Notes

important points :
The Rational Numbers
The rational numbers are those numbers which can be expressed as a ratio between
two integers. For example, the fractions 1/3 and −1111/8 are both rational numbers.
All the integers are included in the rational numbers, since any integer z can be written
as the ratio z/1.
All decimals which terminate are rational numbers (since 8.27 can be written
as 827/100.) Decimals which have a repeating pattern after some point are also
rationals: for example,
0.0833333....= 1/12.
The set of rational numbers is closed under all four basic operations, that is, given any
two rational numbers, their sum, difference, product, and quotient is also a rational
number (as long as we don't divide by 00)
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The Irrational Numbers
An irrational number is a number that cannot be written as a ratio (or fraction). In
decimal form, it never ends or repeats. The ancient Greeks discovered that not all
numbers are rational; there are equations that cannot be solved using ratios of integers.
The first such equation to be studied was 2=x
2
. What number times itself equals 2?

√2 is about 1.414, because 1.4142

=1.999396, which is close to 2. But you'll never hit
exactly by squaring a fraction (or terminating decimal). The square root of 2 is an
irrational number, meaning its decimal equivalent goes on forever, with no repeating
pattern:
√2= 1.41421356237309....
Other famous irrational numbers are the golden ratio, a number with great
importance to biology: (1+√5)/2=1.61803398874989...
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The Real Numbers
The real numbers is the set of numbers containing all of the rational numbers and all
of the irrational numbers. The real numbers are “all the numbers” on the number

line. There are infinitely many real numbers just as there are infinitely many numbers
in each of the other sets of numbers.
Constants and Variables are the two types of symbols in algebra.

Constant:
A symbol which has a fixed numerical value is called a constant.
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Variables:
A quantity which has no fixed value but takes no various numerical values is
called a variable.

For example:
Temperature at different times of a day represents a variable.
The height of a student in your grade is a variable, as it varies from student
to student. A variable is denoted by a letter like x, y, z, u, v etc.
A combination of a constant and a variable is also a variable.
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Intervals
Interval: All the real numbers between two given numbers.
Example: all the numbers between 1 and 6 is an interval
Example: the interval 2 to 4 includes numbers such as:
2.1 2.1111 2.5 2.75 2.80001 π
7
/2 3.7937
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Inequalities
With Inequalities we use:
 > greater than
 ≥ greater than or equal to
 < less than
 ≤ less than or equal to
Like this:
Example: x ≤ 20
Says: "x less than or equal to 20"
And means: up to and including 20
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