Properties
1. Closure:
The result of adding two real numbers is another real number.
A + b 
+ 
The way in which the summands are grouped does not change the result.
(a + b) + c = + (b + c)
The order of the addends does not change the sum.
a + b = b + a
The 0 is the neutral element in the addition because every number added to it gives the same number.
a + 0 = a
+ 0 =
Two numbers are opposites if they are added together and the result is zero.
a + (−a)= 0
e − e = 0
The opposite of the opposite of a number is equal to the same number.
−(−) =
) = Subtracting Real Numbers
The difference of two real numbers is defined as
the sum of the minuend plus the opposite of the subtrahend.
a − b = a + (−b)
Multipying Real Numbers
The rule of signs for the product of integers and rational numbers is still maintained with the real numbers.
Properties
1. Closure:
The result of multiplying two real numbers is another real number.
a · b
2. Associative:
The way in which the factors are grouped does not change the result.
(a · b) · c = a · (b · c)
(e · ) · = e · ( · )
3. Commutative: ) · = e · ( · )
The order of factors does not change the product.
a · b = b · a
The 1 is the neutral element of the multiplication because any number multiplied by it gives the same number.
a · 1 = a
· 1 =
A number is the reciprocal of another if when multiplied by each other, the product is the muliplicative identity.
The product of a number for a sum is equal to the sum of the products of this number for each of the addends.
a · (b + c) = a · b + a · c
· (e + ) = · e + ·
It is the reverse of the distributive property.
a · b + a · c = a · (b + c)
· e + · = · (e + ) Dividing Real Numbers
The division of two real numbers is defined as the product of the dividend by the reciprocal of the divisor.
Powers with an Integer Exponent
Powers with a Fractional Exponent
Properties
1, a0 = 1
2 a1 = a
3. Multiplication of powers with the same base:
It is another power with the same base and whose exponent is the sum of the exponents.
am · a n = am+n
(−2)5 · (−2)2 = (−2)5+2 = (−2)7 = −128
4. Division of powers with the same base:
It is another power with the same base and the exponent is the difference between the exponents.
am : a n = am — n
(−2)5 : (−2)2 = (−2)5 — 2 = (−2)3 = −8
5. Power of a power:
It is another power with the same base and the exponent is the product of the exponents.
(am)n = am · n
[(−2)3]2 = (−2)6 = 64
6. Multiplication of powers with the same exponent:
It is another power with the same exponent, whose base is the product of the bases
an · b n = (a · b) n
(−2)3 · (3)3 = (−6)3 = −216
7. Division of powers with the same exponent :
It is another power with the same exponent, whose base is the quotient of the bases.
an : b n = (a : b) n
(−6)3 : 33 = (−2)3 = −8
Definition of an Interval
An interval is a to set of real numbers between two given points: a and b, which are called ends of the interval.
Open Interval
An open interval, (a, b), is the set of all real numbers greater than a and smaller than b.
(a, b) = {x / a < x < b}
/ a < x < b} 
Closed Interval
A closed interval, [a, b], is the set of all real numbers greater than or equal to a and less than or equal to b.
[a, b] = {x / a ≤ x ≤ b}
/ a ≤ x ≤ b}
Half-Closed Intervals
(a, b] = {x / a < x ≤ b}
/ a < x ≤ b} 
[a, b) = {x / a ≤ x < b}
/ a ≤ x < b} 
Half-closed intervals are also called half-open intervals.
When there are a set of points formed by two or more of these intervals, the sign 
(Union) is used between them.
(Union) is used between them.
The half-line is determined by a number.
x > a
(a, +∞) = {x / a < x < +∞}
/ a < x < +∞} 
x ≥ a
[a, +∞) = {x / a ≤ x < +∞}
/ a ≤ x < +∞} 
x < a
(-∞, a) = {x / -∞ < x < a}
/ -∞ < x < a} 
x ≤ a
(-∞, a] = {x / -∞ < x ≤ a}
/ -∞ < x ≤ a}
The absolute value of a real number, a, is written as |a|. It is the equivalent number if the number is already positive, but the opposite if the number is negative.
|5| = 5 |-5 |= 5 |0| = 0
|x| = 2 x = −2 x = 2
|x|< 2 − 2 < x < 2 x (−2, 2 )
(−2, 2 )
|x|> 2 x< 2 or x>2 (−∞, 2 ) (2, +∞)
(2, +∞)
|x −2 |< 5 − 5 < x − 2 < 5
− 5 + 2 < x < 5 + 2 − 3 < x < 7
− 5 + 2 < x < 5 + 2 − 3 < x < 7
Properties of the Absolute Value
1 Opposite numbers have equal absolute value.
|a| = |−a|
|5| = |−5| = 5
2 The absolute value of a product is equal to the product of the absolute values of the factors.
|a · b| = |a| ·|b|
|5 · (−2)| = |5| · |(−2)| |− 10| = |5| · |2| 10 = 10
3 The absolute value of a sum is less than or equal to the sum of the absolute values of the addends.
|a + b| ≤ |a| + |b|
|5 + (−2)| ≤ |5| + |(−2)| |3| = |5| + |2| 3 ≤ 7
Distance
The distance between two real numbers a and b, which writes d(a, b), is defined as the absolute value of the difference in both numbers:
d(a, b) = |b − a|
The distance between −5 and 4 is:
d(−5, 4) = |4 − (−5)| = |4 + 5| = |9|
