Real Numbers

Real numbers are a crucial part of the number system that we use to represent and perform arithmetic operations on quantities in the real world. They include all rational numbers, which are numbers that can be expressed as the ratio of two integers, and irrational numbers, which cannot be expressed as the ratio of two integers and have an infinite number of decimal places. In general, all arithmetic operations (such as addition, subtraction, multiplication, and division) can be performed on real numbers.

The real numbers are also thought of as the "usual" numbers. Just think of them as all the union between the set of rational numbers and the set of irrational numbers. Real numbers can be used to measure a continuous, one-dimensional quantity, such as temperature, duration, or distance. By continuous, we mean that pairs of values can have arbitrarily small differences. Every real number can be almost uniquely represented by an infinite decimal expansion.

There are infinitely many real numbers, just as there are infinitely many rational or irrational numbers, but it can be proved that the infinity of real numbers is a bigger infinity.

This Venn diagram shows the relationships between various sets of numbers, including whole numbers, integers, rational numbers, and irrational numbers, all of which fall into the category of real numbers.

The set of real numbers includes various categories, such as natural and whole numbers, integers, and rational and irrational numbers. In the table below, all the real number formulas, or the representation of the classification of real numbers, are defined with examples.

The commutative property states that the numbers on which we operate can be moved or swapped from their original position without changing the resulting answer. This property holds true for addition and multiplication of real numbers, but not for subtraction or division.

The commutative property of addition is represented as follows:

$a+b=b+a$ , where a and b are any two real numbers

$5+3=8$ ; $3+5=8$

$2\frac{1}{2}+150=152\frac{1}{2};150+2\frac{1}{2}=152\frac{1}{2}$

$-15+45=30;45+(-15)=30$

The commutative property of multiplication is represented as follows:

$a\times b=b\times a$ , where a and b are any two real numbers that are not zero

$3\times 8=24;8\times 3=24$

$\frac{1}{2}\times 100=50;100\times \frac{1}{2}=50$

$-10\times 12=-120;12\times \left(-10\right)=-120$

With the associative property, when more than two real numbers are added or multiplied, the result remains the same, no matter how the numbers are grouped. The associative property does not hold true for division or subtraction equations.

The associative property of addition:

$\left(x+y\right)+z=x+\left(y+z\right)$

$\left(3+5\right)+16=8+16=24$

$\left(3+5\right)+16=3+21=24$

$\left(-4+12\right)+6=8+6=14$

$-4+\left(12+6\right)=-4+18=14$

The associative property of multiplication:

$p\ast (q\ast r)=(q\ast r)\ast r$

$\left(5\times 4\right)\times 3=20\times 3=60$

$5\times \left(4\times 3\right)=5\times 12=60$

The distributive property explains that operations performed on numbers available in brackets can be distributed for each number outside the bracket. It is one of the most frequently used properties in mathematics. When a factor is multiplied by the sum or addition of two terms, it is essential to multiply each of the two numbers by the factor, and finally perform the addition operation. The property is expressed as:

$a\left(b+c\right)=ab+ac$ , where a, b, and c are three different values

$2\left(4+3\right)=2\times 4+2\times 3=8+6=14$

You can check your work to make sure that this is correct by solving the problem as it is originally written:

$2\left(4+3\right)=2\left(7\right)=14$

Since you get the same solution, you can see that this property works correctly.

There is an additive identity and a multiplicative identity as shown below.

In addition, 0 is the additive identity. The property states that any real number plus 0 is the number itself. $m+0=m$

For multiplication, 1 is the multiplicative identity. The property states that any real number multiplied by 1 is the number itself. $m\ast 0=m$

$46+0=46$

$8347984+0=8347984$

$-19+0=-19$

$\frac{1}{5}+0=\frac{1}{5}$

$8\times 1=8$

$3184\times 1=3184$

$-486\times 1=-486$

$\frac{2}{13}\times 1=\frac{2}{13}$

Base (Number Systems)

Number Systems

Solving Equations

8th Grade Math Flashcards

Common Core: 8th Grade Math Flashcards

MAP 8th Grade Math Practice Tests

8th Grade Math Practice Tests

Understanding number sets, especially the vast amount of numbers that belong in the real numbers set, can be confusing for students. There are a number of properties of real numbers that must be memorized and a number of functions to remember. Working with a private tutor can make these mathematical concepts become much clearer for your student.

A private tutor can spend as much time as needed on concepts that are especially challenging to your student while skimming through the concepts that they understand quickly, making their tutoring time as efficient as it is effective. Private tutors can answer your student's questions right away, so there is no need for your student to muddle through homework assignments without knowing what they are doing or worse, doing the calculations the wrong way. If your student needs help understanding number sets and real numbers, contact the Educational Directors at Varsity Tutors today to learn how to get started.