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The Real Number System
Set -- any group of numbers with one or more attributes in common.
Sets are shown by enclosing the numbers in braces, {}.
Elements -- the numbers in a set.
Example: {0, 2, 4, 6} -- The elements of this set are 0, 2, 4, and 6.
Finite set -- each and every element can be listed.
(See example above.)
Infinite set -- the elements go on indefinitely, without limit (indicated by ellipsis points, ...).
Example: The set of all counting numbers {1, 2, 3, 4, ...}.
Subset -- when every element of one set is also an element of another set.
Example: The set of counting numbers is a subset of the set of integers (see chart below).
Some Basic Sets to Know:
Natural or Counting Numbers: {1, 2, 3, 4, . . . }
Whole Numbers: {0, 1, 2, 3, 4, . . .}
Integers: {. . . , -4, -3, -2, -1, 0, 1, 2, 3, 4, . . . }
Real Numbers
Real Numbers are composed of Rational and Irrational numbers.
a
Rational Numbers can be expressed as the ratio of two integers, , where b does not equal zero.
b
Rational numbers can be expressed as either a terminating or repeating decimal.
Irrational Numbers cannot be expressed as either a terminating or repeating decimal.
Examples: 1.45445444544445 . . . , 5 , and  .
Real Numbers
Rational Numbers
Integers
Whole
Numbers
Natural
Numbers
Irrational Numbers
Radicals
The inverse of squaring a number is finding its Square Root.
Since 6 2 = 36, 6 is a square root of 36.
2
However,  6  = 36, too, so -6 is also a square root of 36.
Every positive real number has one positive square root and one negative square root.
Principal Square Root -- k -- the non-negative square root of a number k.
The negative square root is indicated by  k
Radical Symbol (from Latin word for 'root') --
.
Radicand -- the number under the radical.
Radical -- an expression such as
81 .
Product Property of Square Roots
For all real numbers a and b, where a  0 and b  0,
ab  a b
Quotient Property of Square Roots
For all real numbers a and b, where a  0 and b > 0,
a
a

b
b