Estimating, Comparing , Converting, and Computing Whole Numbers, Fractions, Decimals, and Percents

Rational numbers are numbers which may be expressed in the form a/b
where, a and b are integers and b ≠ 0. The Set of Rational Numbers includes
both positive and negative rational numbers.

Examples: -3, +2, - 5/6, +2/3, -1.50, +1.3, etc.

Irrational numbers are numbers which may not be expressed in the form
a/b, where a and b are integers and b ≠ 0. They are generally expressed as
radicals, radical fractions , or non-terminating, non- periodic decimals .
Irrational numbers may not be located precisely on a number line, except
geometrically.

Examples: 2 , 3 , 5 3 , 5 2 , etc.

Real numbers are all numbers associated with points on a number line, and
the Set of Real Numbers includes all rational and irrational numbers,
although the irrational numbers may not be located precisely on the number
line, except geometrically.

The Set of Counting Numbers C = {1, 2, 3, 4, …}
The Set of Whole Numbers W = {0, 1, 2, 3, …}
The Set of Integers I = {…, -2, -1, 0, +1, +2,…}

Mutual Assured Destruction Rule

If you add together any number and its negative, they cancel each other out
and the result is zero.

a + (-a) = 0 for any value of a

If you take the negative of a negative number, then the result is a positive
number: -(-5) = 5

Absolute Value of a Number

The absolute value of a number is the distance along a number line from that
number to 0. For example, the absolute value of 0 is 0; the absolute value of
3 is 3; and the absolute value of –5 is 5. Note that the absolute value of any
number (except 0) is positive.

The absolute value of a (written as |a| ) can be found from this rule:
if a ≥ 0, then |a| = ?
if a < 0, then |a| = ?

Adding Positive and Negative Numbers

1) When adding numbers of the same sign, we add their absolute values and
give the result the same sign.

Examples:
2 + 5.7 = 7.7
(-7.3) + (-2.1) = -(7.3 + 2.1) = -9.4

2) When adding numbers of the opposite signs, we take their absolute
values, subtract the smaller from the larger, and give the result the sign
of the number with the larger absolute value.

Example: 7 + (-3.4) = ?

The absolute values of 7 and -3.4 are 7 and 3.4. Subtracting the smaller
from the larger gives 7 - 3.4 = 3.6, and since the larger absolute value
was 7, we give the result the same sign as 7, so 7 + (-3.4) = 3.6.

Example: 8.5 + (-17) = ?

The absolute values of 8.5 and -17 are 8.5 and 17. Subtracting the
smaller from the larger gives 17 - 8.5 = 8.5, and since the larger absolute
value was 17, we give the result the same sign as -17, so 8.5 + (-17) = -8.5.

Subtracting Positive and Negative Numbers

Subtracting a number is the same as adding its opposite.

Examples:
In the following examples, we convert the subtracted number to its opposite
and add the two numbers .

7 - 4.4 = 7 + (-4.4) = 2.6
22.7 - (-5) = 22.7 + (5) = 27.7
-8.9 - 1.7 = -8.9 + (-1.7) = -10.6
-6 - (-100.6) = -6 + (100.6) = 94.6

Multiplying Positive and Negative Numbers

1) To multiply a pair of numbers if both numbers have the same sign, their
product is the product of their absolute values (i.e., their product is
positive).

Example: 0.5 × 3 = 1.5

In the product above, both numbers are positive, so we just take their
product.

Example: (-1.1) × (-5) = |-1.1| × |-5| = 1.1 × 5 = 5.5

In the product above, both numbers are negative, so we take the product
of their absolute values.

2) If the numbers have opposite signs, their product is negative.

Example: (-3) × 0.7 = -2.1

In the product above, the first number is negative and the second is
positive, so we take the product of their absolute values, which is
|-3| × |0.7| = 3 × 0.7 = 2.1, and give this result a negative sign: -2.1,
so (-3) × 0.7 = -2.1

Dividing Positive and Negative Numbers

1) To divide a pair of numbers when both numbers have the same sign, divide
the absolute value of the first number by the absolute value of the
second number, and give the result a positive sign.

Example: 7 ÷ 2 = 3.5

In the division above, both numbers are positive, so we just divide as
usual.

Example: (-2.4) ÷ (-3) = |-2.4| ÷ |-3| = 2.4 ÷ 3 = 0.8

In the division above, both numbers are negative, so we divide the
absolute value of the first by the absolute value of the second.

2) To divide a pair of numbers if both numbers have different signs , divide
the absolute value of the first number by the absolute value of the
second number, and give this result a negative sign.

Example: (-1) ÷ 2.5 = -0.4

In the division above, both numbers have different signs, so we divide
the absolute value of the first number by the absolute value of the
second, which is |-1| ÷ |2.5| = 1 ÷ 2.5 = 0.4, and give this result a negative
sign: -0.4, so (-1) ÷ 2.5 = -0.4.

Fractions and Rational Numbers

Regular Fractions consist of a top number (called the numerator) and a
bottom number (called the denominator).

Examples: 1/2, 1/4, or 2/3

A Proper Fraction is one where the numerator is less than the denominator,
and the value of the fraction is between 0 and 1.

An Improper Fraction is one where the numerator is larger than the
denominator, and the value of the fraction is greater than 1.

Equivalent Fractions are two fractions with the same value. If you multiply
both the numerator and denominator of a fraction by the same number, the
value of the fraction stays the same. However, note that you cannot add the
same number to the top and bottom to obtain equivalent fractions.

Multiplying and Adding Fractions

To multiply two fractions, simply multiply the two numerators and the two
denominators:

Examples:

Adding two fractions is easy if they both have the same bottom number.
Then, just add the two top numbers and keep the bottom number the same.

Examples:
3/8 + 2/8 = (3+2)/8 = 5/8
1/2 + 1/2 = (1+1)/2 = 2/2 = 1
3/5 + 1/5 = (3+1)/5 = 4/5

To add two fractions if they don’t have the same denominator, you first
need to convert them so that they both do have the same denominator. To
do this, use the fact that you can multiply both the numerator and the
denominator by the same number and still keep the same value.

The lowest common denominator (LCD) is the lowest number that can be
divided evenly by all denominators in the problem. Once you have found the
LCD and have converted the fractions so that they have the same
denominator, just add the numerators.

Examples:

Reciprocals, Compound Fractions, and Division

The reciprocal of a fraction is found simply by turning the fraction upside
down (in other words, putting the denominator on top and the numerator on
the bottom).

Examples:
The reciprocal of 2/3 is 3/2
The reciprocal of 1/2 is 2/1
The reciprocal of 5/3 is 3/5

Note: If you multiply a fraction and its reciprocal together, then the result
is 1.

A compound fraction (complex fraction) is a fraction that itself has a
fraction on both the top and the bottom.

Examples:

To simplify a compound fraction (in other words, to convert it into a regular
fraction), multiply the numerator and denominator of the compound fraction
by the reciprocal of the fraction in the denominator:

Example:

In other words, to divide two fractions, multiply the first fraction by the
reciprocal of the second fraction:

Prohibition: No Zero Denominators
It is highly illegal to have a fraction whose denominator is zero!

RULES FOR FRACTIONS

Note: In all rules for fractions, it is automatically assumed that
no denominators are equal to zero!