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Applied Math Basics

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Table of contents

1 Applied Math Basics

2 Basic Math Terminology

2.1 The Decimal System
2.2 The Basic Sets of Numerals

3 Basic Whole Number Operations

3.1 Rounding Whole Numbers
3.2 Adding Whole Numbers
3.3 Subtracting Whole numbers
3.4 Multiplying Whole Numbers

3.4.1 Single number times Single number producing a Single number

3.5 Dividing Whole Numbers
3.6 Factoring Whole Numbers

4 Introduction to Fractions

4.1 Simplifying Fractions
4.2 Improper and Mixed Fractions
4.3 Adding Fractions

4.3.1 Adding Fractions With The Same Denominator
4.3.2 Adding Fractions With Different Denominators

4.4 Subtracting Fractions

4.4.1 Subtracting Fractions With The Same Denominator
4.4.2 Subtracting Fractions With Different Denominators

4.5 Multiplying Fractions
4.6 Dividing Fractions

5 Decimals

5.1 Adding Decimal Numbers
5.2 Subtracting Decimal Numbers
5.3 Converting Fractions to Decimal Numbers
5.4 Multiplying Decimal Numbers
5.5 Dividing Decimal Numbers

6 Measurements

6.1 Converting English Measurements
6.2 The Metric System

6.2.1 Length
6.2.2 Mass
6.2.3 Time
6.2.4 Temperature
6.2.5 Volume

6.3 Metric Length Measurement Conversions
6.4 Measure Metric Quantities
6.5 Converting English and Metric Measurements

7 Trade-based Calculations

8 Glossary

Applied Math Basics

This book covers basic arithmetic concepts including whole numbers, decimals, fractions, and English and metric measurements. Instructional emphasis is on application to typical problems one would encounter in the workplace. Simple and yet effective algorithms are used to solve them. Furthermore you should notice that all digits here involve the base 10 numbering system (note: human beings should naturally have 10 fingers).

Basic Math Terminology

The Decimal System

The decimal system, sometimes referred to as base 10, contains a total of ten identifiers called digits. The decimal system is widely used because humans have ten fingers on which to count. For now, we will disregard other number systems for the sake of simplicity with the understanding that the decimal system is not unique.

The ten digits of the decimal system, arranged from lowest to greatest, are:

0 (zero)
1 (one)
2 (two)
3 (three)
4 (four)
5 (five)
6 (six)
7 (seven)
8 (eight)
9 (nine)

The decimal system uses positional notation to represent numbers larger than 9. This means that a digit's position in relation to other digits affects its meaning. Digits in the furthest right position represent the number of ones being counted, while digits in the second position from the right represent the number of tens. Digits in the third position from the right represent the number of hundreds, and digits in the fourth position from the right represent the number of thousands. This pattern can continue forever; for more information, see orders of magnitude.

For example, the number 535,254 means 5 hundreds of thousands, 3 tens of thousands, 5 thousands, 2 hundreds, 5 tens, and 4 ones. We would say this number as "five hundred thirty-five thousand two hundred fifty-four".

The Basic Sets of Numerals

Counting numbers are the numbers we use every day to count things. Mathematicians sometimes refer to this set of numbers as the Natural Numbers.

$1,2,3,4...\$

Whole numbers include all the counting numbers and zero.

$0,1,2,3,4...\$

Basic Whole Number Operations

Rounding Whole Numbers

Rounding is the process of finding the closest number to a specific value. You round a number up or down based on the last digit you are interested in.

${{down}\atop\overbrace{0,1,2,3,4,}} {{up}\atop\overbrace{5,6,7,8,9}}$

For example, rounding the number 245 to the nearest tens place would round up to 250, while the number 324 rounded to the nearest tens place would be rounded down to 320.

Adding Whole Numbers

First, arrange the numbers in columns so say 134+937

134
+937
___

Add the first column (starting on the right)

  134
+937
     1
     1
Note the 10's digit put under the next column Now add the next column and the number underneath
  134
+937
    71
      1
Finish it off with the other columns
  134
+937
1071
  1  1 So the answer to 134+937 is 1071

Subtracting Whole numbers

To subtract numbers think of a basket of oranges. If you have ten oranges in a basket and you remove eight oranges you are left with two oranges. For example:

 10
- 8
_____
  2

If you have have ten oranges in a basket and you remove all ten then you will no longer have any oranges so you are left with zero oranges. For example:

 10
-10
____
  0

Multiplying Whole Numbers

Single number times Single number producing a Single number

Take the first number as ' $n 1$ ' Take the second number as ' $n 2$ ' Repeat ' $n 1$ ' ' $n 2$ ' times and add the ' $n 1$ 's together

$n_1 = 3 \times 10^0$

$n_2 = 2 \times 10^0$

$(n_1)\times(n_2)= 2(3 \times 10^0) = 6 \times 10^0 = 6$

Dividing Whole Numbers

Dividing whole numbers is the process of representing fractions as decimal numbers (eg. 0, 2.2, 4.55).

$4 \div 2 = 2$

In the above example, 2 goes in to 4 twice; therefore, the answer (or the quotient) would be 2.(for more information about fractions, refer to Introduction to Fractions)

Divisions are often represented as fractions. For example,

$68 \div 43 = \frac{68}{43}$

Factoring Whole Numbers

Factoring is the process of determining what prime numbers when multiplied will give a specific number. This process of factoring is very important in reducing fractions, which is covered in the Fractions chapter of this book. For example:

$4 = 2 \times 2\$

Or a more complicated example:

$180 = 2 \times 2 \times 3 \times 3 \times 5\$

Introduction to Fractions

A fraction is a special way of representing a quantity that is not a whole number.

For example, say one has a pizza and wishes to split it up equally between two people. We would say that each person should get one half of the pizza. One half is represented numerically as:

$\frac{1}{2}$

Relating these two numbers back to the pizza problem:

$\frac{1 \mbox{ pizza}}{2 \mbox{ person}}$

In math terminology, the number 1 is called the numerator while the 2 is called the denominator.

$\frac{numerator}{denominator}$

The line between the numerator and denominator is called the fraction bar. This way of representing fractions is called display representation.

Another way of representing fractions is by using a diagonal line between the numerator and the denominator.

$1/2\$

In this case, the separator between the numerator and the denominator is called a slash, a solidus or a virgule. This method of representing fractions is called in-line representation, meaning that the fraction is lined up with the rest of the text. You will often see in-line representations in texts where the author does not have any way to use display representation.

Simplifying Fractions

Oftentimes when doing operations using fractions, one will obtain a result that looks like this:

$\frac{12}{16}$

While this may be a correct result of your calculations, it can be confusing for some. To avoid this confusion, we reduce the fraction to its lowest terms.

$\frac{12}{16}=\frac{3}{4}$

In order to do this, we have to be able to find the greatest common factor between the numerator and the denominator. We do this by breaking up both the numerator and the denominator into their prime factors:

$\frac{12}{16}=\frac{2 \times 2 \times 3}{2 \times 2 \times 2 \times 2} = \frac{2\!\!\!/ \times 2\!\!\!/ \times 3}{2\!\!\!/ \times 2\!\!\!/ \times 2 \times 2} = \frac {3}{4}$

It is implied that any parts is multiplied by one. If we divide 2s out of the factorized fraction, we are left with one 2 one the denominator.

$\frac{4}{8} = \frac{1 \times 2\!\!\!/ \times 2\!\!\!/}{2 \times 2\!\!\!/ \times 2\!\!\!/} = \frac{1}{2}$

Improper and Mixed Fractions

An improper fraction is a fraction where the numerator is greater than the denominator.

$\frac{4}{3},\frac{124}{23}, \frac{53}{3}$

A mixed fraction is a fraction plus a whole number.

$1\,\!\frac{1}{3}, 5\,\!\frac{9}{23}, 17\,\!\frac{2}{3}$ (This is equal to $\frac{4}{3},\frac{124}{23}, \frac{53}{3}$ above respectively).

Adding Fractions

Adding Fractions With The Same Denominator

In order to add fractions with the same denominator, you only need to add the numerator while keeping the original denominator for the sum.

$\frac{1}{5} + \frac{3}{5} = \frac{1+3}{5} = \frac{4}{5}$

Adding fractions with the same denominator is the rule but it begs the question why? Why can’t (or shouldn't) I add both numerators and denominators?

$\frac{1}{4} + \frac{1}{4} = \frac{1+1}{4+4} = \frac{2}{8}$

To make sense of this try taking a 12 inch ruler and drawing a 3 inch horizontal line (1/4 of a foot) and then on the end add another 3 inch line (1/4 of a foot). What is the total length of the line? It should be 6 inches (1/2 a foot) and not 2/8 of a foot (3 inches). In essence it seems we can only add like items and like items are terms that have the same denominator and we add them up by adding up numerators.

Adding Fractions With Different Denominators

When adding fractions that do not have the same denominator, you must make the denominators of all the terms the same. We do this by finding the least common multiple of the two denominators.

Least common multiple of 4 and 5 is 20; therefore, make the denominators 20:

$\frac{1}{4} + \frac{2}{5} = \frac{1 \times 5}{4 \times 5} + \frac{2 \times 4}{5 \times 4} = \frac{5}{20} + \frac{8}{20}$

Now that the common denominators are the same, perform the usual addition:

$\frac{5+8}{20} = \frac{13}{20}$

Subtracting Fractions

Subtracting Fractions With The Same Denominator

To subtract fractions sharing a denominator, take their numerators and subtract them in order of appearance. If the numerator's difference is zero, the whole difference will be zero, regardless of the denominator.

$\frac{6}{13} - \frac{2}{13} = \frac{6-2}{13} = \frac{4}{13}$

Subtracting Fractions With Different Denominators

To subtract one fraction from another, you must again find the least common multiple of the two denominators.

Least common multiple of 4 and 6 is 12; therefore, make the denominator 12:

$\frac{3}{4} - \frac{1}{6} = \frac{3 \times 3}{4 \times 3} - \frac{1 \times 2}{6 \times 2} = \frac{9}{12} - \frac{2}{12}$

Now that the denominator is same, perform the usual subtraction.

$\frac{9-2}{12} = \frac{7}{12}$

Multiplying Fractions

Multiplying fractions is very easy. Simply multiply the numerators and the denominators together. (Note: Don't forget to simplify the answers (a fraction) of the result whenever possible)

$\frac{1}{2} \times \frac{2}{3} = \frac{1 \times 2}{2 \times 3} = \frac{2}{6} = \frac{1}{3}$

Dividing Fractions

To divide fractions, simply invert the numerator and the denominator of the second term in the problem, then multiply the two fractions.

$\frac{1}{2} \div \frac{3}{4}$

Invert the numerator and the denominator in the second term:

$\frac{1}{2} \times \frac{4}{3}$

Multiply:

$\frac{1 \times 4}{2 \times 3} = \frac{4}{6}$

As an optional final step, you can simplify the resulting fraction:

$\frac{4\!\!\!/}{6\!\!\!/} = \frac{2}{3}$

Decimals

Decimals are basically fractions expressed without a denominator, rather replaced by a power of ten, and then the decimal point is inserted into the numerator at a position corresponding to the power of ten of the denominator. It is usual to add a leading zero to the left of the decimal point when the number is less than one.

$\frac{2}{5} = \frac{2 \times 2}{5 \times 2} = \frac{4}{10^1} = 0.4$

Adding Decimal Numbers

Add decimal numbers much the same way you would add integers. Line up decimal points, and then proceed to add each column and carry at the top. The decimal point in the answer should line up with all of the others. Here is an example:

$12.3 + 24.2 =$

 12.3
+24.2
 ----
 36.5

Subtracting Decimal Numbers

Subtract as you would integers, but remember to follow all the rules from addition of decimals.

Converting Fractions to Decimal Numbers

To convert a fraction to a decimal number, divide the numerator by the denominator.

$\frac{3}{4} = 0.75$
$\frac{10}{3} = 3.333333333333333...$

Multiplying Decimal Numbers

Multiplying decimal numbers can be tricky at times, but most of the times, it is similar to multiplying any integers. Although there are easier methods of multiplying, this is one of the methods.

You can make both decimal numbers have same multiple of a power of ten.

$0.6 \times 0.75 = (60 \times 10^{-2}) \times (75 \times 10^{-2})$

Then multiply the first terms together, and the second terms.

$(60 \times 75) \times (10^{-2} \times 10^{-2}) = 4500 \times (10^{-4})$

Then insert the decimal point into a corresponding power of ten.

$4500 \times (10^{-4}) = 450 \times (10^{-3}) = 45 \times (10^{-2}) = 4.5 \times (10^{-1}) = .45 \times (10^{0}) = .45 \times (1) = .45$

Dividing Decimal Numbers

Dividing decimal numbers is similar to multiplying them.

Make both decimal numbers have same multiple of a power of ten.

$0.3 / 0.4 = (3 \times 10^{-1}) / (4 \times 10^{-1})$

Then divide the first terms together, and the second terms.

$(3 \times 10^{-1}) / (4 \times 10^{-1}) = (3/4) / 10^{0}$

Then insert the decimal point into a corresponding power of ten.

$(3/4) / 10^{0} = 0.75 / 1 = 0.75\,\!$

Alternatively, you can make the numbers integers (if the decimal is finite) and perform a simple division.

$0.3 / 0.4 = (0.3 \times 10) / (0.4 \times 10) = 3/4 = 0.75$

Measurements

Converting English Measurements

Here are some conversions that you should know in the English system:

1 foot = 12 inches 1 yard = 3 feet 1 mile = 5280 feet 1 gallon = 16 cups 1 gallon = 4 quarts

The Metric System

Metric system is based on the SI (Système International d'Unités) units.

Length

Length is measured in metres (m). There are 100 centimetres (cm) in 1 metre, and 10 millimeters (mm) in each centimetre. Therefore, there are 1000 millimeters in each metre. There are 1000 metres in each kilometre (km).

Mass

Mass is measured in kilograms (kg). The kilogram was originally defined as the mass of one litre of pure water at a temperature of 4 degrees Celsius and standard atmospheric pressure.

Time

Time is measured in seconds (s). Second is defined as the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the Cesium-133 atom at zero kelvins. There are 60 seconds in a minute, 3600 seconds in as hour (or 60 minutes), and 86,400 seconds in a day (or 24 hours, or 1,440 minutes).

Temperature

Temperature is measured in kelvins or degrees Celsius (°C) in SI; the latter is used more frequently in general applications. The degree Celsius is a unit of temperature named after the Swedish astronomer Anders Celsius (1701–1744), who first proposed a similiar system in 1742. The Celsius temperature scale was designed so that the freezing point of water is 0 degrees Celsius, and the boiling point is 100 degrees Celsius at standard atmospheric pressure.

Volume

Volume is measured cubic metre (m³). The volume of a solid object is a numerical value given to describe the three-dimensional concept of how much space it occupies. One-dimensional objects (such as lines) and two-dimensional objects (such as squares) are assigned zero volume in the three-dimensional space. Litre (L and l) is used commonly. 1000 L makes a cubic metre.

Metric Length Measurement Conversions

1 meter= 10 decimeters= 100 centimeters= 1000 millimeters 1 decimeter= 10 centimeters= 100 millimeters 1 centimeter= 10 millimeters 1 kilometer= 1000 meters= 10,000 decimeters= 100,000 centimeters= 1,000,000 millimeters 1 kilogram= 1000 grams= 1,000,000 milligrams 1 gram= 1000 milligrams

Measure Metric Quantities

Converting English and Metric Measurements

Trade-based Calculations

Glossary

Conversion

Greatest Common Factor

Improper Fraction

Least Common Multiple

Litre - The basic unit of volume in the metric system

Multiplication

Meter - The basic unit of length in the metric system.

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Applied Math Basics - eMyTextBooks

Free online books - just read it!

Applied Math Basics

Applied Math Basics

Basic Math Terminology

The Decimal System

The Basic Sets of Numerals

Basic Whole Number Operations

Rounding Whole Numbers

Adding Whole Numbers

Subtracting Whole numbers

Multiplying Whole Numbers

Single number times Single number producing a Single number

Dividing Whole Numbers

Factoring Whole Numbers

Introduction to Fractions

Simplifying Fractions

Improper and Mixed Fractions

Adding Fractions

Adding Fractions With The Same Denominator

Adding Fractions With Different Denominators

Subtracting Fractions

Subtracting Fractions With The Same Denominator

Subtracting Fractions With Different Denominators

Multiplying Fractions

Dividing Fractions

Decimals

Adding Decimal Numbers

Subtracting Decimal Numbers

Converting Fractions to Decimal Numbers

Multiplying Decimal Numbers

Dividing Decimal Numbers

Measurements

Converting English Measurements

The Metric System

Length

Mass

Time

Temperature

Volume

Metric Length Measurement Conversions

Measure Metric Quantities

Converting English and Metric Measurements

Trade-based Calculations

Glossary

Also helps finding: AppliedMathBasics, AppliedMath, MathBasics, applyed, meth, asics, aplied, hath, basicse, applie, ath, basica, appled, kath, basicz