Download Product: A number that is the result of multiplication

Rational Number: A number that can be written as a/b where a and b are integers, but b is not
equal to 0.
There are many different ways we can classify numbers. In fact, many of the classifications
are subsets of other classifications. The following are depictions of how the classifications
relate.
Natural
Numbers
Examples
What's new
1, 2, 3, 4,...
Whole
Numbers
0, 1, 2, 3, 4,...
0
Integers
Rationals
Irrationals
…, -3, -2, -1, -1/4, 6/5, 0.125,
, 2.345678...
0, 1, 2, 3, ...
-3.272727
Fractions,
Opposites of
repeating and non-terminating
natural
terminating
decimals
numbers
decimals
Together the rationals and irrationals form a larger set of numbers called the Real Numbers
Fraction: A number that can be written as a quotient of two quantities.
Different Meanings of Fractions and Ratios
Numbers such as
can have different meanings depending on the context
in which whether they represent fractions or ratios. The following are three possible
meanings for fractions and/or ratios.
Part-Part
In this meaning, a ratio represents a relationship between one part of a whole and
another part of the whole. For example, the ratio of eighth graders at Smith Middle
School who are in the school band (40) to those who are not (110) can be represented
by the part-part ratio 40/110.
Part-Whole
In this meaning, fraction represents a relationship between a part and its whole. The
numerator indicates the number of parts of the whole to be considered. For example, if a
cake is cut into 12 equal pieces and 5 pieces remain, we can represent this situation
using the part-whole meaning of fraction as 5/12. Using the example about band
members at Smith Middle School, we can represent the number of bands members out
of the total number of eighth graders at the school using a part-whole representation:
40/150.
Whole-Part
In this meaning, a ratio represents a relationship between a part and its whole. For
example, the ratio of all eighth graders at Smith Middle School to those eighth graders
in the school band could be represented by the whole-part ratio: 150/40. Using the cake
example, we could represent that whole-part ratio of the number of pieces originally cut
to the number of pieces that were left over as 12/5.
Numerator: The term above the line in a fraction. The numerator tells how many parts
are being talked about or considered.
In the following example, the denominator, 3, indicates the number of equivalent pieces
that a whole (e.g., a whole circle, a whole candy bar, a whole group of people) is
divided into. The numerator, 2, indicates the number of these pieces to be considered
(i.e., the part).
Denominator: The number below the line in a fraction. The denominator indicates what
kind or size of parts the numerator counts.
Numerator & Denominator
A "fraction" is used to indicate the number of parts of a whole. The denominator of the
fraction indicates the number of equivalent pieces the whole is divided into. The
numerator indicates the number of these pieces to be considered (i.e., the part).
3/4: How many units of size 4 can I break 3 into?
4/2: How many units of size 2 can I break 4 into?
5/2.5: How many units of size 2.5 can I break 5 into?
Denominators
What is the common denominator for the fractions below?
2
5
and 3
4
First, find common multiples of the denominators, 5 and 4.
denominator
multiples
5
5
10
15
20
25
30
35
40
4
4
8
12
16
20
24
28
32
The common multiples of 4 and 5 are 20, 40, 60, . . . So, 20, 40, 60, . . . are candidates
for a common denominator of our fractions. How do we convert the fractions to
fractions with common denominators? For example, let's make the common
denominator equal to 20. Since 20 is the fourth multiple of 5, we need to multiply 2/5
by 4/4 to get 8/20. Since 20 is the fifth multiple of 5, we need to multiply 3/4 by 5/5 to
get 15/20.
Why do we multiply by 2/5 by 4/4? To ensure that 2/5 and 8/20 measure the same
amount - that is, that they are equivalent fractions. 2/5 can be represented by
and 8/20 can be represented by
Also, note that 2/5 = 0.4 and 8/20 = 0.4; since 2/5 and 8/20 have the same decimal form,
they are equivalent.
Why do we care about common denominators? Because fractions with common
denominators (called like fractions) can easily be added and subtracted.
For example,
2
+ 3
=
5
4
8
20
and 15
20
=
23
20
or graphically,
+
=
+
=
In the example above, any of the common multiples could have been used. 20 was used
because it's the smallest multiple, or the least (smallest) common denominator.
A non-example of equivalent fractions. 0.3333... and 0.3 are not equal, so 1/3 and 3/10
are not equivalent fractions.
Equivalent Fractions: Fractions that have the same decimal form.
For example, 1/2 = 12/24 since both are equal to 0.5 in decimal form.
Improper Fraction: A fraction where the numerator is equal to or larger than the
denominator.
Fractions such as 4/3 and 6/5 are improper fractions.
Mixed Number: A fraction that contains both a whole number and a fraction.
The fractions 2 1/2 and 3 1/4 are both examples of mixed fractions. They are also
frequently referred to as mixed numbers or mixed numerals.
Simplify: To use the rules of arithmetic and algebra to rewrite an expression as simply as
possible.
Often in mathematics, we are asked to "simplify expressions." In this case, we can
combine like terms, etc., to write the expression in simplest form.
For example, simplify 3x + 4y + 2 - x + 2y - 8. In this case, we can combine like terms
like the 3x and -x, 4y and 2y, and 2 and -8. Using the commutative and associative laws
of addition, we can rewrite the sentence as (3x - x) + (4y + 2y) + (2 - 8) which
simplifies to 2x + 6y - 6.
Fundamental Theorum of Fractions
An equivalent fraction can be formed by multiplying or dividing both the
Numerator and denominator by the same number.
Fraction Bar
The name of the line that separates the divisor from the dividend in a standard
division question.