THE DIFFERENT TYPES OF NUMBERS
REAL NUMBERS
Real numbers can be thought of as points on an infinitely long number line.
Real numbers include all numbers except imaginary numbers.
Real numbers include whole numbers, zero, fractions, decimals, rational numbers,
irrational numbers, negative numbers, positive numbers, percentages, percentiles etc.
Real numbers are endless and they go to the point of infinity.
RATIONAL NUMBERS
The rational numbers (ℚ) are included in the real numbers (ℝ), and in turn include the
integers (ℤ), which include the natural numbers (ℕ)
Rational numbers, simply stated, is any number that can be expressed as a fraction or as
a ratio. Rational numbers include whole numbers, zero, positive numbers, negative
numbers, and decimal numbers.
Irrational Numbers: All numbers except rational numbers. An example of an irrational
number is the square root of a number.
Some examples of rational numbers are:
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2
5,000,876,876
1/2
1.098
IRRATIONAL NUMBERS
Set of real numbers (R), which include the rational (Q), which include the integers (Z),
which include the natural numbers (N). The real numbers also include the irrational
(R\Q).
Irrational numbers include all numbers except rational numbers.
The mathematical constant pi (π) is an irrational number that is much represented in
popular culture.
Examples of irrational numbers are:
pi (π) which mathematically is 3.141592. All square roots that do not have a perfect
square root, like the square root of 5, 3, 11, etc.
pi (π), squares and square roots are mathematical values that are used in some
geometric and arithmetic calculations. pi (π) is automatically converted to its
mathematic equivalent which is 3.141592 and rounded off to the nearest hundredth is
3.14
Squares are mathematically calculated by multiplying a number by itself. Below
are examples of how squares are calculated:
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22 which means 2 squared is 2 x 2 which is 4
62 which means 6 squared is 6 x 6 which is 36
1002 which means 100 squared is 100 x 100 which is 10,000
-32 which means negative 3 squared is -3 x -3 which is +9 or 9
+32 which means positive 3 squared is +3 x +3 which is +9 or 9
As you can see in the examples above:
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A positive number x a positive number = a positive number
A negative number x a negative number = a positive number
A positive number x a negative number = a negative number
A negative number x a positive number = a negative number
A square root of a number is the number that when multiplied by itself gives you the
original number.
Some numbers like the ones below are perfect squares because the square root is a
whole number without any decimal points.
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4 The square root is 2 because 2 x 2 = 4.
9 The square root is 3 because 3 x 3 = 9.
16 The square root is 4 because 4 x 4 = 16.
25 The square root is 5 because 5 x 5 = 25.
36 The square root is 6 because 6 x 6 = 36.
49 The square root is 7 because 7 x 7 = 49.
64 The square root is 8 because 8 x 8 = 64.
81 The square root is 9 because 9 x 9 = 81.
100 The square root is 2 because 10 x 100 = 100.
The square roots of other numbers like 3 and 6, for example, have decimal points. For
example, the square root of 3 is 1.73 and the square root of 6 is 2.44.
During the TEAS test you can use a calculator to calculate squares and square roots.
POSITIVE NUMBERS
Positive numbers on the number line below are those numbers equal to and greater
than zero.
NEGATIVE NUMBERS
Negative numbers on the number line above are those numbers less than zero.
INTEGERS
Integers are all positive and negative numbers including zero but NOT fractions or
decimals.
EVEN NUMBERS
Even numbers are all positive and negative numbers including zero that are evenly
divisible by 2.
Some examples of even numbers are:
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2
4
6
-10
0
-222
ODD NUMBERS
Odd numbers are all positive and negative numbers not including zero that are NOT
evenly divisible by 2.
Some examples of odd numbers are:
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1
3
5
-11
-231
ARABIC NUMERALS
Arabic Numerals, or numbers are the numbers that we use every day. All the numbers
above, as described with the different types of numbers, are Arabic numerals or
numbers.
For example, these are the Arabic numbers:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, -4, -345 and -653
ROMAN NUMERALS
Roman numerals are typically not used in our everyday life, however, they are often
used as an alternative way to express the year on the calendar and in pharmacology.
The basic Roman numbers and their Arabic equivalents are:
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I which is Arabic number one (1)
V which is Arabic number five (5)
X which is Arabic number ten (10)
L which is Arabic number fifty (50)
C which is Arabic number one hundred (100)
D which is Arabic number five hundred (500)
M which is Arabic number one thousand (1,000)
Roman numerals, other than those listed above, can be determined with the use of
simple addition or subtraction.
For example, the Arabic number 6 is VI as a Roman numeral. The V is 5 and the I is
number 1. The I or one is added to the V or 5 because the I comes after the V.
On the other hand, the Arabic number IV is 4 because the I or one comes before the V
or 5 and 5 - 1 is 4.
Addition is performed when the Roman numeral starts with the largest number
and then it proceeds with decreasing numbers like the below:
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XVI is equivalent to the Arabic number 16 because these Roman numerals from left
to right are in descending and decreasing order so the addition is 10 + 5 + 1 = 16
XIII is equivalent to the Arabic number 13 because these Roman numerals from left
to right are in descending and decreasing order so the addition is 10 + 1 + 1 + 1 =
13
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DXVI is equivalent to the Arabic number 13 because these Roman numerals from left
to right are in descending and decreasing order so the addition is 500 + 10 + 5 + 1
= 516
Similarly, Arabic numbers can be converted to Roman numerals as follows:
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321 is equivalent to the Roman numeral CCCXXI because C = 100 and 100 +100 +
100 = 300 and 20 is XX and one is I. Note, that the numbers 3, 2 and 1 are in
descending order so you would do addition to arrive at this Roman numeral.
Although years are somewhat more difficult than the examples done above, these
conversions follow the same principles of addition and subtraction.
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The year MCLV is equivalent to the Arabic year 1155 because M is 1000, C is 100, L is
500 and V is 5 and, because all these numbers are in decreasing order, addition is
the operation that you would perform to convert to an Arabic year.
1000 + 100 + 50 + 5 = 1155
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Similarly, you would convert the Arabic year 1025 to the Roman numeral year of
MXXV because M is 1000, and 25 is XXV and, because all these numbers are in
descending or decreasing order, addition is the operation that you would perform to
convert to an Roman numeral year.
1000 + 25 = 1025
When Roman numerals are paired with a number in increasing order, the
operation that is performed is subtraction, so:
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Arabic 4 is equivalent to Roman numeral IV because 5 – 1 = 4
Arabic 9 is equivalent to Roman numeral IX because 10 – 1 = 9
Arabic 40 is equivalent to Roman numeral XL because 50 – 10 = 40
Arabic 90 is equivalent to Roman numeral XC because 100 – 10 = 90
Arabic 400 is equivalent to Roman numeral CD because 500 – 100 = 400
Arabic 900 is equivalent to Roman numeral CM because 1000 – 100 = 900
When a Roman numeral has more than 3 consecutive identical Roman numeral symbols,
subtraction rather than adding 4 numbers is indicated. For example, the Arabic
conversion for 4 is NOT Roman numeral IIII, but instead, it is IV. Similarly, Arabic 96 is
NOT Roman numeral XCIIIIII but instead XCVI.
Subtraction is used to convert Roman numerals to Arabic numbers and Arabic
numbers to Roman numerals, as shown below:
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The Arabic equivalent for the Roman numeral 99 is XCIX because XC is 90 (100 - 10)
and IX is 9 (10 – 1), so 90 + 9 = 99
The Arabic equivalent for the Roman numeral CDXCV is 495 because CD is 400 (500
- 100), XC is 90 (100 – 10), V is 5, so 400 + 90 + 5 = 495
The Arabic equivalent for the Roman numeral 1925 is MCMXXV because M is 1000,
MC is 900 (1000 – 100 = 900), XX is 20 and V is 5. 1000 + 900 + 20 + 5 = 1925
The Roman numeral equivalent for the Arabic number 444 is CDXLIV because CD is
400 (500 – 100), XL is 40 (50 – 10) and IV is 4 (5 – 1), so 400 + 40 + 4 = 444
The Roman numeral equivalent for the Arabic number 1386 is MCCCLXXXVI because
M is 1000, CCC is 300, LX is 50, XXX is 30 and VI is 6, so 1000 + 300 + 50 + 30 + 6 =
1386
The Roman numeral equivalent for the Arabic number 2097 is MMXCVII because
MM is 2000 (1000 + 1000), XC is 90 (100 – 10) and VII is 7 (5 + 2), so 2000 + 90 + 7
= 2097
FRACTIONS: PROPER FRACTIONS AND IMPROPER FRACTIONS AND MIXED NUMBERS
PROPER FRACTIONS
IMPROPER FRACTIONS
MIXED NUMBERS
CHANGING MIXED NUMBERS INTO FRACTIONS
CHANGING IMPROPER FRACTIONS INTO MIXED NUMBERS
ADDING FRACTIONS
SUBTRACTION WITH FRACTIONS
MULTIPLYING WITH FRACTIONS
DIVIDING WITH FRACTIONS
FINDING THE COMMON DENOMINATOR
ADDING FRACTIONS WITH FINDING COMMON DENOMINATORS
REDUCING FRACTIONS
DECIMALS
ROUNDING OFF DECIMALS
ADDING DECIMALS
SUBTRACTION WITH DECIMALS
MULTIPLICATION OF DECIMALS
DIVISION WITH DECIMALS
PERCENTAGES
CONVERTING AMONG FRACTIONS, DECIMALS AND PERCENTAGES
MULTIPLYING BY TEN AND MULTIPLES OF TEN
ALGEBRAIC EQUATIONS
SOLVING ALGEBRAIC EQUATIONS
SETTING UP AND SOLVING ALGEBRAIC EQUATIONS
SIMPLIFIED ALGEBRAIC EQUATIONS
ALGEBRAIC WORD PROBLEMS
BASIC TERMS AND TERMINOLOGY RELATING TO TRANSLATING PHRASES AND SENTENCES INTO
EXPRESSIONS, EQUATIONS AND INEQUALITIES
MATHEMATICAL SYMBOLS
EXPRESSIONS, EQUATIONS AND INEQUALITIES
EXAMPLES OF EXPRESSIONS
EXAMPLES OF EQUATIONS
EXAMPLES OF INEQUALITIES
COMPOSING EXPRESSIONS
COMPOSING EQUATIONS
COMPOSING INEQUALITIES
BASIC TERMS AND TERMINOLOGY RELATING TO SOLVING PROBLEMS INVOLVING RATIOS AND RATES OF
CHANGE
RATES OF CHANGE
CALCULATIONS USING RATIO AND PROPORTION
THE CONVERSION OF PERCENTAGES INTO RATIOS AND CONVERTING RATIOS INTO PERCENTAGES
THE CONVERSION OF FRACTIONS INTO RATIOS AND CONVERTING RATIOS INTO FRACTIONS
BASIC TERMS AND TERMINOLOGY RELATING TO APPLYING ESTIMATION STRATEGIES AND ROUNDING
RULES FOR REAL WORLD PROBLEMS
THE FRONT END ESTIMATION STRATEGY
FRONT END ESTIMATIONS WITH ADDITION
FRONT END ESTIMATIONS WITH ADDITION WITH LARGER NUMBERS
FRONT END ESTIMATIONS WITH THE ADDITION OF EVEN LARGER NUMBERS
FRONT END ESTIMATIONS WITH SUBTRACTION
FRONT END ESTIMATIONS WITH MULTIPLICATION
MEASUREMENT SYSTEM CONVERSIONS TO THE NEAREST ESTIMATION OR APPROXIMATION
ROUNDING RULES FOR REAL WORLD PROBLEMS
THE POSITIONS OF NUMBERS AND THEIR MEANINGS: WHOLE NUMBERS: ONES, TENS, HUNDREDS,
THOUSANDS, ETC
THE POSITIONS OF NUMBERS AND THEIR MEANINGS: DECIMAL PLACES
THE POSITIONS OF NUMBERS AND THEIR MEANINGS: WHOLE NUMBERS WITH DECIMAL PLACES
PERCENTAGES AND THEIR MEANING
CONVERTING AMONG FRACTIONS, DECIMALS, RATIOS AND PERCENTAGES
CALCULATING PROBLEMS INVOLVING PERCENTAGES
BASIC TERMS AND TERMINOLOGY RELATING TO SOLVING ONE OR MULTI STEP PROBLEMS WITH
RATIONAL NUMBERS
REAL NUMBERS
RATIONAL NUMBERS
IRRATIONAL NUMBERS
POSITIVE NUMBERS
NEGATIVE NUMBERS
INTEGERS
EVEN NUMBERS
ODD NUMBERS
ARABIC NUMERALS
ROMAN NUMERALS
FRACTIONS: PROPER FRACTIONS AND IMPROPER FRACTIONS AND MIXED NUMBERS
PROPER FRACTIONS
IMPROPER FRACTIONS
MIXED NUMBERS
CHANGING MIXED NUMBERS INTO FRACTIONS
CHANGING IMPROPER FRACTIONS INTO MIXED NUMBERS
ADDING FRACTIONS
SUBTRACTION WITH FRACTIONS
MULTIPLYING WITH FRACTIONS
DIVIDING WITH FRACTIONS
FINDING THE COMMON DENOMINATOR
ADDING FRACTIONS WITH FINDING COMMON DENOMINATORS
REDUCING FRACTIONS
DECIMALS
ROUNDING OFF DECIMALS
ADDING DECIMALS
SUBTRACTION WITH DECIMALS
MULTIPLICATION OF DECIMALS
DIVISION WITH DECIMALS
PERCENTAGES
CONVERTING AMONG FRACTIONS, DECIMALS AND PERCENTAGES