion here, the math world has defined which operation should get priority over others. This is super important. You won't really be able to do any involved math if you don't get this clear. But don't worry, this tutorial has your back.

2 + 3 = 3 + 2, 6 x 4 = 4 x 6. Adding zero to a number does not change the number. Likewise, multiplying a number by 1 does not change it.
You may already know these things from working through other tutorials, but some people (not us) like to give these properties names that sound far more complicated than the property themselves. This tutorial (which we're not a fan of), is here just in case you're asked to identify the "Commutative Law of Multiplication". We believe the important thing isn't the fancy label, but the underlying idea (which isn't that fancy).

The distributive property is an idea that shows up over and over again in mathematics. It is the idea that 5 x (3 + 4) = (5 x 3) + (5 x 4). If that last statement made complete sense, no need to watch this tutorial. If it didn't or you don't know why it's true, then this tutorial might be a good way to pass the time :)

More numbers than you probably imagine can be represented as the ratio of two integers. We call these rational numbers. But there are also really amazing numbers that can't. As you can guess, we call them irrational numbers.