# The 14 Properties of Real Numbers

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Has to deal in the learning of proofing equations and understanding how to solve equations

### In the world of Real Number there are 14 properties

each one relates to its own rules and is used when proofing an equation

### Proofing

basically when you proof an equation, you are proving that you have obtained the right answer by completing the right steps in the right order in the right way

the sum of any number and zero is itself; X+0=X

### Example;

if the property states that X+0=X then that means 5+0=5 and 103+0=103

### Multiplicative Identity Property

the product and any number and one is itself; x·1=x

### Example;

if the property states that X·1=X then that means 5·1=5 and 1097473·1=1097473

the sum of two opposite numbers is zero; x+(-x)=0

### Example;

if the property states that x+(-x)=0 then that means 3+(-3)=0 and 75+(-75)=0

### Multiplicative Inverse Property

the product of a number and its reciprocal is one; ¼·4=1 or ½*2=1

### Multiplicative Property of Zero

the product of any number and zero is zero; X·0=0

### Example;

if the property states that X·0=0 then that means 7·0=0 and 13·0=0

### Reflexive Property

anything is equal to itself; X+1=X+1; X=X

### Example;

if the property states that anything is equal to itself then that means 5=5, 5·1=5·1, 7-22=7-22, and 8+12=8+12

### Symmetric Property

if you switch the place of the variable from the left to the right&from the right to the left, the value stays the same; if X=B then B=X

### Example;

if the property states that if X=B then B=X and the value stays the same, that means if 5=X then X=5, and the value stays the same

### Transitive Property

a statement that includes if, and, and then; it takes two true statements and creates another sentence that is assumed to be true because it was formed with the information collected from the first two statements

### Example;

if a=5+c and b=5+c then a=b; we see that the variables [a] and [b] both equal [5+c] so we can assume that [a=b]; if a=5x+b and x=2+3 then a=25+b

### Substitution

simply for any two numbers/variables that are [added/multiplied/divided] and do not apply to any other property then it is substitution; you are substituting two numbers and an operation to a simplified value

### Example;

if you have the equation 5x+(3·4)=62 and you multiply 3 and 4 you get 12, so you substitute it into the equation and it is now 5x+12=62

### Distributive Property

for any number a, b,& c; a(b+c)=(a·b)+(a·c)

### Example;

so applying the property to this equation [5(x+9)] you get this result; 5(x+9)=(5·x)+(5·9) which simplifies to 5x+45

when using addition throughout the whole equation, changing the order of the numbers does not affect the value of the sum; a+b+c=b+a+c

### Example;

5+3+9=3+5+9; so 5+3+9 has the same sum/value as 3+5+9

### Communtative Property of Multiplication

when using multiplication throughout the whole equation, changing the order of the numbers does not affect the value of the product; a·b·c=b·a·c

### Example;

5·2·4=2·5·4; so 5·2·4 has the same product/value as 2·5·4

when using addition throughout the whole equation, changing the grouping does not affect the value of the sum; (a+b)+c=a+(b+c)

### Example;

(5+10)+2=5+(10+2); so (5+10)+2 has the same sum/value as 5+(10+2)

### Associative Property of Multiplication

when using multiplication throughout the whole equation, changing the grouping does not affect the value of the product; (a·b)c=a(b·c)

### Example;

(5·2)·8=5(2·8); so if (5·2)·8 has the same product/value as 5(2·8)

### The Next 4 Properties

Later on in Algebra 1 you will be introduced to 4 new properties; the properties of equality. But those do not apply to these 14 properties of Real Numbers.

Example: