TX PACT: Essential Academic Skills 700
(Subtest III: Mathematics - 703)
Real numbers
are numbers that have a specific value. Real numbers can be rational or irrational. Within rational
numbers, there are integer, whole, and natural numbers.
Rational numbers
are numbers that can be expressed as a ratio (fraction or comparison) of two integers, a/b where b ≠ 0
(because zero can't be the denominator of a fraction). All rational numbers can be written exactly in both
fraction and decimal forms, and can be graphed on a number line. Note that exact is a formal term in
mathematics meaning a value that has not been rounded or approximated.
rational numbers
ex. mixed numbers - like 4½ which have both a whole number part, like the "4," and a fractional part, like
the "½"
ex. decimal numbers that end, such as 0.25
ex. decimal numbers that repeat the same pattern forever, such as 0.727272... (also written 0.\
overline{72}0.72), which is 8/11 when written in fraction form
Natural Numbers
real numbers that can be pointed to in nature - one tree, two people, three dogs. This set is also known
as the counting numbers; it starts at 1 and goes up by 1s in the same way that children first learn to
count, yielding the infinitely large set of values 1, 2, 3, 4, 5,...
Whole Numbers
the set of natural numbers and zero; it starts at 0 and goes up by 1s to create the infinite set 0, 1, 2, 3,
4,...
Integers
positive and negative counting numbers and zero; the infinite set ...-3, -2, -1, 0, 1, 2, 3,...
Irrational Numbers
, All include a symbol such as square root or pi
ex. must be written a symbol or radical.
real numbers that CANNOT be represented exactly as a ratio of two integers.
In order to be expressed exactly (without rounding), irrational numbers are often written with symbols
(like π, e, or \sqrt{2}2).
When irrational numbers are written using only ordinary numerals, they must be rounded.
Irrational numbers, by definition, cannot be written in fraction form, and when expressed as decimals,
they have infinitely many, non-repeating digits.
Irrational numbers can be approximated (π \approx 3.14≈3.14 \approx≈ 22/7; e \approx 2.72≈2.72, and \
sqrt{2} \approx 1.412≈1.41) and then positioned on a number line.
Real numbers
can be represented in a variety of ways, with the most common being fractions, percents, and decimals.
Fractions
usually represent partial numbers but can represent whole numbers.
½ means one half of one unit
4/1 means 4 of one unit
The top part of a fraction is referred to as the fraction's numerator. The bottom of a fraction is its
denominator. You can remember this with the mnemonic "denominator is down." The fraction bar itself
should be understood to be a division symbol.
Percentages
usually represent part of a whole but can represent larger values.
40% means 4 out of 10 parts of 1 unit or 40 out of 100 units.
200% means twice as much as 1 unit
Decimals
are numbers less than 1 displayed using place values and the powers of ten. They are usually first
introduced as decimal fractions, which are fractions with denominators that are powers of 10.
For example, 1/10 is shown to be equivalent to 0.1.
When turning a fraction into a decimal, be sure to divide in the correct order:
(Subtest III: Mathematics - 703)
Real numbers
are numbers that have a specific value. Real numbers can be rational or irrational. Within rational
numbers, there are integer, whole, and natural numbers.
Rational numbers
are numbers that can be expressed as a ratio (fraction or comparison) of two integers, a/b where b ≠ 0
(because zero can't be the denominator of a fraction). All rational numbers can be written exactly in both
fraction and decimal forms, and can be graphed on a number line. Note that exact is a formal term in
mathematics meaning a value that has not been rounded or approximated.
rational numbers
ex. mixed numbers - like 4½ which have both a whole number part, like the "4," and a fractional part, like
the "½"
ex. decimal numbers that end, such as 0.25
ex. decimal numbers that repeat the same pattern forever, such as 0.727272... (also written 0.\
overline{72}0.72), which is 8/11 when written in fraction form
Natural Numbers
real numbers that can be pointed to in nature - one tree, two people, three dogs. This set is also known
as the counting numbers; it starts at 1 and goes up by 1s in the same way that children first learn to
count, yielding the infinitely large set of values 1, 2, 3, 4, 5,...
Whole Numbers
the set of natural numbers and zero; it starts at 0 and goes up by 1s to create the infinite set 0, 1, 2, 3,
4,...
Integers
positive and negative counting numbers and zero; the infinite set ...-3, -2, -1, 0, 1, 2, 3,...
Irrational Numbers
, All include a symbol such as square root or pi
ex. must be written a symbol or radical.
real numbers that CANNOT be represented exactly as a ratio of two integers.
In order to be expressed exactly (without rounding), irrational numbers are often written with symbols
(like π, e, or \sqrt{2}2).
When irrational numbers are written using only ordinary numerals, they must be rounded.
Irrational numbers, by definition, cannot be written in fraction form, and when expressed as decimals,
they have infinitely many, non-repeating digits.
Irrational numbers can be approximated (π \approx 3.14≈3.14 \approx≈ 22/7; e \approx 2.72≈2.72, and \
sqrt{2} \approx 1.412≈1.41) and then positioned on a number line.
Real numbers
can be represented in a variety of ways, with the most common being fractions, percents, and decimals.
Fractions
usually represent partial numbers but can represent whole numbers.
½ means one half of one unit
4/1 means 4 of one unit
The top part of a fraction is referred to as the fraction's numerator. The bottom of a fraction is its
denominator. You can remember this with the mnemonic "denominator is down." The fraction bar itself
should be understood to be a division symbol.
Percentages
usually represent part of a whole but can represent larger values.
40% means 4 out of 10 parts of 1 unit or 40 out of 100 units.
200% means twice as much as 1 unit
Decimals
are numbers less than 1 displayed using place values and the powers of ten. They are usually first
introduced as decimal fractions, which are fractions with denominators that are powers of 10.
For example, 1/10 is shown to be equivalent to 0.1.
When turning a fraction into a decimal, be sure to divide in the correct order:
