Basic Terms and Terminology Relating to Different Types of Numbers and Algebra

  • Real numbers: 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: Any number that can be expressed as a fraction or as a ratio. These 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.
  • Positive numbers: Numbers equal to and greater than zero.
  • Negative numbers: Numbers less than zero.
  • Integers: All positive and negative numbers including zero but NOT fractions or decimals.
  • Even numbers: All positive and negative numbers including zero that are evenly divisible by 2.
  • Odd numbers: All positive and negative numbers not including zero that are NOT evenly divisible by 2.
  • Arabic numerals: The numbers that we use every day
  • Roman numerals: Alternative numbers sometimes used to express the year on the calendar and in pharmacology.
  • Square: The product of a number multiplied by itself
  • Square root: The number that when multiplied by itself gives you the original number
  • Perfect square: The square root is a whole number without any decimal points

The Different Types of Numbers

Most people think of numbers as simply the numbers that they recite when they are counting. Although the numbers that are recited when counting are one type of number, there are several other kinds of numbers.

The different types of numbers are:

  • Real numbers
  • Rational numbers
  • Irrational numbers
  • Positive Numbers
  • Negative Numbers
  • Integers
  • Even Numbers
  • Odd Numbers
  • Arabic numerals, or numbers
  • Roman numerals

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 (-)

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:

  • 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:

  • 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:

  • 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.

  • 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:

  • 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:

  • 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:

  • 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:

  • 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
  • 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:

  • 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.

  • 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

  • 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:

  • 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:

  • 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

Fractions, like decimals, are less than a whole and less than one. Numbers less than one can be expressed as a fraction or a decimal.

There are two types of fractions:

  • Proper fractions
  • Improper fractions

Proper Fractions

Proper fractions are less than 1; and improper fractions are more than 1. All fractions have a number above the "divide line" ( / or ----) and a number under the "divide line" ( / or ----).

As shown above, fractions consist of two numbers, a numerator and a denominator. The numerator is the top number of the fraction. In the proper fraction ¼, the numerator is 1. The numerator tells us how many equal parts of the whole there are under consideration. Similarly, in the proper fraction ¼, the denominator is 4. The denominator tells us how many total equal parts there are under consideration. The fraction ¼ indicates, therefore, that one part of 4 equal parts is under consideration.

Here are some examples of proper fractions, all of which are less than 1:

  • 1/2: where 1 is the numerator and 2 is the denominator
  • 2/5: where 2 is the numerator and 5 is the denominator
  • 88/345: where 88 is the numerator and 345 is the denominator

Improper Fractions

Here are some examples of improper fraction, all of which are more than 1:

  • 5/3: where 5 is the numerator and 3 is the denominator
  • 19/4: where 19 is the numerator and 4 is the denominator
  • 564/324: where 564 is the numerator and 324 is the denominator

Did you notice that the numerators in the three above proper fractions are less than the denominators? All of these fractions are less than 1. For example, the 2/5 fraction represents that there are 5 parts in the whole and you have only 2 of the 5 equal parts of the whole which is less than a whole, or less than 1.

Did you also notice that the numerators in the three above improper fractions are more than the denominators? For example, the 5/3 fraction represents that there are 3 parts in the whole and you have 5 parts, which is more than the whole, or greater than 1.

When the numerator and the denominator are identical, the fraction is equal to 1. For example, the fraction 6/6 is equal to 1 and the fraction 987/987 is equal to 1. The numerators and denominators are identical. Of the 6 parts in the whole, you have all 6 parts; and, of the 987 parts in the whole, you have all of the 6 parts and you also have all of the 987 parts, therefore, it is a whole or 1 for both of these fractions.

The denominator is the bottom number in the fraction. Above, the denominator is 2 and this denominator tells us how many total equal parts there are in the whole. In this case, there are only 2 equal parts in the whole.

The fraction 5/8 has 5 as the numerator and 8 as the denominator. The numerator of 5 indicates that there are 5 equal parts in the whole that are under consideration and the denominator of 8 indicates that there are a total of 8 equal parts in the whole. Similarly, the fraction 8/10 has 8 as the numerator and 10 as the denominator. The numerator of 8 indicates that there are 8 equal parts in the whole that are under consideration and the denominator of 10 indicates that there are 10 total equal parts in the whole.

Looking at the cake below, determine the fraction that indicates how much of this cake is gone and also determine the fraction that indicates how much of this cake is left.

The fraction that indicates how much of this cake is gone is ¼, since only one piece of the 4 piece pie is gone. The fraction that indicates how much of this cake is left is ¾ since 3 of the original parts of the cake totaling 4 remains after ¼ of the cake is taken.

A fraction such as 3/3, 77/77, 88/88 are all equal to the same number which is 1. When you have 3 equal parts of a whole and the whole has a total of 3 equal parts, you still have the whole or 1. Similarly, when you have 77 equal parts of a whole and the whole has a total of 77 equal parts, you still have the whole or 1; and when you have 88 equal parts of a whole and the whole has a total of 88 equal parts, you still have the whole or 1.

This concept goes even further when you look at each of these fractions with the / indicating division, as was discussed above with division. So, 3 divided by 3 is 1; 77 divided by 77 is 1; and 88 divided by 88 is 1.

  • 1/2: where 1 is the numerator and 2 is the denominator
  • 2/5: where 2 is the numerator and 5 is the denominator
  • 88/345: where 88 is the numerator and 345 is the denominator

When the numerator and the denominator are identical, as the one immediately above, the fraction is equal to 1. For example, the fraction 6/6 is equal to 1 and the fraction 567/567 is equal to 1. The numerators and denominators are identical. Of the 6 parts in the whole, you have all 6 parts; and, of the 567 parts in the whole, you have all 567 parts, therefore, it is a whole or 1.

As you will learn in two of the sections below, fractions can also be represented as decimals, with percentages and ratios.

For example, the fraction ¼ is equivalent to 0.25 or .25, 25% and 1:4 and 2/16 can be expressed as 0.125 or .125, 12.5% or 2:16.

Mixed Numbers

Mixed numbers are a mix of a whole number and a fraction. For example, 2 1/2 teaspoons and 4 5/8 tablespoons are both mixed numbers. As you can see from these two examples, these mixed numbers are more than 1 or more than a whole. For example, 2 1/2 teaspoons is 2 whole teaspoons plus 1/2 a teaspoon and 4 5/8 tablespoons is 4 tablespoons plus 5/8 of a tablespoon.

You have to convert all mixed numbers into improper fractions, which are also more than 1, before you can perform any calculations with them and other fractions or other mixed numbers. Additionally, you will also have to convert all mixed numbers into whole numbers plus decimal places, which are also more than 1, before you can perform any calculations with them and other decimal numbers.

Below are some examples of mixed numbers:

  • 34 6/8
  • 278 6/8
  • 5 6/9
  • 999 13/15

You can change mixed numbers into fractions and into other mixed numbers with reducing the fraction portion of the mixed number. When a mixed number is changed into a fraction, the fraction will always be an improper fraction because mixed numbers are more than one and improper fractions are also more than one.

For example, the mixed number 2 1/2 can be changed into 5/2 and the mixed number 4 5/8 can be changed into 37/8. Neither of these mixed numbers can be changed into another mixed number by reducing the fraction portion of the mixed number because 5 and 2 as well as 37 and 8 have no common denominator that can be used to reduce them to their lowest possible terms.

Changing Mixed Numbers Into Fractions

To change mixed numbers into fractions, you must:

  • Multiple the denominator by the whole number; then
  • Add the numerator to it; and then finally
  • Divide this number by the denominator.

Here are some examples:

2 1/2

Multiple the denominator by the whole number, add the numerator to this product (the answer to the multiplication) and then finally, divide this number by the denominator, as shown be.

2 1/2 = (2 x 2 +1)/2 = 5/2

In the above example, you multiply the denominator of 2 by the whole number of 2 and then add 1, the numerator of the fraction. Finally, you divide by the denominator of 2. So, 2 x 2 = 4; 4 + 1 = 5; and 5 is then placed over the denominator of 2.

Changing Improper Fractions Into Mixed Numbers

The reverse calculation can also be done. For example, you can convert improper fractions into mixed numbers.

As you may see, the improper fraction of 5/2 can be converted back to a mixed number by dividing the numerator of the improper fraction by the denominator of the improper fraction.

5/2 means 5 / 2

5 / 2 = 2 1/2

When you convert, or turn, mixed numbers back into improper fractions and improper fractions back into mixed numbers, you can easily check your mathematical calculations. If your original calculation gives you 5/2 and you convert this back to a mixed number, you should see the original mixed number.

NB: Proper fractions CANNOT be converted into mixed numbers like improper fractions can because proper fractions are always less than 1 and mixed numbers are always more than one.

Here is an example of turning or converting mixed numbers back into improper fractions and improper fractions back into mixed numbers:

2 5/8 = (8 x 2 + 5)/8 = 21/8

Next, check your answer by changing the improper fraction back to the original mixed number.

21/8 means 21 / 8

21 / 8 = 2 5/8

Adding Fractions

Adding fractions is similar to addition of other numbers, as discussed above, with one additional warning and difference. When fractions are added together, the denominators must all be identical. If they are NOT identical before you do addition, you must reduce them to "common terms", as fully discussed below.

Here are some examples of adding fractions that are already in common terms and with identical denominators:

  • 12 1/4 + 32 2/4 = 44 3/4
  • 12 1/18 + 2 2/18 = 14 3/18
  • 11 1/4 + 32 = 44 1/4

The examples above have numbers which have the same identical denominators. This does not always occur. When you are doing calculations with different denominators, it is necessary to find a common denominator to all the different denominators in order to solve these problems. Apples and oranges cannot be added together because they are different; and different denominators cannot be added together because they, too, are different. Only apples and apples, and oranges and oranges can be added together.

Subtraction With Fractions

Similar to the addition of fractions, the subtraction of fractions is calculated by setting up the fractions and the whole numbers in the same manner. Again, the examples below have numbers which have the same identical denominators. This does not always occur. When you are doing calculations with different denominators, it is necessary to find a common denominator to all the different denominators in order to solve these problems.

See the examples below:

  • 9 3/4 - 2 2/4 = 7 1/4
  • 12 1/18 - 2 1/18 = 10
  • 23 2/8 - 2 = 21 2/8

Multiplying With Fractions

The multiplication and division of fractions are a little bit more difficult than adding and subtracting fractions but we are sure that you can master it.

To multiply fractions, you have to multiply the numerators and then multiply the denominators.

Here are some examples:

  • 3/4 x 2/3 = 6/12 In the next section you will learn how to reduce this fraction
  • 1 /2 x 2/3 = 2/6 In the next section you will learn how to reduce this fraction
  • 2/4 x 2/3 = 4/12 In the next section you will learn how to reduce this fraction

Dividing With Fractions

The division of fractions is a bit more difficult than multiplying with fractions. Instead of multiplying the numerators and then multiplying the denominators as is done with the multiplication of fractions, the first numerator is multiplied by the first denominator to get the numerator in the answer and then the denominator of the first fraction is multiplied by the second denominator to get the denominator of the answer.

Below are some examples of the division of fractions and the steps that you have to do in order to perform these calculations accurately:

  • 7/2 / 1/2 = 7 x 2 over 2 x 1 (The numerators and denominators are then crisscross multiplied.) = 14/2 or 14 / 2 = 7
  • 5/7 / 2/10 = 5 x 10 over 7 x 2 (The numerators and denominators were crisscross multiplied.) = 50/14 or 50 or 50 / 14 = 3.57
  • 2/3 / 7/8 = 2 x 8 over 3 x 7 (The numerators and denominators were crisscross multiplied.) = 16/21 or 16 or 16 / 21 = 0.76

An alternative way to divide fractions is to:

  1. Invert and turn the second fraction upside down; this gives you the reciprocal of that fraction.
  2. Multiply the numerator of the first fraction by the numerator of the second inverted, reciprocal fraction.
  3. Multiply the denominator of the first fraction by the denominator of the second inverted, reciprocal fraction.
  4. Solve it as done above and as below.

Here are some examples:

  • 7/2 / 1/2 = 7/2 x 2/1 The first fraction was inverted for its reciprocal = 14/2 = 7
  • 5/7 / 210 = 5/7 x 10/2 = 50/14 or 50 or 50 / 14 = 3.57
  • 2/3 / 7/8 = 2/3 x 8/7 = 16/21 or 16 or 16 / 21 = 0.76

Finding the Common Denominator

Proper and improper fractions can be reduced to their lowest common denominator, which is also referred to as reducing fractions to their lowest terms. Reducing fractions to the lowest terms make them easier to work with.

A common denominator is defined as a number that is a multiple of different denominators. In other words, a common denominator is a number that each and every one of the denominators can be divided into evenly.

For example, when you are asked to find a common denominator for the fractions below, the common denominators for each is:

  • 1/4 and 1/8 : One common denominator is 8 because 4 and 8 can both be evenly divided into 8; another common denominator is 16 because 4 and 8 can both be evenly divided into 16; and still another common denominator is 32 because 4 and 8 can both be evenly divided into 32.Please notice that 32 is the product of 8 and 4; 8 x 4 = 32 and both the 8 and the 4 can be evenly divided into 32 because multiplied by each other the answer is 32.

As you can see in the above example, many fractions have more than one common denominator and ALL fractions have at least one common denominator that is the product of the denominators in the equation. Larger denominators with multiple digits tend to have only one common denominator that is the product of the different denominators in the equation.

  • For example, 2/9 and 5/18: One common denominator is 18 because 9 and 18 can both be evenly divided into 18; and another common denominator is 162 because 9 and 18 can both be evenly divided into 162.

And

  • 2/786 and 3/50: One common denominator is 39,300 because 786 multiplied by 50 is 39,300; and there are no other common denominators because another

When there is more than one common denominator, it is best to work with the least of the all the common denominators, however, it is NOT necessary to find this least common denominator with your first reduction. You can continue to reduce fractions using multiple steps and still find the least of the all the common denominators.

Here is an example of using multiple steps to find the least of the all the common denominators:

60/90 can be reduced by 2 as follows: 30/45 and can be further reduced by 5 as follows: 6/9 and can be reduced by 3 as follows: 2/3

You could have immediately recognized the fact that the fraction 60/90 can be reduced by 30 and get the same reduced fraction of 2/3.

Adding Fractions With Finding Common Denominators

Now, here are some addition of fractions calculations that require that we find a common denominator before adding them together.

1/4 and 1/8:

If 1/4 = 2/8 and 1/8 = 1/8, so 2/8 + 1/8 = 3/8

In the above example, the steps for this calculation are:

  1. Determining the common denominator of 8 because both 8 and 4 can be evenly divided into 8
  2. Dividing the denominator of 4 into 8 which is 2
  3. Then multiplying this 2 by the numerator in this fraction which is 2 x 1 = 2
  4. Using the common denominator of 8 and dividing the second denominator of 8 into the common denominator of 8 which is 1
  5. Then multiplying this 1 by the numerator in this fraction which is 1 x 1 = 1
  6. Adding the fractions together which is 2/8 plus 1/8 is 3/8.

Similarly, you could solve the same calculation of these fractions using a different common denominator of 32 instead of 8 because, again, both 4 and 8 can be divided evenly into 32 because the product of 4 and 8 is 32. Both calculations, using 8 and using 32, will both give the correct answer; however, the larger the numbers, the more difficult the calculations.

If 1/4 = 8/32 and 1/8 = 4/32, then 8/32 + 4/32 = 12/32

The steps for this calculation are:

  1. Determining the common denominator of 32 because both 8 and 4 can be evenly divided into 32
  2. Dividing the denominator of 4 into 32 which is 8
  3. Then multiplying this 8 by the numerator in this fraction which is 8 x 1 = 8
  4. Using the common denominator of 32 and dividing the second denominator of 8 into the common denominator of 32 which is 4.
  5. Then multiplying this 4 by the numerator in this fraction which is 4 x 1 = 4
  6. Adding the fractions together which is 8/32 plus 4/32 is 12/32.

Although both of the two calculations above used different common denominators (8 and 32) and they will come up to different answer, that is 3/8 and 12/32; both of these answers are correct because they are equivalent; however, the larger the numbers, the more difficult the calculations. For this reason, many people prefer to use the LEAST or smallest common denominator rather than just any common denominator and by reducing fractions to make them smaller.

As shown in the example immediately above, the answer of 12/32 can be further brought to its lowest terms by dividing both the numerator of 12 and the denominator of 32 by 4, giving you the fraction of 3/8 which is reduced to its lowest terms.

Reducing Fractions

Another mathematical procedure that can be used to make cumbersome and large fractions more manageable is to reduce the fractions. Reducing fractions entails dividing the numerator and the denominator by the same number to make them smaller. Reducing fractions does not alter the value, it simply makes the fraction smaller and more manageable.

For example, reducing fractions is done in the following manner:

24/48 can be reduced to 12/14 when the numerator and denominator are both divided by 2.

12/24 can be further reduced when the numerator and denominator are both divided by 2 or by 6 as shown below.

12/24 = 6/12 when the numerator and denominator are divided by 2

12/24 = 2/4 when the numerator and denominator are divided by 6. This fraction 2/4 can be further reduced when the numerator and denominator are divided again by 2, giving us the lowest fraction possible of 1/2

Please note that the original fraction of 24/48 could be reduced with 24 which would yield the lowest fraction possible of 1/2

Fractions that do NOT have a common denominator cannot be reduced.

Other examples of reducing fractions are shown below.

Reducing fractions involves recognizing a number that can be evenly divided into both the numerator and denominator. For example, if the fraction is 3/9, both the numerator (3) and the denominator (9) can be evenly divided by 3 without anything left over. When you reduce 3/9, you divide the numerator of 3 by 3; and then you divide the denominator of 9 by 3, as shown below:

3/9 / 3/3 = 1/3

Likewise, you can reduce 68/124, as shown below:

68/124 / 2/2 = 34/62

66/124 = 34/62 / 2/2 = 17/31

As you can see in the example above, even numbers can be reduced several times by 2. It could have been immediately reduced by 4, as shown below, but you will still get the same number if you just repeatedly divide by 2.

68/124 / 4/4 = 17/31

Decimals

Ten fingers on two hands, the possible starting point of the decimal counting.

It should not surprise you that the numbers we use are based on the number 10 when you know that we have 10 fingers and 10 toes.

Decimals are another way of expressing a proper fraction, an improper fraction, mixed numbers and percentages. Deci means ten (10). All decimals are based on the system of tens or the "power of ten". For example, 0.7 is 7 tenths; 8.13 is 8 and 13 hundredths; likewise, 9.546 is 9 and 546 thousandths.

Whole numbers are also based on the system of tens. The numbers below and their meaning in the system of tens is shown below:

  • 8: 8 ones
  • 23: 2 tens and 3 ones
  • 567: 5 hundreds, 6 tens and 7 ones
  • 1345: thousands, 3 hundreds, 4 tens and 5 ones

Decimals, because they are less than one, are based on the system of tens in terms of:

  • Tenths
  • Hundredths
  • Thousandths
  • Ten thousandths
  • Hundred thousandths

Below is a chart that shows the meaning of decimal places:

Decimal Place(s) After the Decimal Point

Meaning

Example and Equivalent

1st

Tenths

2.3 = 2 and 3 tenths

2nd

Hundredths

21.98 = 21 and 98 hundredths

3rd

Thousandths

0.985 = 985 thousandths

4th

Ten thousandths

2.4444 = 2 and 4,444 ten thousandths

5th

Hundred thousandths

0.77777 = 0.77,777 hundred thousandths

When the decimal point (.) is preceded with a zero (0), the number is less than 1; when there is a whole number before the decimal point, the number is more than one or equal to one. Numbers with decimal points are readily converted into fractions and mixed numbers, as well as percentages which will be covered below.

For example:

  • 2.3 = Two and 3 tenths or 2 3/10
  • 21.98 = 21 and 98 hundredths or 21 98/100

Rounding Off Decimals

Decimal numbers are often rounded off when pharmaceutical calculations are done so be prepared to be able to round numbers off when you take your TEAS examination.

When you have to round off to the nearest hundredth, you must look at the next number, or thousandths, and determine if it is less than 5, equal to 5 or more than 5. In the examples below, the tenth place will be highlighted with green, the hundredth place will be highlighted in pink, the thousandths place will be highlighted in yellow and the ten thousandths place will be highlighted in turquoise so you can readily see and distinguish among these decimal places.

The number 45.7589 is 45 and 7589 ten thousandths.

This number rounded off to the nearest thousandth is 45.759 because the number in the ten thousandth place is more than 5. You round off a number to the nearest thousandth by looking at the number in the next place, which is the ten thousandth place. If this number is five or more, you would increase the 8 in the thousandth place to 9. If the number in the ten thousandth place had been less than 5, this number would be rounded to the nearest thousandth as 45.758; the 8 would not be increased by one when the number in the ten thousandth place is less than 5.

Now, here are some numbers rounded off to the nearest tenth. Remember, if the hundredth place, or 2nd number after the decimal, is 5 or more, the tenth place is increased by 1; and if the 2nd number after the decimal is less than 5, the number in the tenth place remains the same.

Here are some numbers rounded off to the nearest tenth. Please note the pink number in the hundredth place (2 numbers after the decimal) is the one that determines whether or not the number in the tenth place (the first number after the decimal) moves up 1 or remains the same. If it is 5 or more, the number in the tenth place is increased by 1; and if the number in the hundredth place is less than 5, the number in the tenths place remains the same.

  • 3.44 = 3.4
  • 4.01 = 4.0 which is simply written as 4 without a decimal at all
  • 78.09 = 78.1
  • 8.67 = 8.7
  • 22.09 = 22.1
  • 99.22 = 99.2
  • 987.53 = 987.5

Here are some numbers rounded off to the nearest hundredth. Please note the yellow number in the thousandth place (3 numbers after the decimal) is the one that determines whether or not the number in the hundredths place (the second number after the decimal) moves up 1 or remains the same. If it is 5 or more, the number in the hundredths place is increased by 1; and if the number in the thousandth place is less than 5, the number in the hundredths place remains the same.

  • 3.447 = 3.45
  • 4.011 = 4.01
  • 78.099= 78.1
  • 8.675 = 8.68
  • 22.092 = 22.09
  • 99.229 = 99.23
  • 987.532 = 987.53

Here are some numbers rounded off to the nearest thousandth. Please note the turquoise number in the ten thousandth place (4 numbers after the decimal) is the one that determines whether or not the number in the thousandths place (the third number after the decimal) moves up 1 or remains the same. If it is 5 or more, the number in the thousandths place is increased by 1; and if the number in the ten thousandth place is less than 5, the number in the thousandths place remains the same.

  • 3.4478 = 3.448
  • 4.0111 = 4.0111
  • 78.0999= 78.1
  • 8.6754 = 8.675
  • 22.0923 = 22.092
  • 99.2299 = 99.23
  • 987.5328 = 987.533

Adding Decimals

Performing arithmetic calculations with decimals is considerably less difficult and less cumbersome than calculating with fractions.

When adding and subtracting with decimals, it is extremely important that you line up the numbers in an accurate and precise manner. For example, all of the thousandth place numbers must be placed and lined up carefully with one thousandth place number on top of the other without any shifting to the right or the left of the thousandth place in other places reserved for other numbers; all of the hundredth place numbers must be placed and lined up carefully with one hundredth place number on top of the other without any shifting to the right or the left of the hundredth place into other places reserved for other numbers; and all of the tenth decimal place numbers must be placed and lined up carefully with one tenth place number on top of the other without any shifting to the right or the left of the tenth place into other places reserved for other numbers.

Similarly, whole number with ones, tens, hundreds and thousands places must also be lined up accurately in the correct place.

For example, when you are required to calculate the sum of 22.3, 9.987 and 334.21, it must be set up as below:

22.3 + 9.987 + 334.21

You would then simply add all of these numbers together to get the sum, which is the answer to addition calculations.

22.3 + 9.987 + 334.21 = 366.497

Some people prefer to add zeros to hold empty places. Any added zeros that are used to hold places when lining up calculations for the addition and subtraction of decimals do NOT change the value of the number or the correct answer. See an example of this below.

022.300 + 009.987 + 334.210 = 366.497

If, on your TEAS examination, you are asked to round a number like 366.497 off, the question may ask you to round it off to the nearest whole number, the nearest tenth, and the nearest hundredth.

Here are some examples:

Round 366.493 to the nearest whole number.

Answer: 366

  1. 493 rounded off to the nearest whole number is 366 because the number in the tenth place, which is 4, is less than 5.

Round 366.493 to the nearest tenth

Answer: 366.50 or 366.5

366.493 rounded off to the nearest tenth is 366.5 because the number in the hundredth place, which is 9, is more than 5. Because that number in the hundredth place is greater than 5, the number in the tenth place increases by 1 from 4 to 5.

Round 366.493 to the nearest hundredth.

Answer: 366.49

366.493 rounded off to the nearest hundredth is 366.49 because the number in the thousandth place, which is 3, is less than 5. Because that number in the thousandth place is less than 5, the number in the hundredth place remains unchanged as a 9.

Subtraction with Decimals

Subtraction with decimals is similar to the addition of decimals in terms of lining these problems up correct and it is also the same as regular subtraction in term of the calculations that are used.

Here are some examples:

Example 1:

22.3 - 9.987

Zeros are placed in the two places after 3 to enable subtraction without altering the value of the number. 22.3 is the same as 22.300.

22.300 - 9.987 = 12.313

Example 2:

334.21 - 22.30

Again, a zero is placed in the place after the 3 to enable subtraction without altering the value of the number. 22.3 is the same as 22.30.

334.21 - 22.30 = 311.91

Multiplication of Decimals

The multiplication of decimals is relatively simple to do and master. Just remember that the total number of decimals places in the answer must be equal to the total number of decimal places for both of the numbers that you are multiplying with.

For example, if you are multiplying 4.5 x 5.98, the answer will have three decimal places from the left and when you multiply one decimal number with a whole number like 2 x 1.2, the rule is the same and the decimal place moves only one space to the left so the answer is 2.4.

Other than moving the decimal point in the answer to the multiplication calculation, the calculation of these multiplication calculations are the same as regular multiplication that is done with whole numbers without decimal places.

Here are some examples:

23.4 X 21

491.4 and because the total number of decimal places is one, the decimal is moved one place from the right to the left leaving you with the answer of 491.4.

Here are a couple of more examples:

34.01 X 21.1

717.611 and because the total number of decimal places is 3, the decimal is moved 3 places from the right to the left leaving you with the answer of 717.11.

3.941 X 21.11

83.19451 and because the total number of decimal places is 5, the decimal is moved 5 places from the right to the left leaving you with the answer of 83.19451.

Division with Decimals

Similar to multiplication with decimals, the simplest way to divide with decimals is to temporarily ignore them and just to division. When you have calculated the answer or quotient you would then simply move the correct decimal place into the answer.

The dividend is the number in a division calculation that is divided up into equal parts, the divisor is the number that is divided into the dividend and the quotient is the answer to the division calculation.

See this division terminology below:

Quotient /­­­­(Divisor ) Dividend

For example, when you divide 12 by 4, it is set up as follows:

3 (Quotient)/­­­­­­­4 (Divisor) ) 12 (Dividend)

Here are some examples of how to divide with decimals.

Example 1:

Divide 6.6 by 0.6/(0.6 ) 6.6 (Note: There is one decimal place in the divisor. This one decimal place in the divisor will be considered in the final step of calculating division with decimal numbers.

Step 1: Ignore the decimals temporarily and calculate the division without any decimals whatsoever.

11/(6) 66

Step 2: Place the decimal in the answer one place to the right because there was one decimal place in this calculation with the divisor which was 0.6.

Answer: 11/(6) 66 = 11

Divide 6.666 by 0.66

Step 1: Ignore the decimals temporarily

101/(66) 6666

Step 2: Place the decimal in the answer two places to the right because there were a total of three decimal places in this calculation. One was in 0.6 and the other two decimal place were in 6.66.

11.1/(0.6)6.66

Percentages

Percentages are a number with a % sign that represent numbers in comparison to 100. As shown in the picture below, of all of the persons, or 100% of people, using a web browser to access Wikipedia, 20.03 %, or 20.03 people out of every hundred people, used Chrome and 19.26% of all of the persons using a web browser to access Wikipedia, 19.26%, or 19.26 people out of every hundred people, used Firefox, etc.

100% is the whole and it is equal to 1. Percentages less than 100% are less than 1 and percentages more than 100% is more than one.

A pie chart showing the percentage by web browser visiting Wikimedia sites (April 2009 to 2012).

Percentages have equivalents in terms of a fraction, a decimal point number and as a ratio. See the table below for these equivalents:

PERCENTAGE

Equivalent Fraction

Equivalent Decimal Point Number

Equivalent Ratio

10%

1/100

0.10

1 : 10

20%

20/100

0.20

20 : 100

30%

30/100

0.30

30 : 100

55%

55/100

0.55

55 : 100

23%

23/100

0.23

23 : 100

76%

76/100

0.76

76 : 100

125%

125/100

1.25

125 : 100

133%

133/100

1.33

133 : 100

142%

142/100

1.42

142 : 100

125%

125/100

1.25

125 : 100

As you can see in the chart above, percentages less than 100% are less than 1 or the whole; and percentages more than 100% are more than one. For example, 76% is less than 1 or the whole; 76% is 0.76 which is less than 1 or the whole and 76/100 which is a proper fraction and, therefore is also less than 1 or the whole.

On the other hand, percentages more than 100% are more than 1 and more than the whole. 142% is more than one; 142% is 142/100 which is an improper fraction that is more than 1 and 142% is equal to 1.42 which reads one and 42 hundredths which is also more than 1.

Converting Among Fractions, Decimals and Percentages

You may have noticed a pattern in the chart above suggesting how to convert among fractions, decimals and percentages.

The conversion of percentages into fractions is done by simply placing the percentage number over 100. Regardless of whether or not the percentage number is less than or greater than 100, the denominator of the fraction is always 100.

Here are some examples:

  • 12% = 12/100
  • 120% = 120/100
  • 220% = 220/100
  • 2222% = 2222/100

The conversion of percentages into decimal numbers is done by moving the percentage number's decimal place 2 places to the left. As previously discussed, dividing by 100 can be done simply by moving the decimal place two places to the left.

Here are some examples:

  • 12% = 0.12
  • 120% = 1.20
  • 220% = 2.20
  • 2222% = 22

The conversion of percentages into ratios is done by placing the percentage number and then : (colon) and then 100. A ratio is read as 12 is to 100 when you see 12 : 100. There are 12 per hundred. Ratios will be further discussed in a later section below.

Here are some examples:

  • 12% = 12 : 100
  • 120% = 120 : 100
  • 220% = 220 : 100
  • 2222% = 2222 : 100

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