# 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.

## Place value

You've been counting for a while now. It's second nature to go from "9" to "10" or "99" to "100", but what are you really doing when you add another digit? How do we represent so many numbers (really as many as we want) with only 10 number symbols (0, 1, 2, 3, 4, 5, 6, 7, 8, 9)?
In this tutorial you'll learn about place value. This is key to better understanding what you're really doing when you count, carry, regroup, multiply and divide with mult-digit numbers. If you really think about it, it might change your worldview forever!

Finding place value

Sal finds the place value of 3 in 4356.

Writing a number in standard form

Sal writes six hundred forty-five million, five hundred eighty-four thousand, four hundred sixty-two with numbers.

Writing a number in expanded form

Sal writes 14,897 in expanded form.

Intro to place value

Practice thinking about the value of each digit in a number.

Representing numbers

Comparing place values

Sal compares numbers in different place values.

Understanding place value

Sal discusses how a digit in one place represents ten times what it represents in the place to its right.

Place value when multiplying and dividing by 10

Recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right.

Place value relationships example

## Rounding whole 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, you'll learn about conventions for rounding whole numbers. Very useful when you might not need to (or cannot) be completely precise.

Rounding whole numbers 1

Sal rounds 24,259 to the nearest hundred.

Rounding whole numbers 2

Sal rounds 423,275 to the nearest thousand.

Rounding whole numbers 3

Sal rounds 1585 to the nearest ten.

Round whole numbers

Practice rounding whole numbers to the nearest hundred or thousand.

## Understanding whole number representations

Whether with words or numbers, we'll try to understand multiple ways of representing a whole number quantity. We'll even play with place value a good bit to make sure that everything is clicking!

Regrouping numbers into various place values

Sal regroups 4500 different ways.

Comparing whole number place values

Sal compares whole numbers written in expanded form, written form, and number form.

Creating the largest number

Sal arranges digits to make the largest possible number.

Whole number place value challenge

Practice problems to challenge your understanding of whole number place value

## Regrouping whole numbers

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).

Regrouping whole numbers: 675

Sal regroups 675 into various addition problems.

Regrouping whole numbers: 72,830

Sal regroups 72,830 by its place values.

Regrouping whole numbers: 430

Sal regroups 430 five different ways.

Regroup whole numbers

Regroup numbers in each place value (e.g., regroup 1 ten as 10 ones) to express whole numbers in different ways.

## Rational and irrational numbers

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.

Intro to rational & irrational numbers

Learn what rational and irrational numbers are and how to tell them apart.

Classifying numbers: rational & irrational

Given a bunch of numbers, learn how to tell which are rational and which are irrational.

Classify numbers: rational & irrational

Practice identifying whether numbers are rational or irrational.

Approximating irrational number exercise example

Comparing irrational numbers with a calculator

Practice using a calculator to find the approximate decimal values of irrational numbers.

## Order of operations

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.

Intro to order of operations

This example clarifies the purpose of order of operations: to have ONE way to interpret a mathematical statement.

Order of operations example

Work through a challenging order of operations example with only positive numbers.

Order of operations example

Simplify this tricky expression using the order of operations. Expression include negative numbers and exponents.

Order of operations: PEMDAS

Work through another challenging order of operations example with only positive numbers.

Order of operations

Practice evaluating expressions using the order of operations.

Order of operations with negative numbers

Practice evaluating expressions using the order of operations. Numbers used in these problems may be negative.

## The distributive property

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 :)

Distributive property over addition

Learn how to apply the distributive law of multiplication over addition and why it works. This is sometimes just called the distributive law or the distributive property.

Distributive property explained

Distributive property over subtraction

Learn how to apply the distributive property of multiplication over subtraction and why it works. This is sometimes just called the distributive property or distributive law.

Distributive property algebraic expressions

Here we have some algebraic expressions to which we need to apply the distributive property. Now we're beginning to see how useful this property can be!

Distributive property exercise examples

You'll be a pro applying the distributive property once you've solved these exercise examples with us.

Factor with the distributive property

Practice applying the distributive property to factor numerical expressions (no variables).

## Arithmetic properties

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).

Commutative law of addition

Commutative Law of Addition

Commutative property for addition

Commutative Property for Addition

Commutative law of multiplication

Commutative Law of Multiplication

Associative law of addition

Associative Law of Addition

Associative law of multiplication

Associative Law of Multiplication

CA Algebra I: Number properties and absolute value

1-7, number properties and absolute value equations

Properties of numbers 1

Number properties terminology 1

Identity property of 1

Identity Property of 1

Identity property of 1 (second example)

Identity property of 1

Identity property of 0

Identity property of 0

Inverse property of addition

The simple idea that a number plus its negative is 0

Inverse property of multiplication

Simple idea that multiplying by a numbers multiplicative inverse gets you back to one

Properties of numbers 2