## What Is Mental Math?

Here is the short and the most common answer: mental math is preforming math computations mentally without using pencil/pen and paper or electronic devices such as a calculator or other technology-based systems. In this definition, math problems could be solved quickly, and in many cases, be even faster than a calculator with practice and proficiency. Mental math skills are further enhanced through core principles based on standard math. And just as important, mental math opens the mind to alternative ways or insights in solving math problems through reverse engineering (manipulation of numbers), and the understanding of the relationships between numbers directly involved, and those interconnected with them allowing for additional options to be made available. The underlining effects of mental math are real. Besides academic benefits not only in math but other essential subjects as well, it increases the processing power of the mind, particularly improvement of logical thinking and reasoning capabilities. In addition, the mind can hold information more efficiently, improving recalling features.

Compare how the math problem below is solved: (i) standard math principles and (ii) mental math methodologies.

Mental math is also defined another way, though mostly unknown: the ability to solve math problems mentally that would otherwise be unlikely without the aid of writing or electronic devices. Under this concept, the question then becomes why learn such methods. It’s simple really. For example, in sports such as football and boxing to name only a couple, some exercises are developed to increase performance even though those exercises have nothing to do with throwing and catching a ball, tackling a person, rushing the quarterback, or other football-related skills, or jabbing your opponent, throwing hooks, and countering as used in boxing. These types of exercises are designed for enhancing certain skills. As for mental math, there are techniques created for improving thought processes and challenging the mind to finding unique solutions, thus, increasing creativity and opening the mind to many other possibilities – some standard math core principles due to their own nature are too ridged, inherently restricting other possibilities though not on purpose, and stagnating the potential of the mind.

## We Are Already Using Mental Math

Believe it or not, most children and adults have been using mental computations for as long as they can remember. Children are likely to be performing mental math before doing it on paper. Everyone else has already at one time or another done estimations. For example, estimating how much can be spent when entering a convenient or grocery store is one such case. Accurate mental computations are necessary in class room settings but in most real-life situations outside of the classroom, mental estimations are often applied. Another example involves computing how many gallons can be bought with a certain amount of money. If we have a $20 bill, and gas price per gallon is $2.59, right away, we buy less than 10 gallons since multiplying $2.59 by 10 is an easy computation: 10 X $2.59 = $25.90. Using $2.60 in place of $2.59, getting over 7 gallons is possible because 2 times 7 gives 14, and $0.60 times 7 is about $4.20 with both totaling $18.20. We may not consciously know it, but mental computations are an essential part of life. These are only a couple of examples of many.

The terms “mental math” may sound intimidating, but it really isn’t; in fact, mental computations are largely based on elementary math level, and a small portion on some middle school mathematics (in terms of this eBook). Mental computations are also performed in higher-level math classes such as calculus and other areas, though not as prevalent unless in competition environments. It has been around for a long time but remains virtually unknown in the mainstream, especially in western societies. Increasing popularity of mental math in the west, also called Vedic math or Abacus math in the east (concepts or principles may be slightly different), has grown tremendously in the last 10 years and will continue to increase even bigger in the following 5 to 10 years, evident by the many online courses, ongoing written books and eBooks, and thousands of video tutorials. In terms of growth in the modern era, post Internet birth, the market is still in its infancy – an area largely untapped. Though in a niche environment at the moment, it s potential is going to be huge, mainly for two reasons in no particular order: (i) increasingly more people are noticing the importance and benefits of it, and (ii) accessibility and exposure throughout the Internet. Math in general along with being able to read and write are probably the three most important essential knowledge to have for survival in the industrialized world.

The eBook story begins with the author, a former Information Technology (IT) Network Security engineer, with 8+ years of mental math, including competition experience. Mental math has been a large part of the author’s life not only personally but as well as the professional environment. The mental aspects developed from mental math has greatly enhanced his job skills and performances due to faster processing power of the mind from the years of engaging in mental math. The benefits are real. Once again, the greatest attribute is improved brain power and efficiency of the mind, leading to increased performances even where math may or may not be directly involved.

## Short Story Of The Mental Math Unleash The Power eBook

The journey in completing the 1500+ pages eBook, the 9+ hour video tutorials, and the MM Bonus Package (in progress) took about five years working almost daily including weekends and holidays, even working over 24 and up to 36 hours straight at times in this massive undertaking. Having a passion in doing something makes it seemed like there isn’t enough time in a day for work. However, more importantly, the eBook was written from the standpoint of a beginner in mind, someone who has no experience in mental math. The author takes great pride when a customer can perform mental math using the tools in the eBook to benefit him or herself. Too often when books or eBooks are bought even from people who are experts in their respective field, some materials tend to skip a few steps here and there mainly because the person writing the material forgets the level of his or her customers or may assume the customers should already know this or that, or they should be able to figure the missing parts on their own. Yes, there are certain conditions or foundations required in learning anything – here, basic math is pretty much all that is needed (elementary school math ). Bottom line, when the Mental Math Unleash The Power eBook is purchased with its FREE items, a customer gets a massive knowledge base, in addition to knowing that the materials were written with the fullest passion and determination as possible by the author, and to the best ability of the author.

## eBook Structure

Fourteen math subjects are provided with many topics in the Mental math Unleash The Power eBook (watch the eBook video on the HOME page or click here for a summary). This eBook makes learning mental math Easier… Better… Quicker.

Each topic with an exception of a few is made of three parts: (i) NOTES TO CONSIDER, (ii) EXAMPLES, and (iii) EXERCISES. Depending on the topic, the NOTES TO CONSIDER and EXAMPLES sections may consist of more than a page, while the EXERCICES part is only a page. Two or more examples are provided in the working EXAMPLES section since different scenarios affect the methods. For example, a technique for computing the product between two numbers close to 100 but under 100 applies the same principles but there are cases where the technique needs to be modified just a bit so it can accommodate every type of situations.

The recommendation in learning a technique for each topic is to start with the information in the NOTES TO CONSIDER before moving onto the EXAMPLES section. Useful information pertaining to the respective technique of the section is provided here. Understanding the NOTES TO CONSIDER is not required for comprehending the mental math technique.

The NOTES TO CONSIDER is where the introduction to a mental math technique re-sides and may include but not limited to the following types of information:

- Math terms and principles
- General instructions of the technique
- Format to the answer or arithmeticprocess
- Information of the technique as well as examples for illustrating how the technique is applied
- Types of numbers which are applicable/non-applicable to the technique

The EXAMPLES section contains the actual instructions and computations for a particular technique or techniques.

Due to the detailed step by step instructions and explanations of a technique, some examples may be longer than a page but a technique is designed to solve for the answer in one line in most cases. The EXAMPLES section consists of four subparts (see below).

- Chapter Heading
- Topic and Instructions
- Actual Examples
- General Information and Answer

Following the EXAMPLES section is an EXERCISES page, if any (only some topics do not have any), containing random problems related to the current mental math technique.

Some exercises contain problems from more than one technique when related techniques are taught one after another.

The exercises help to enhance the learning process. Moreover, the answers are shown at the bottom of the same page, thus, making it convenient to check for the answers without having to go to the back of the eBook.

## RECOMMENDATION

Recommendations of how to use the materials are provided in the eBook including its structures and their purposes. Due to the many different topics, review the table of contents to get a summary of the methods being taught – it is a good way to get a solid overview of the materials in the eBook. The written instructions are based from the standpoint of performing math mentally, thus, the techniques are structured for that purpose. However, the methods are flexible enough so some work also can be written if necessary, increasing efficiency and manageability. Learn the methods from the mental approach and adapt it accordingly.

In terms of what to learn initially, begin with math addition and move onto subtraction. Some of the fundamentals taught in these two chapters are used in many parts of the eBook. The first chapters (Addition, Subtraction, Multiplication, and Division) are very important and learning those materials is helpful for the other remaining subjects. Learning and being able to apply every technique is the goal but since there are so many of them, either try to learn them all or chose certain ones. This just means we can learn at our own pace.

The eBook was written for a wide general class of people because the math materials are “basic” level when compared to geometry, trigonometry, calculus, and other math subjects. The author, when producing this eBook and other related materials, assumed customers do not have any experience, except mostly basic math as learned in elementary school with some level of middle school. Therefore, children 8 years or above can learn the materials; the future plan is to createanother mental math eBook or set of them, specifically for children. If anyone who has not learned algebra and numberbases, leave those subjects alone for the meantime until when ready. The great thing about this eBook is the knowledge it contained can be used for a lifetime since math unlike technology remains virtually in tack. Changes in technology in terms do not affect math (as far as we know). For example, adding 1 to 1 is 2, and the change in technology does not affect the principles involved in getting the answer. However, the way math is delivered may be different due to increasingly changes in technology.

## MATH SUBJECTS AND TOPICS

Since there are too many techniques to list, only a few are provided for each subject (14 subjects are given) to get a little sense of the materials in the Mental Math Unleash The Power eBook. The list of the techniques themselves is roughly between 15 and 16 pages in the table of contents.

### ADDITITION

##### Number Complements

Adding numbers together using number complements makes the arithmetic process easier because the principles involve adding values ending in zero/zeroes where possible. Working with zero ending numbers is generally simpler than with non-zero values. This technique or method is often combined with other techniques.

##### Smaller Manageable Parts (SMP)

Adding numbers using smaller manageable parts or (SMP) is another principle that creates as many parts ending in zero/zeroes as possible from one number, making the addition process easier. SMP is one of the most important mental math principles to learn as it is applied in subjects such as subtraction, multiplication, division, and others respective of each of the environments it is used in. SMP is essentially what mental math entails at the core level – mental math in many cases is really about breaking down mathematical processes into smaller parts.

##### Place Values

Adding numbers together using place values or sometimes called partial sums can be applied to add two numbers. However, probably the greatest feature is the ability to add as many numbers to each as can be managed. Keep in mind techniques can be combined or used as a stand-alone method. Depending on each situation, one technique may be better than another. So the same is true whether they are combined or not.

### SUBTRACTION

##### Smaller Manageable Parts (SMP)

Subtracting between two numbers using smaller manageable parts or (SMP) essentially involves breaking down one number into parts with at least one part ending in zero/zeroes. Basically, the same types of principles are applied as in math addition but based on the requirements of subtraction. However, in subtraction the subtrahend or number being subtracted is the one simplified, while in addition, either the minuend (number doing the subtraction) or the subtrahend can be simplified.

##### Splitting

Subtracting between two numbers using splitting is very effective under the right situations even allowing for subtraction between large numbers to be done quickly. This technique is based on using place values where the places values are split accordingly for easier computing. For example, with splitting we can subtract this math problem (subtract 8114 from 19726) in 3-4 seconds mentally given that proficiency has been attained (even in minutes of learning the technique). As with many techniques, the splitting technique can be learned in minutes. Furthermore, it is also used in other math subjects throughout the eBook but tailored to those environments.

##### Rounding (Base Amounts)

Subtracting between two numbers using rounding or base amounts applies the same types of principles as for math addition but specific for subtraction. Depending on each situation, one or more numbers can be rounded (or base amounts can be applied to) so the answer can be computed easier. For example, why subtract 199 from 2092 the conventional way by subtracting starting from the units digit, one digit at a time, moving left along with borrowing and other tasks, when we can subtract 200 from 2100 to get 1900, and subtract 3 to get 1893. Even though several scenarios exist in the eBook, the technique can be adapted for any situations to the types of math problems in the same section.

### MULTIPLICATION

##### Three-Digit Number By One

Multiplying three-digit numbers by a single-digit multiplier is a technique that is essential to almost all the other techniques in the multiplication chapter, expanding over 300 pages. This technique uses smaller manageable parts (SMP), meaning the number is simplified or broken down to parts before multiplication is applied. For example, a math problem such as multiplying 257 by 3 can be solved several ways under this technique. Two options are shown here. First, the number 257 is broken down into 200, 50 and 7, and each part is multiplied by 3 one part at a time with adding the products to each other as they are determined. Secondly, the number 257 is simplified as 200, 60, and -3. Then multiplication, addition, and subtraction are applied where appropriate.

##### Vertical And Crosswise (called standard here)

Multiplying numbers together using the “standard” technique in the eBook or known as the vertical and crosswise method is a good overall mental math multiplication strategy. Three variants are used for multiplying three different sets of numbers: (i) two by two, (ii) three by two, and (iii) three by three. Many multiplication techniques are taught in the eBook but one of the advantages of this technique is every multiplication process is between two single-digit values from both numbers, making the computations manageable. Additional variants of this technique are in the Mental Math Bonus Package for multiplying between two numbers of different sizes up to eight digits each.

##### Ending In 5

Multiplying numbers ending in 5 to each other consists of three variants: (i) two numbers of the same value and ending in a 5, (ii) two numbers with a difference 10 between them and ending in a 5, and (iii) two numbers with a difference greater than 10 between them and ending in a 5. For example, the first variant is for multiplying sets of numbers as: 35 X 35; 85 X 85; 115 X 115; 295 X 295; 1025 X 1025; 1225 X 1225 – and many more. The second variant is for sets like these: 15 X 25; 35 X 45; 105 X 115; 455 X 465; 1125 X 1135; 1565 X 1575; and other sets of numbers meeting the same requirements. Finally, the third variant is for these sets or similar: 15 X 35; 25 X 85; 105 X 145; 625 X 85; 1225 X 245; 185 X 25 – and many others. Each variant is unique with some similarities between them. However, using one variant to solve the math problems of another variant will not work but all three are easy learn.

##### By 11

Multiplying numbers by 11 is one of the easiest techniques to learn, especially when there are two-digit numbers, and when the sum is less than 10 – it’ll probably take only about 2-3 seconds to multiply 45 by 11. With this technique, very large numbers can be multiplied by 11 with ease. The basis of this technique comes from the fact that 11 is the sum between 10 and 1. With this technique, for instance, the answer from multiplying a large number such as 23421 by 11 can be performed in seconds mentally. Even though this is a multiplication technique, the arithmetic involves only simple math addition. Though not discussed here, multiplying by 11 is extended to multiplying by 111 with some slight modifications. Basically, once multiplying by 11 is mastered, usually in a short time, the same types of principles (with some changes) can be used on even larger multipliers with 1s such as 1111, 11111, 11111, and many others.

##### By 12 – 19

Multiplying numbers by multipliers 12 through 19 is a technique designed for long multiplication or multiplying numbers of five digits or more. There is never more than one multiplication process to perform at a time, allowing for the arithmetic process to be manageable even when large numbers are involved. Of course, the technique can be used on smaller numbers as well. Similar principles are applied for all the multipliers in this range with the difference being the units digits. All this means is that the multipliers are applied in the same manner, and since each units digit is unique among the multipliers, different values are produced because of them. For instance, if 43251 is multiplied by 12 and 19, the identical math processes are involved but with two totally contrasting results due to the units digits.

##### By Bases

Multiplying numbers using the base amounts of the numbers consist of many variants. Understanding base amounts is a crucial requirement in being able to apply these variants efficiently and effectively. Many variants are available for two and three-digit numbers and even larger numbers. The numbers can be in the same bases such as 95 and 98, or they can be in different bases such as the following sets of numbers: 1995 and 89; 898 and 96; 12899 and 19; 4993 and 398; and so forth. The way the variants are used is dependent on the relationships between the numbers and other numbers.

### DIVISION

##### By 5 → (0.005, 0.05, 0.5, 50, 500, 5000, etc.)

Dividing numbers by 5 or members of its family (see title above) involves the same types of principles for the divisors in the family. The members of this divisor (5) need to be proportional to 5. For example, the divisor of 50 is ten times the amount of 5, or 5 is ten times the amount of the divisor 0.05. Basically, from a current divisor such as 5 (using 5 in this case), the divisor (0.5) just smaller than 5 is 10 times less than 5, while the divisor (50) just larger than 5 is 10 times greater than 5. Only some family members are listed in the title but more can be used under the same environmentas along as they meet the conditions of the family of 5. This means one principle can accommodate many member divisors in the family. The difference lies in moving the decimal point, if required, either to the right or left from a value computed from the “initial computation” – therefore, the same math process is required for any member, which means only one computation is needed for all the divisors in this family. So dividing 451 by 5 incorporates the identical “initial computation” as for dividing 451 by 0.005. The difference involves moving the decimal point of the result produced from the “initial computation,” which is based on a divisor’s value dependent on each its proportion to 5. In this way, a new set of rules is not needed for each divisor in the family.

##### By 9

Dividing numbers by 9 makes use of base 10, allowing for simple computations to be made in finding the answer for any math division problem. This technique can be used to divide any (whole) number, including large numbers of five digits or more by 9. The arithmetic process is manageable and involves basically simple math. And for smaller numbers such as two and three-digit values, it is so easy it may even be surprising.

##### Three By Two

Dividing three-digit numbers by two-digit divisors is a good starting point before learning how to divide large numbers with four digits or greater. Math division is one of the “harder” math subjects to learn but the math problems are simplified into smaller parts, making it easier to work with. The eBook contains many mental math division techniques, and this particular technique is capable of finding the solutions quickly mainly because applying division is not necessary.

##### Long Division: Close To Base 100/1000

Dividing numbers by divisors close to 100 or 1000 is a technique dependent on the base amounts of the divisors, and this technique is designed specifically for dividing large numbers. There are two variances of this technique where base 100 and 1000 are used as reference numbers or the “actual” or implied divisors. Using these divisors with one for each of the respective variants is simpler than applying the actual divisors since dividing by 100 or 1000 is easier than say 93 or 991. For example, if the math problem consists of dividing 12364 by 97, the divisor is converted to 100 instead and adjustments are made to offset the differences between the original divisor 97 and 100. The same is true if 1000 is used as the divisor. With a math problem of dividing 32546 by 995, the divisor of 995 is not applied in the arithmetic process unless necessary. Therefore, the math problem becomes dividing 32546 by 1000.

##### Long Division: Three-Digit Divisor (Round Up/Down)

Dividing numbers by three-digit divisors is something not often performed in mental math, but it is taught in the eBook. It is great that we can learn to divide using divisors close to a base as in the technique and its variances above but what about those far away from it. Dividing a number such as 45261 by 98, a value close to a base of 100, is simpler than dividing the same number by a divisor like 134, a value considered far from either 100 or 200. The divisors here are pretty much somewhere between two full base amounts. This technique is well suited for these types of math problems.

##### Parts Of The Dividend

Dividing numbers based on parts of the dividend is manageable using this technique under the right situations. Depending on the math problem and the relationship between the dividend and divisor, this technique use can use either parts of the dividend or the whole dividend to be divided by the divisor. Other division techniques involve working with a digit of the dividend, one at a time, and this in turn makes the arithmetic process manageable. However, this technique uses a part or the whole of the dividend, thus, working with two or more digits at any one time. Solving the problem in this way means the dividend is used up more quickly. Here, use dividing 12109 by 12 as an example, the whole dividend can be used at once or parts of 121 or 1210 can be applied. If the part of 1210, for instance, is used, only the last digit of 9 in the dividend is left to process because the digits 1, 2, 1, and 0 are the first four digits in the dividend of 12109. So the strategy is different than going from one digit to the next of the dividend until the units digit has been reached.

### SQUARES

##### Two-Digit Number

Squaring two-digit numbers has four versions (method 4 contains two variants) in the eBook. Each technique has its own pros and cons. The three versions and the fourth with two variants mentioned are for general squaring purposes, which means each can be used to square any two-digit numbers (whole numbers). There are also techniques for squaring only specific types of two-digit numbers, and in many cases, they do a better job than general-purpose ones. Keep in mind some techniques may get a little use to when compared to another, but it doesn’t mean it is not a good technique. In some cases, a technique that takes more time getting use to may turn out to be the superior one because through trails and practice, its principles and application are better suited for one’s own way of reasoning.

##### Three-Digit Number

Squaring three-digit numbers has four versions in the eBook. As with squaring two-digit numbers, each technique is different and offers unique perspectives on doing the same tasks. These four versions are used for squaring any three-digit whole numbers but specific ones also exist to be applied on only certain types of three-digit numbers. For example, a general-purpose technique can square a number such as 387 with no issues. However, in cases with a number like 387, a value near to a base of 400 in this instance, a specific technique may be more efficient since it is designed for squaring three-digit numbers close to a base amount.

##### Close To 100 But Over 100

Squaring numbers close to 100 but over 100 is made more efficient and manageable using this technique since only certain numbers meet the conditions required. It is better suited than general-purpose squaring techniques for these types of numbers – the closer the numbers are to 100 but over makes the arithmetic process easier. On the other hand, the further and above a number is away from 100 makes the computations more challenging due to greater differences between 100 and the number far from 100. The best thing about this technique is it that is easy to learn and apply, and it can be extended to squaring larger numbers close to their respective bases such as 1002, 10004, 100006, and so forth.

##### Repeating 9s

Squaring numbers with just 9s is so easy it seems crazy. The numbers must only have the digit 9 in them such as in 99, 999, 9999, 99999, and so forth. Why use a general-purpose squaring technique or standard method when squaring these types of numbers can be done without even using multiplication? The answers of the squares from these types of numbers contain four different digits (9801); however, there may be more than one of each.

### SQUARE ROOTS

##### Three To Four-Digit Perfect Square Numbers

Finding the square roots of three to four-digit perfect square numbers is relatively easy with the tools in the eBook. A perfect square number is a whole number such as 16, 25, 100, and so forth – decimal values are not acceptable. The idea here is to determine a number when multiplied to itself, which gives the perfect square number. If determining the square root of 81 is required, for instance, it involves finding a whole number when multiplied to itself that gives 81 as the product or result. Because of its simplicity, finding the square roots for these types of numbers is easy as stated earlier. Note the square roots of a number are both positive and negative numbers. For example, the square roots for 196 are 14 and -14: the product between 14 and 14 is 196 or multiplying -14 by -14 also creates 196.

##### Five To Six-Digit Perfect Square Numbers (Estimations)

Finding the square roots of five to six-digit perfect square numbers involves similar principles as with three to fourdigit numbers. Estimating the answers is the goal of this technique. The exact answers in some cases could also be computed but require additional steps. As part of the arithmetic process, eliminating some potential numbers as answers is one feature of this technique. The technique works by separating the number into smaller parts, and the answers are estimated based on those values and or others. Note the square roots of a number are both positive and negative numbers.

##### Two-Digit Non-Perfect Square Number (Estimation)

Finding the square roots of two-digit non-perfect square numbers is the purpose of this technique. First, define what is a non-perfect square number; it is a number created from the product of two same decimal numbers, thus, its square root is a decimal number. For example, 17 is a two-digit non-perfect square number because the product of two whole numbers does not give 17: multiplying 4 by 4 gives 16, and the product between 5 and 5 is 25. Due to decimal values and rounding requirements, the square roots are close but usually not 100% accurate. However, the technique uses simple arithmetic in computing the square roots; basic math is all that is required such as addition, subtraction, multiplication, and division, and not even all four are needed. Note the square roots of a number are both positive and negative numbers.

##### Three-Digit Non-Perfect Square Number Or Larger (Estimation)

Finding the square roots of three-digit non-perfect square numbers is an extension to the technique right above. So how is the square root of a non-perfect square number such as 300 found mentally? First, determine two decimal numbers of the same value when multiplied to each other gives 300 or close to it (the number of digits after the decimal point and rounding affect the answer). Keep in mind the three-digit non-perfect number is a whole number, while the square roots are decimals values. Again, only simple computations are used in finding the answers. Here, the square root of 300 is roughly about 17.321 or -17.321 with three digits after the decimal point and rounding. In another environment, for example, with a two-digit requirement, the square roots would then be slightly different. If either 17.321 or -17.321 is multiplied to themselves, the answer of 300.017 is barely over 300. Note the square roots of a number are both positive and negative numbers.

### CUBES

##### Two-Digit Number

Cubing two-digit numbers as with squaring two-digit numbers has four versions (method 4 contains two variants). These four versions are for cubing any two-digit whole numbers from 10 through 99, thus, they are considered general-purpose methods. We can choose to learn all four and use them, or pick the one we are most comfortable with. From a mental perspective, cubing two and three-digit numbers can be a daunting task. So what does it mean to cube a number in mathematics? In short, the number appears three times in a multiplication expression (28 X 28 X 28) where, 28 in this instance, is multiplied to itself a total of three times. Fortunately, the eBook provides easy step by step procedures for cubing in a manner we can learn quickly. As with many mental math techniques, the core methodologies are based on simplifying the problem into smaller portions or parts, reducing the load and increasing manageability and efficiency as much as possible.

##### Close To 100 But Under 100

Cubing two-digit numbers when the numbers are close to 100 but just under 100 is a technique specifically designed for certain types of numbers. The general-purpose techniques as those mentioned above are more than capable of performing the tasks here; however, they are as efficient when the numbers meet the conditions required in this particular environment. This technique takes advantage of where the number is, close to 100. Using a reference point of 100 allows for the computations to be managed more easily. For example, multiplying a number such as 96 to itself a total of three times (or 96 X 96 X 96) is more difficult with the standard approach or others. The technique can be extended to for cubing larger numbers meeting the bases or reference numbers with some modifications.

##### Three-Digit Number

Cubing three-digit numbers is challenging but the challenge can be met with the tools in the eBook. This is another general-purpose technique, which is used for cubing numbers from 100 to 999. Before learning this technique or any other for cubing three-digit numbers, we need to have proficiency in adding, multiplying, and squaring and cubing two -digit numbers. Cubing three-digit numbers mentally may perhaps be one of the most difficult tasks to perform. To make things as easy as possible, the math problems are broken down into smaller parts for easier computing.

### CUBE ROOTS

##### Four To Six-Digit Perfect Cube Numbers

Finding the cube roots of four to six-digit perfect cube numbers involves straight-forward math processes. A perfect cube number is a value produced from multiplying a number to itself three times such as in this expression “45 X 45 X 45,” giving 91125. The number 91125 is a five-digit perfect cube root number (decimal values are not involved). The answers for the cube roots, unlike finding square roots, do not have positive and negative values. If the expression “-45 X -45 X -45” is acted upon, the answer would be -91125, which is different than 91125 from multiplying three positive 45s to each other as shown in the previous expression. The answers are easily found by separating the perfect cube numbers into parts and working one part at a time. Therefore, we should be able to find the cube root for a number like 658503 within a few simple steps.

##### Seven-Digit Perfect Cube Numbers (Estimations)

Estimating the cube roots of seven-digit perfect cube numbers uses related principles from the techniques above. Even though estimations are taught, in certain cases, the exact answers can be computed. This technique will show how estimations are performed where the first and last digits of the answers are computed accurately. The only real work is in estimating the middle digits since the cube roots for seven-digit perfect cube numbers are three-digit values. Therefore, the arithmetic process contains three major parts.

##### Estimations Of Non-Perfect Numbers (1000 Or Less)

Estimating the cube roots of non-perfect cube numbers with values less than 1000 is the purpose of this technique. The cube roots are decimal numbers, meaning that three decimal values are required to be multiplied to each other in creating the non-perfect cube number (this value is a whole number). Anytime decimal numbers are involved, each different requirement affects the answers slightly. How can we find the cube root of 765 mentally? In finding the answer, other numbers are needed, but they are easy to work with. The technique takes advantages of numbers close to the non-perfect number and simple arithmetic in order to compute its cube root.

### FRACTIONS

##### Convert Repeating Decimal To Fraction

Converting repeating decimals to fractions is as easy as counting from 1 to 10, and this technique is perfect for the job. Two techniques are available in the eBook. The first technique forms the basis of the second. A repeating decimal number is just like any other decimal number but with one or more digits after the decimal point repeating. Several notations are used to write a repeating decimal number. One notation is represented here with three dots following the last digit: 1.222…, 0.3444…, 234.0929292…, and so forth. The three dots denote that a digit or more is repeating. Here are the repeating digits in the three examples given: the digit 2 in 1.222…, the digit 4 in 3.444…, and the digits 9 and 2 in 234.0929292…. Sometimes it may be hard to differentiate the number of repeating digit(s) as in these two numbers (1.222… and 1.222222222…). Notice will be given in the eBook so we can accurately determine the answers correctly. Other notations may have an over-bar over the repeating digit(s) or the repeating digit(s) are underlined. With the tools provided in the eBook, convert repeating decimals to fractions is made easy. The conversions may seem daunting, especially with a number like 3625.00987987987…, but performing the conversions is even easier than thought possible.

##### Sum Of Two Fractions

Adding two fractions together with this technique does not involve using the lowest common denominator (LCD) as in standard or conventional mathematics. Typically, the LCD needs to be determined from the denominators before two fractions can be added to each other in order to compute the answer, which also requires other math processes along the way. This means the LCD is required to be computed first, and then values dependent on how the LCD was created are multiplied to the respective numerators, and finally the two new fractions are added to one another. With this technique, those math processes are not needed since it uses straight multiplication and addition to determine the answer. Being a general-purpose method, this technique is used to add proper and improper fractions to each other. Here are some samples: 1/2 to 6/7, 8/13 to 6/11, 21/8 to 3/17, 6/5 to 11/3, and so forth. Notice mixed fractions are not included – other techniques are available in the eBook to add, subtract, multiply, and divide mixed fractions.

##### Sum Of Two Fractions Whose Numerators Equal One

Adding two fractions when the numerators are both 1 together can be done by this technique, which is designed specifically for these types of fractions: 1/2 + 1/8, 1/9 + 1/4, 1/11 + 1/7, and others similar. When the fractions meet the conditions, this technique is more efficient than using a general purpose one. Besides having many characteristics of a general-purpose technique such as not requiring the LCD, additionally, it incorporates simpler methods in finding the answer due to the numerators being a value of 1 in both fractions.

##### Difference Between Two Fractions

Subtracting one fraction from another employs the same types of principles as adding two fractions together but with subtraction features. Therefore, the lowest common denominator (LCD) is not required along with any other math processes relating to the LCD. The answer is derived from multiplication processes between the numerators and denominators (with a difference between them) and the products of the denominators.

##### Multiply Two Mixed Fractions

Multiplying two mixed fractions to each other can be done using the standard method with a mental approach since in some cases, the standard way may be more manageable and efficient. With the standard method, both the mixed fractions are converted into improper fractions or fractions whose numerators are greater than their respective denominators, followed by applying multiplication to both numerators and then denominators. The mental math technique does not require converting the mixed fractions into improper fractions, and in some cases, is more efficient than other methods.

##### Divide Between Fractions

Dividing one fraction by another mentally is by far a more efficient method than using the standard approach. For one, the second fraction or the fraction on the rightmost side is not required to be flipped or inverted, saving time in the process. Using the standard method means the second fraction must be inverted before applying any computations.

### PERCENTS

##### Convert Fraction To Percentage

Converting a fraction into a percent is another easy technique to learn and apply. In fact, all mental math techniques are based on elementary math, only done mentally. The math processes themselves for the conversion are easy to perform – it is one of those math methods that is so simple it often gets overlooked.

##### Percentage Increased Of A Number

Finding the increased of a number by percentage is easy to perform using this technique. When these types of problems are presented, the main thing is knowing exactly what is asked before the appropriate arithmetic can be applied to ensure the answers are computed accurately. For example, these two problems are different: (i) finding 20% of a number and (ii) finding the increased percentage of a number from 50 to 60. The latter is the type of problem solved by this technique.

##### Number Decreased By A Percentage

Finding the value after a number has been decreased by a certain percentage is the purpose of this technique. For example, if we have a number of 100, the question is how to determine the value after a percentage of 100 has been subtracted. The problem does not focus on how much 100 has been decreased, but rather what is left after the decreased. So understanding what is meant by the questions is a key component. For example, these two problems are different: (i) finding 30% of 100 and (ii) what is the value after 30% has been taken or subtracted from 100.

### DECIMALS

##### Percent To Decimal

Converting a percentage to a decimal number can be done easily with this technique. The arithmetic process of the technique makes it simple to perform these types of conversions with ease mentally. So how would we convert 34% into a decimal value, or something like 99.09%? The math involved is simple and straight forward.

##### Common Fractions To Decimals: Sevenths

Converting fractions to decimal numbers based on common fractions to decimals numbers is done quickly with this technique. Several techniques for other conversions based on common fractions are also taught in the eBook with this being one of them. If sevenths are understood, converting 1/7, 2/7, 6/14, 8/14, 6/7, or others in the sevenths family (denominator is 7 or can be simplified into 7) into decimal numbers can be done is easily and quick. For example, convert 11/7 into a decimal number. In short, keep the quotient of 1 and use 0.571428 as the decimal portion, and that’s it. So the answer is 1.571428. Understanding sevenths makes computing the decimal portion simple.

##### Common Fractions To Decimals: Ninths

Converting fractions to decimal numbers based on ninths may be the easiest conversions from common fractions to decimal numbers. The denominator of 9 or 9s is an important concept in performing these types of conversions. The math behind the technique is solid, and the application is easy. Converting a fraction like 2/9 takes 2 seconds or less. Even more complex fractions involving a denominator of 9 or 9s such as 7/99, 65/99, 123/99, 456/999, and many others are simple to convert into decimal numbers. Without applying this technique, converting the fractions just mentioned would be tedious since standard math division would have to be used – this means to convert 456/999, we need to divide 456 by 999 until the desired number of digits after the decimal point has been achieved.

### NUMBER SEQUENCES

##### Sum Of Consecutive Numbers Starting From One Or Ending In One

Finding the sum for a series of numbers where the starting or ending point is 1 is made simple without having to add every number in the sequence. Imagine finding the sum in this series (from 1 to 158) by adding all the digits to each other. Understanding this technique is crucial for other techniques involving different number sequences such as a series of numbers from 3 to 299 where the starting point is not 1. The technique offers more than one way to solve the answer depending on the series of numbers. The condition required is that the numbers must be in consecutive order, meaning the difference (sometimes called common difference) between any two consecutive terms is 1. For example, in the number sequence from 1 to 10, a common difference of 1 exists between two consecutive terms. Here is the consecutive number sequence starting from one: 1, 2, 3, 4, 5, 6, 7,8, 9, and 10. The difference between 1 and 2 is 1, between 4 and 5 is 1, and between 7 and 8 is 1.

##### Sum Of Consecutive Even/Odd Numbers

Finding the sum for a consecutive even/odd number sequence is just as easy. Two types of number sequences can be solved under this technique: (i) consecutive even series of numbers and (ii) consecutive odd series of numbers. The series can contain as many numbers as possible as long as the conditions set forth by the technique are met. With these types of consecutive even/odd number sequences, the common difference between two consecutive even or odd terms is 2. For example, in this even number series (2, 4, 6, 8, 10, 12) and the odd number series (5, 7, 9, 11, 13, 17, 19), the common difference between either two consecutive even or odd terms is 2. So the difference between 2 and 4 is 2, and 2 is the difference between 8 and 10 in the consecutive even number series or sequence. In the consecutive odd number sequence, the difference between 7 and 9 is 2, and between 17 and 19 is also 2. The math involved has some similarities to the technique above. In either case, both techniques are written in a way that makes learning simple.

##### Find The Consecutive Numbers Given The Sum And Number Of Terms

Finding the consecutive numbers in a number sequence given a sum and the number of terms consists of two versions: (i) odd number of terms and (ii) even number of terms. An odd number of term number sequence is one with an odd valued number of terms such as this sequence: 1, 2, 3. The numbers must be in consecutive order and the number of terms, 3 in this case, must be odd. While in this sequence(1, 2, 3, 4), there are four consecutive terms or even number of terms since 4 is an even number. Modifications are required when the technique is used to find the answers between these two types of number sequences. The objective is to determine the actual terms in the number sequence, keeping in mind, the terms are in consecutive order. With this technique, we’ll be able to do both with simple computations.

### NUMBER BASES

##### Convert Two-Digit Numbers From Other Bases To Base 10

Converting two-digit numbers from others bases to base 10 is only a matter of applying multiplication and addition. Preforming math in other number bases may seem daunting – for the most part, it is relatively simple. Base 10 is the number system, often called decimal system, we used almost daily. However, other number systems or bases also exist, and they have their own purposes. The values in one base is different than another. For example, the number of 45 in a base 9 environment or others is commonly written with a subscript beneath the 5 in 45 – this is not the same amount of 45 in base 10. The purpose of this technique is to convert two-digit numbers in other bases into base 10 values, so we can understand them better, especially if we have no experience working in other bases. As mentioned earlier, the math involved in the conversion is basic math, so these conversions can be done mentally quickly.

##### Convert From Base 10 To Any Base

Converting a number from base 10 to other bases is the purpose of this technique. This technique does the opposite of the technique mention right above. The nice thing about these types of conversions is they are based on three of the four fundamental subjects in mathematics (addition, subtraction, multiplication, and division). Number bases seemed difficult to comprehend for many people even when performed manually with written materials. So how would we convert a number of 11 into a number in base 4, 5, or any other bases (except base 10)? The eBook provides the necessary tools for understanding what number bases are and how to convert them accordingly.

##### Same Base Addition

Adding numbers from other bases together is similar to adding numbers in base 10 (the decimal system). The decimal system consists of the numbers we normally use. This technique mimics the standard way of adding two numbers to each other. As in the conventional or standard method, the arithmetic process is started from the units digits of both numbers, moving left, one digit at a time. If carryover values are not present, the mental math and standard methods are basically the same. The difference lies in how a carryover value is handled. The math processes for adding two numbers together in other bases are easy – it involves adding, subtracting, and dividing in some cases but with small values.

### ALGEBRA

##### Multiply Two Binomials

Multiplying two binomials to each is really no different from multiplying two expressions together. For example, multiplying the two expressions (called binomials) of (a + 2) (a + 5) involves multiplying each term in one expression to all the terms in the other expression. Then basic math is applied where needed to form the answer. This technique shows how to multiply these terms with a method different than FOIL, which increases performances since the terms relating to FOIL are not required to be memorized. FOIL is a common method used for multiplying two binomials to each other but requires memorizing the terms and to where it needs to be applied. In using FOIL, we need to know these meanings: first (F), outer (O), inner (I), or last (L).

##### Square A Binomial

Squaring a binomial mentally is difficult for many. However, once this technique is learned and applied, squaring a binomial is not much different from squaring a number. The technique allows for squaring a binomial in three easy steps with the fourth involving addition/subtraction. For example, squaring the binomial (x + 1) mentally should only take 5 seconds or less with proficiency. Note squaring (x + 1) is the same as multiplying these two expressions to each other: (x + 1) (x + 1).

##### Cubing A Binomial

Cubing a binomial mentally at first seemed impossible. To cube a binomial means to multiply a binomial to itself a total of three times – the same concept as cubing a number like 8 with the number 8 appearing three times in a multiplication expression: 8 x 8 x 8. So in cubing a binomial (made up of two terms) such as (x + 1), the multiplication expression is “(x + 1) (x + 1) (x + 1).” Even when done on paper with pencil/pen, the math process takes time. By understanding the strategies in the eBook, cubing binomials should not be a problem.