Real Number Definition

If you’re interested in operations involving real numbers, it is important to understand what a real number is. Real numbers are defined as all numbers that are:

  • Whole numbers and Integers
  • Rational numbers
  • Irrational numbers

Take a look at some of the definitions below.

Whole Number Integer
Definition Positive number that doesn’t have a decimal or fraction Positive or negative whole numbers
Example 0, 2, 30, 160 -4, -90, 0, 3, 88

Now that you know the difference between whole numbers and integers, let’s compare rational and irrational numbers.

Rational Irrational
Definition Positive or negative number that has a decimal or fraction Positive or negative numbers that can’t be written as a decimal or fraction
Example 0.55, -0.88, \frac{4}{5} \pi, \sqrt{2}, \sqrt{3}

Basic Maths Operations

Basic math operations involve the following:

  • Addition
  • Subtraction
  • Multiplication
  • Division

Addition is when two numbers are combined as a new sum.

 

addition_rules

 

Subtraction is when one number is taken away from another.

 

subtraction_rules

 

Multiplication can be thought of a faster way to add. It is when we take a number and add it however many times are specified.

 

multiplication_rules

 

Division is when we see how many times one number can be divided into groups of another.

 

division_rules

 

Operations with Positive and Negative Numbers

There are a couple of rules you should remember with respect to signs. Signs tell us whether a number is positive (+) or negative (-).

When you perform addition or subtraction, signs can change depending on the absolute value of the numbers. Take for example two scenarios:

 

addition_signs_changing

 

Rule Example Absolute Value
A 10 + (-5) = 5 10 > |-5| -> 10 > 5
B 10 + (-11) = -1 10 < |-11| -> 10 < 11

Multiplication and division rules are much easier.

 

multiplication_division_rules

Rule Example
A 4 x 4 = 16
B 3 x -2 = -6
C 4 \div 2 = 2
D 15 \div -3 = -5

Order of Operations

When we have a combination of operations in a single expression, it can be confusing to understand what operation we should do first. This is where the acronym PEMDAS comes in handy.

Description Symbol
P Parenthesis ()
E Exponent x^{2}
M Multiplication x
D Division \div
A Addition +
S Subtraction -

In order to solve this, let’s take a look at an example:

    \[ (5 + 3) \div 2 * 2 - 3 + 1 \]

The steps are below.

Step Operation Why Remaining
1 (5 + 3) = 8 P 8 \div 2 * 2 - 3 + 1
2 2 * 2 = 4 M 8 \div 4 - 3 + 1
3 8 \div 4 = 2 D 2 - 3 + 1
4 3 + 1 = 4 A 2 - 4
5 2 - 4 = -2 S

Commutative Law

The commutative law states that order does not matter in an operation. Note that this only applies to addition and multiplication.

 

Addition

In addition, the order in which you add each number does not matter. This can be written as the following:

 

associative_addition

 

As an example, take a look at the image below.

 

associative_example

 

Multiplication

In multiplication, the order in which you multiply each number does not matter. This can be written as the following:

 

associative_multiplication

 

As an example, take a look at the image below.

 

associative_multiplication_example

Associative Law

The associative law states that how you group numbers that have the same operation does not matter. Like the commutative law, this does not apply to subtraction and division. The associative law only applies when we have the same operation between multiple numbers.

 

Addition

In addition, the order in which you add each number does not matter no matter how you group them. Grouping numbers usually comes in the form of putting parentheses around them. This can be written as the following:

 

commutative_rules

 

As an example, take a look at the image below.

 

commutative_example_addition

Multiplication

In multiplication, the order in which you multiply each number does not matter no matter how you group them. This can be written as the following:

 

commutative_addition

 

As an example, take a look at the image below.

 

commutative_multiplication

Distributive Law

The distributive law can be thought of as a combination of the other laws. It can be applied to:

  • Multiplication
  • Addition
  • Subtraction

In order to understand the distributive law, we can look at the following image:

 

distributive_rules

 

The table below summarizes the steps:

Step 1 You start with an operation between some numbers, here its addition a + b
Step 2 You multiply this operation with a number (a+b) x c
Step 3 You ‘distribute’ the c to every number inside the parenthesis Distribute c to a and b
Step 4 Add the result ac + bc

As an example, take a look at the image below.

 

distributive_example

Multiplicative Identity

A multiplicative identity sounds complicated, but is really just a number that, when multiplied, leaves the original number as it is. For rational numbers, the multiplicative identity is 1.

    \[ b * 1 = b \]

As you can see, when you multiply any rational number by 1, it is simply itself. See an example below.

    \[ 15 * 1 = 15 \]

Multiplicative Inverse

A multiplicative inverse is the reciprocal of a number. To understand what a reciprocal is, take the image below as an example.

 

multiplicative_inverse

As you can see, a reciprocal can be thought of the inverse of a number. Since 3 can be written as \frac{3}{1}, it’s inverse is simply \frac{1}{3}. Notice that when a number and its inverse are multiplied together, they equal to one. Take a look at some examples below.

Number Reciprocal Result
5 \frac{1}{5} 5 * \frac{1}{5} = 1
\frac{1}{10} \frac{10}{1} \frac{1}{10} * \frac{10}{1} = 1
4 \frac{1}{4} 4 * \frac{1}{4} = 1
 
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Danica

Located in Prague and studying to become a Statistician, I enjoy reading, writing, and exploring new places.