Real number properties are simple mathematical principles which illustrate how real numbers interact when doing certain mathematical operations such as addition, subtraction, multiplication, and division. These properties make it easier for one to simplify equations, solve equations, and do algebra.
However, what are the properties of real numbers and how can one identify them in a mathematical equation? There are a number of essential properties including the closure property, commutative property, associative property, distributive property, identity property, and the inverse property. All these properties are used to show different ways of manipulating and using numbers without changing a mathematical equation.
In this article, we shall discuss the various properties of real numbers with formulas and examples.
What Are Real Numbers?
eal numbers are all those numbers which can be shown on a number line. They consist of positive numbers, negative numbers, zero, fractions, decimals, and irrational numbers.
For example:
- 5
- -8
- 0
- 3/4
- 2.5
- √2
- π
There are two main categories of real numbers: Rational numbers and Irrational numbers. The rational numbers are the ones that can be expressed as fractions, whereas the irrational numbers cannot be expressed as fractions.
The significance of real numbers lies in the fact that their characteristics are used in arithmetic and algebra.
What Are the Properties of Real Numbers?
Properties of real numbers are basically laws that define how real numbers interact with each other when they are added, multiplied, divided, or subtracted.
For instance:
4 + 7 = 7 + 4
Even though the order of the numbers has been changed, the result is still 11. This is an example of commutative property of addition.
Now let’s examine the most widely used properties.
1. Closure Property
The closure property implies that when you perform an operation on two real numbers, you get another real number, providing the operation is well-defined.
Examples:
5 + 6 = 11
The numbers 5 and 6 are real numbers, and so is 11.
The set of real numbers is closed under operations of addition, subtraction, and multiplication. Division operation is also closed for real numbers if we do not divide any real number by zero.
2. Commutative Property
The commutative property means that changing the order of the numbers will not affect the final result.
In case of addition:
a + b = b + a
Example:
3 + 8 = 8 + 3
In case of multiplication:
a × b = b × a
Example:
4 × 5 = 5 × 4
Division and subtraction are not commutative properties.
Example:
8 − 3 ≠ 3 − 8
3. Associative Property
The associative property involves grouping of numbers; it means that regrouping will not affect the result.
For addition:
(a + b) + c = a + (b + c)
Example:
(2 + 3) + 4 = 2 + (3 + 4)
Both are equal to 9.
For multiplication:
(2 × 3) × 4 = 2 × (3 × 4)
Both are equal to 24.
Subtraction and division are not associative.
4. Distributive Property
The distributive property deals with multiplication in terms of addition or subtraction.
The formula is:
a(b + c) = ab + ac
Example:
3(4 + 2)
According to the distributive property:
3 × 4 + 3 × 2 = 12 + 6 = 18
It also proves helpful in problems involving variables.
Example: 5(x + 2) = 5x + 10
5. Identity Property
The identity property states that some specific numbers do not change the value of another number.
There are two types of identity properties.
Additive Identity
Zero is the additive identity.
a + 0 = a
Example: 9 + 0 = 9
Multiplicative Identity
One is the multiplicative identity.
a × 1 = a
Example: 9 × 1 = 9
6. Inverse Property
The inverse property involves two numbers that combine to give an identity value.
Additive Inverse
Opposite of a number added to itself gives zero.
a + (−a) = 0
Example:
7 + (−7) = 0
Hence, the additive inverse of 7 is −7.
Multiplicative Inverse
Product of a nonzero number and its reciprocal is one.
a × 1/a = 1
Example:
5 × 1/5 = 1
Hence, the multiplicative inverse of 5 is 1/5.
How to Find Properties of Real Numbers
Having knowledge about how properties of real numbers can be identified will enable you to determine the property that is illustrated in an equation.
First, check whether the order of numbers has been altered. In case it has been altered, it could mean that the commutative property is illustrated.
Second, check whether there has been a change in the way numbers are grouped. In case the parentheses have shifted while the order is still the same, then it is most likely the associative property.
Third, check whether a number outside the parenthesis is multiplied to all the numbers within the parentheses. In this case, it means that the distributive property is illustrated.
Also, you can check for zeros, one, opposites, or reciprocals.
Examples:
6 + 0 = 6
Additive identity property
8 × 1 = 8
Multiplicative identity property
Properties of Real Numbers: Quick Comparison
| Property | Main Idea | Example |
|---|---|---|
| Closure | Result remains a real number | 3 + 4 = 7 |
| Commutative | Order changes | 2 + 5 = 5 + 2 |
| Associative | Grouping changes | (2 + 3) + 4 = 2 + (3 + 4) |
| Distributive | Multiply across terms | 2(3 + 4) = 6 + 8 |
| Identity | Number remains unchanged | 7 + 0 = 7 |
| Inverse | Produces an identity value | 5 + (−5) = 0 |
Why Are These Properties Important?
Properties of real numbers are important as they form a basis for algebra and other forms of advanced mathematics. Properties allow simplification of expressions, solving of equations, organization of mathematical operations, and determining if the mathematical steps have been done properly.
For instance, using the distributive property of real numbers allows transforming
2(x + 5)
into
2x + 10.
Example: Find the Surface Area of a Cuboid
Length = 8 cm
Width = 5 cm
Height = 3 cm
Then we get:
SA = 2(lw + lh + wh)
= 2[(8 × 5) + (8 × 3) + (5 × 3)]
= 2(40 + 24 + 15)
= 158 cm²
Thus, the surface area of a cuboid is 158 cm².
Recommended Reading: What are Co-prime Numbers?
Conclusion
Real number properties are critical guidelines for grasping the essence of numbers. Keep in mind some of the simplest differences: Commutative is a property that alters order; associative is a property that alters grouping; distributive distributes multiplication; identity does not change a number at all; and inverse results in identity. After these properties become known, solving many problems becomes much easier.
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