Properties of Real Numbers

A real number is a value that represents a quantity along a continuous number line.

• Real numbers can be ordered. 
• The symbol for the set of real numbers is , which is the letter R in the typeface "blackboard bold".

• counting (natural) numbers (){1, 2, 3, ... }
• whole numbers {0, 1, 2, 3, ... }
• integers (){... , -3, -2, -1, 0, 1, 2, 3, ...}
• rational numbers () (such as -½, 6.25, )
•  and irrational numbers

FYI: The series of three dots ( ... ), seen within the sets above, is called an ellipsis.

As used in mathematics, the ellipsis means "and so forth".

• will prove useful when working with equations, functions and formulas in Algebra,
• as they allow for the creation of equivalent expressions which will often aid in solving problems.
• In addition, they can be used to help explain or justify solutions.

1. Distributive Property

• "multiplication distributes across addition"

• 3 • (4 + 5) = 3 • 4 + 3 • 5

a + b = b + a

• "commute = to get up and move to a new location : switch places"

• 3 + 4 = 4 + 3

a • b = b • a

• "commute = to get up and move to a new location: switch places"

• 3 • 4 = 4 • 3

a + (b + c) = (a + b) + c

• "regroup - elements do not physically move, they simply group with a new friend."

• 3 + (4 + 5) = (3 + 4) + 5

a • (b • c) = (a • b) • c

• "regroup - elements do not physically move, they simply group with a new friend."

• 3 • (4 • 5) = (3 • 4) • 5

a + 0 = a

• "the value that returns the input unchanged"

• 4 + 0 = 4

a • 1 = a

• "the value that returns the input unchanged"

• 4 • 1 = 4

a + b is a real number

10 + 5 = 15 (a real number)

"the sum of any two real numbers

is another real number"

a • b is a real number

10 • 5 = 50 (a real number)

"the product of any two real numbers is another real number"

If a = b, then a + c = b + c.

If x = 10,

then x + 3 = 10 + 3

"adding the same value to both sides of an equation will not change the truth value of the equation."

If a = b, then a - c = b - c.

If x = 10,

then x - 3 = 10 - 3

"subtracting the same value from both sides of an equation will not change the truth value of the equation."

If a = b, then a • c = b • c.

If x = 10,

then x • 3 = 10 • 3

"multiplying both sides of an equation by the same value will not change the truth value of the equation."

If a = b, then a / c = b / c, assuming c ≠ 0.

If x = 10,

then x / 3 = 10 / 3

"dividing both sides of an equation by the same non-zero value will not change truth value of the equation."

a = a

12 = 12

"a real number is always

equal to itself"

If a = b, then b = a.

"quantities that are equal can be read forward or backward"

If a = b and b = c,then a = c.

If 2a = 10 and 10 = 4b,

then 2a = 4b.

"if two numbers are equal to the same number, then the two numbers are equal to each other"

Exactly ONE of the following holds:

 a < b, a = b, a > b

If 8 > 6, then 8  6 and

8 is not < 6.

"for two real numbers a and b, a is either equal to b, greater than b, or less than b." (common sense)

a = a

12 = 12

"a real number is always

equal to itself"