Online Math Center

6th-Grade Rational Numbers: Sets of Numbers

By the time young math students have reached the sixth grade, they know that math is not just a simple case of addition and subtraction anymore. Young math learners have started to explore topics that go beyond our basic notions of mathematics. 

 

One such topic that introduces students to new ways of approaching, thinking about, and, ultimately, solving math problems, is the topic of number sets.

 

We’re going to take a look and what exactly number sets are, as well as dig into some of the most common number sets that 6th-grade math students will need to get comfortable with.

 

What Is A Number Set?

Number sets are collections of numbers classified as groups according to the values of their elements. In simple terms, number sets are collections of numbers that are grouped together.

 

These individual numbers are assigned to a group based on their characteristics. They will be grouped together with other numbers that share these key characteristics.

 

What are some of the most common sets of numbers, then? And what are these characteristics that math students must look out for?

Natural Numbers

Natural numbers are the numbers that each and every one of us is familiar with. The set of natural numbers is represented by the letter N, and is written like this:

 

= {1, 2, 3, …}

 

These are the numbers that we count with, which is why they are sometimes known as the counting numbers. This set of numbers begins at 1 and is infinite, continuing forever.

 

The set of natural numbers is said to be closed under addition and multiplication. What this means is that if you add a natural number to another natural number, you will always get a natural number as the answer. Similarly, if you multiply a natural number by another natural number, you will get a natural number as the answer.

Integers

The next set of numbers that young math students will be using frequently is the set known as integers. 

 

The set of integers is quite similar to the set of natural numbers, with some very important differences. The set of integers is denoted by the letter Z, and is written as:

 

ℤ = {…, -3, -2, -1, 0, 1, 2, 3, …}

 

As you can see from the set, it includes all of the positive natural numbers, zero, and all of the negative natural numbers. 

 

Just like the natural numbers, the set of integers is closed under multiplication and division. The set of integers, however, is also closed under subtraction, meaning if you subtract any integer from any other integer, the answer will be an integer.

Rational Numbers

The set of rational numbers contains all of the numbers you can make by dividing one integer by another (but not dividing by zero). In other words, rational numbers are fractions

 

That means, the set of rational numbers includes: 

  • Integers
  • Positive natural numbers
  • Zero
  • Negative natural numbers
  • Fractions

 

The set of rational numbers is represented by the letter Q, and we denote it like this:

 

ℚ = {m/n: m, n are integers}

 

This means that the set of rational numbers is equal to m divided by n, where both m and n are integers. 

 

Introducing Irrational Numbers

Another number set that 6th graders are going to meet – and thoroughly get to know – in math class this year is the set of irrational numbers.

 

The best way to imagine irrational numbers is to imagine the number line. Now, picture the number line if we plotted all of the rational numbers – all integers, natural numbers, zero, and fractions – we would still have many, many points on the number line. 

 

How can we identify these points? These points are what we call irrational numbers.

 

Irrational numbers are those that are neither whole numbers, nor are they fractions. The decimal of irrational numbers never repeats and never ends. 

 

The most famous irrational number is one we’ve looked at quite recently in another article: Pi. As you may remember, pi is a number that helps us identify certain characteristics of circles, such as their circumference, or their area. Pi can be written as π, or it can be written as:

3.14159265358979323846264338327950288419…

 

Don’t let irrational numbers slip out of your mind, as we’ll be looping back around to them for some upcoming articles in the very near future.

Be Rational, Turn to OMC Math Courses

The new school year is well underway. It’s time to make sure your children are staying ahead of the curve and not developing any gaps in their math knowledge. Enroll them today in an OMC Middle School Math Course to make sure they stay on top of number sets and every other topic that will cross their path this school year.


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