Mnemonics to Memorize Mathematical Concepts

Mnemonic (pronounced neh-MAHN-ik ) is a learning technique that aids information retention in our memory. It can be a rhyme, abbreviation or mental image that helps you in memorizing something which is otherwise difficult to commit to our memory.

In mathematics we come across lot of formulae, concepts and values that are torturous to memorize, especially when they are humungous in number. Let’s delve into few useful and handy mnemonics to memorize some very essential mathematical concepts / formula.

Mnemonic to memorize the value of pi (π)

Memorize till 7 decimal places: 3.1415926

May I have a large container of coffee?

mnemonic to memorize value of pi 7 place

Memorize till 10 decimal places: 3.1415926535

May I have a large container of coffee ready for today?

value of pi 10 places of decimals

Mnemonic to memorize the value of e (exponential function)

‘e’ (exponential function) is extensively used in calculus and in problems related to continuous growth.

If you wish to understand “e” in very simple terms, click here.

Value of e up to 15 places of decimal is 2.718281828459045

Value of e can be memorized by just breaking the value in chunks and memorizing it, like we do with mobile numbers. Here it goes –

mnemonic to memorize value of e exponential

2.7 – 1828 – 1828 – 45 – 90 -45

Mnemonics to Memorize Roman Numerals in Order

  • I = 1
  • V = 5
  • X = 10
  • L = 50
  • C = 100
  • D = 500
  • M = 1000

I Value Xylophones Like Children Drink Milk

OR

I Viewed Xerxes Loping Carelessly Down Mountains

Memorize Roman Numberals

Mnemonic to memorize the sequence to carry out the operations of arithmetic

There is a sequence in which we need to carry out arithmetical operations. This sequence can be memorized by the abreviation – BODMAS.

BODMAS

Position of division and multiplication can be interchanged as you know division is nothing but multiplication with inverse or reciprocal of the given number, example 13 ÷ 7 = 13 x 1/7

Position of addition and subtraction can be interchanged as subtraction is nothing but addition by changing the sign of the other number, example, 13 – 7 = 13 + (-7)

Mnemonic to memorize trigonometric Ratios

Often times it’s mind-numbing to memorize trigonometry ratios. Worst part is it keeps escaping out of our memories and we keep struggling.

Here I wrote about a way to easily remember trigonometry ratios

Some People Have, Curly Brown Hair, Turned Permanently Black

Memory trick for trignometric ratios

Do you think these mnemonics are interesting and helpful?
Do you have any such mnemonic to share?

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Learn to Solve Simultaneous Equations Mentally

Do you struggle with solving simultaneous equations?

By simultaneous I mean equations with multiple unknown variables. Generally the number of equations given will be equal to the number of equations.

Let’s take an example,
3x + 4y = 18
5x + 7y = 31

Methods Taught at Schools

In our schools, we are taught to solve for x by equating the co-efficient of y by multiplying both the equations by some constants in such a manner that you get the same resultant value and then subtracting one equation from the other.

For instance in this case, to find the value of x, we will multiply first equation by 7 and second equation by 4 and get 28 in both the cases. This is done so that we get a zero on subtracting.
3x + 4y = 18 …………………(i) x 7
5x + 7y = 31 …………………(ii) x 4

We get 2 equations, where co-efficient of y is same
21x + 28y = 126
20x + 28y = 124

Subtracting second equation obtained above from the first one we get
(21x – 20x) + (28y – 28y) = 2
Hence, x = 2

Plugging the value of x in equation (i) we get y = 3

Problems with the Above Methods

  • This method can become quite laborious, especially when the co-efficients of the unknowns are such that they have to be multiplied by large numbers to make them equal to eliminate one of the unknown by adding or subtracting as the case may be.
  • The above calculations become cumbersome, when the co-efficient(s) are large prime numbers.
  • This method involves multiple steps where we need to do multiplication and addition/subtraction.
  • Also, there is no chance of using this method to solve the problem mentally as one has to keep track of the equations and various computations.

Solving Simultaneous Equations the Smarter Way

Our new method will give us the final answer in fractions, i.e. in Numerator and Denominator for both the variables: x and y.

First, we need to find the numerator of the value of x in the above case, take the simple following steps:
Step #1: Cross-multilply the coefficient of y in the first equation by the constant term (RHS) of the second equation
Step #2: Subtract from it the cross-product of the y coefficient in the second equation and the constant term (RHS) of the first equation.

Solving Simultaneous equation

So the numerator is 4×31 – 18×7 = 124-126 = -2.

Second, we need to find the denominator of the value of x:
Step #1: Cross-multiply the coefficient of y in the first equation by the coefficient of x in the second equation
Step #2: Subtract from it the cross-product of the y coefficient in the second equation and x coefficient in the first equation.

Solving linear equations

Hence the denominator is 4×5 – 7×3 = -1
Hence, the value of x = -2/-1 = 2

Now, let’s try with a simpler example,
x+2y = 8
3x + y = 9

Using the above method, in a single line calculation you can say,
x = (2×9 – 1×8)/(2×3 – 1×1)
x = 10/5 = 2
Therefore, y = 3

Isn’t this amazingly simpler? With some practice you can comfortably apply this technique to solve simultaneous equation mentally.

If you liked this method, you must explore another Vedic Mathematics trick of solving a special class of simultaneous equations in seconds.

Do you find simultaneous linear equations difficult to solve? Do you think you can start using above method in solving equations?

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4 Awesome Math Tricks You Didn’t Know Existed

maths

No matter how much you hate it but the importance of mathematics in every day life can never be ignored. Math is all around us, in everything we do. It is the building block for everything in our daily lives which includes mobile devices, engineering, architecture, money, art and even sports.

When I was a student, I loved studying mathematics, while some of my friends really hated it. Math is all about understanding the concept of a formula and applying it in order to solve a problem and get an appropriate answer.  I was easily able to grasp the concept of formulas, while my friends struggled badly. Instead of understanding, they tried learning all sorts of formulas by heart, which in mathematics is not right.

However, apart from using standard formulas to solve equations, there are several other cool mathematics tricks which people never know about. Here are some of the coolest tricks you can use and share with your friends:

 

1)    Multiplying by 4

This trick is very simple and yet logical. To multiply any number by 4, just multiply it by 2 and then double the answer. The basic concept of this trick is that you can solve a multiplication problem by multiplying by its factors. Let’s assume the number is 25.

Then:

25 x 4

25 x 2 = 50

50 x 2 = 100

25 x 4 = 100

This trick is based on a very simple fact:

2 x 2 = 4

Therefore:

25 x 4 = 25 x (2 x 2)

=>25 x 2 x 2

=>50 x 2

= 100

 

2)    Hate 8 (Eight)

Yep, you read it right. This lovely mathematical trick is known as hateful eight. I am sure you will love it like I did. Grab a calculator and read on:

Ask your friend to choose a number between 1 and 9, but he is not allowed to choose the number 8.

For example: 5

Now, multiply the number by 9:

5  x  9   =  45

Multiply the answer by 12345679 (Please note that number 8 should not be included in the multiplication)

Therefore:

45  x  12345679  =  Magic!

The answer that you will get is 55555555. That is 8 times number 5.

Try another number!

 

3)    Pick a Number and Know Your Age

I don’t know but this trick gives me a great feeling of happiness and excitement. Try it with your friends. I am sure they will like it too.

First, pick a number between 2 and 9

I pick: 5

Next, multiply this number by 2

5  x  2  =  10

Add 5 to the answer:

10  +  5  =  15

Now multiply this number by 50:

15  x  50  =  750

Now, here comes the tricky part.

If you have already celebrated your birthday this year then, add 1764.

If you haven’t yet celebrated your birthday this year then, add 1763.

In my case I have already celebrated my birthday this year, so:

750  +  1764  =  2514

The last step is to subtract your year of birth:

2514  –1985  =  529

The answer you get is magical. Here’s how:

The first number represents the number that you picked, i.e. 5

The last two numbers represents your age, i.e. 29

See, this is called a classic mathematical trick. Try it!

 

4)    Result  1 , 2 , 4 , 5 , 7 , 8

I don’t have any specific name for this trick. Anyway, here it goes:

First pick a number between 1 and 6

For example: 2

Multiply the number by 9:

2  x  9  =  18

Multiply the result with 111:

18  x  111  =  1998

Multiply the result with 1001:

1998  x  1001  =  1999998

In last step divide the result by 7:

1999998  /  7  =  285714

The answer has all the above numbers present, i.e. 1 , 2 , 4 , 5 , 7 , 8.

Conclusion

With a little practice and preparation, you can memorize these mathematical tricks easily and show them to your friends and family just for fun.

Employed as a Board of Advisors at Assignment Plus, Miley Adalson offers help to need learners who wonder, “who can do my assignment”. She also runs a blog that talks about fresh grads, freshman career, etc.

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Shortcut to Find the Fourth Power of any Two Digit Number

In this article we will explore a shortcut to find the fourth power of any 2 digit number. The approach will be similar to the shortcut to find the cube of any 2 digit number. I strongly suggest that you should check that first. 

Generic Form of 4th Power of 2 Digit Numbers

The generic form of fourth power of any two digit number can be algebraically expressed as:

(a + b)4 = a4 + 4a3b + 6a2b2 + 4ab3 + b4

We will split the above result in 2 lines. We start with 4th power of 1st digit and then keep multiplying with ratio of 2nd digit: 1st digit = b/a. The remaining will go in the next line. This can be written as –

= a4 +  a3b  +  a2b2   +  ab3   + b4   

             3a3b + 5a2b2 + 3ab3

__________________________________  Adding the above 2 lines, we get original result.

= a4 + 4a3b + 6a2b2 + 4ab3 + b4

Example of Shortcut to Find the 4th power of 2 Digit Numbers

To try this shortcut, let us consider a simple example i.e. 124

Here the first digit is 1 and its fourth power is also 1. The ratio of 2nd to the 1st digit is 2 (Ratio = b/a)

The remaining 4 terms can be obtained by multiplying each of the previous terms as shown :

124 =  1      2       4      8     16

6      20    24

_____________________
=  1    8      24     32    16

=  1    8      24     32    16

=  1    8      24     33      6

=  1    8      27      3      6

=  2    0       7      3      6

= 20736

Let us try another example. We will find out the 4th power of 91.

914 = 6561            729             81            9           1

2187           405            27

____________________________________

6561        2916             486         36           1

Keeping single digit in each step and carrying the remaining digits and adding to number on the left, we get the answer as 68574961

Tip to use the shortcut: If we memorize the fourth power of all the single digits (0-9), it will come very handy for using this trick of calculating fourth power of any two digit numbers. Find below the table of fourth power of first ten natural numbers –

Number

Fourth Power

1

1

2

16

3

81

4

256

5

625

6

1296

7

2401

8

4096

9

6561

10

10000

Do you think this shortcut can be applied for calculating fourth power of a number?

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Learn Adding Numbers in Seconds

How would you add two numbers?

Let’s say I have to add 35 and 59.

35 + 59 would mean first add 5+9 = 14. Write 4 and carry over1. Then add this 1 to 3 and 5 which comes to 9. So our answer is 94. This is how we were taught in school. Now we will learn to do this at magical speed. However, before we delve into the quick addition method discussed in this post, check out how to avoid carryover while adding numbers.

We draw this line separator that divides the numbers into units, tens, hundreds and so on. We then add numbers in the same place or rather column together moving from left to right rather than right to left.

In our example we draw a line between 3 and 5 in 35 and 5 and 9 in 59. We add 3 and 5 first which is 8. Since it is at tens place this 8 is not 8 but 80. Now add 5 + 9 = 14. Now 14 = 10 +4, so we write 4 in the units place and carry over 1 or 10 to 3+5 or 30 + 50. And it becomes 80 + 10 +4.

3 / 5
5 / 9
——-
80
14
——-
94

At this stage you might be wondering what I am doing. Complicating things unnecessarily?

Well as of now you just added two digit numbers, once you get the hang of the concept you will be able to add 4 digit numbers also in a matter of seconds. Let me take some bigger numbers to demonstrate that.

Example: 986 + 37

98 / 6
3 / 7
——–
101
13
——-
1023

As you see 1010 + 10 +3 = 1023
To add 98 + 3 is a lot easier than adding 986 +37.
Now let us try adding two 3 digit numbers. Example: 575 + 789

57 / 5
78 / 9

Here we again break them i.e.shortcut for addition
5/7/5
7/8/9
———
12+
15+
14 this 4 is written in the units place without thinking

So it becomes
4
6 (5+1 i.e. 1 from 14 and 5 from 15)
13 (12+1 i.e. 12 and 1 from 15)
———–
1364

Now isn’t this method making much more sense? Similarly, if you wish to add consecutive numbers in a jiffy, here’s the shortcut trick to add consecutive numbers.

With some practice you will not even need to scribble this on paper.
Practice 20 to 30 additions and you will be doing it all mentally.  Don’t believe me. Give it a try and see it for yourself.

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How to Quickly Calculate Square of Three Digit Numbers?

Learn to Quickly Calculate Square of Any Three Digit Number

The method of squaring any 3 digit number is an extension of my last post on finding square of any two digit number. To understand and appreciate squaring of 3 digit numbers you should be well versed with shortcut of squaring any 2 digit number. Let us start learning this with the help of an example.

What is the square of 384?

Step 1: To begin with ignore the 3 of 384. You are left with only 84, a two digit number. Using the method of squaring 2 digit numbers, find the square of 84. We get the answer as

Square of 8  |   twice of 8 X 4   |  square of 4

64            64           16

7056 (consolidating the result obtained above)

Step 2: This step is new and different from what we’ve learned in the previous post for squaring 2 digit numbers. Watch carefully.

We have to multiply the first and last digits of our original number and double it. Essentially, that is multiplying together 3 and 4 and then doubling it. Hence we get 24.

Add this number directly to the two left hand digits of our number obtained from the first step.

7056

Add 24 to 70. 70+24=94. So 7056 gets converted to 9456.

Step 3: In the first step we left out the first digit of our number and squared the last two digits. Now we will forget about the unit’s digit 4 and square the first two digits i.e. 38 as before just omitting to square the last digit 8.

Square 38 as a regular 2 digit number, except that you omit the 8 squared.

Square of 3 | twice of 3 X 8

9       |   48

Step 4: Consolidating this with the result obtained in step 2,

9   |  48   |  9456

14        7       456

Hence the answer is 147456.

I’ve shared this method of squaring 3 digit numbers as an extension to the shortcut of squaring 2 digit numbers. Initially you might feel that the traditional method is quicker than having to memorize and execute these steps. However, this method can prove to be quicker than the useful one only if you master this technique with lot of practice.

Do you think this method will help you in reducing the time to calculate square of 3 digit numbers?

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How to Quickly Calculate Square of Any Two Digit Number?

In this post I’ve tried to improvise on the method of squaring presented in one of the earlier posts on Quickermaths.com itself. This trick of squaring any two digit number with ease is inspired by squaring techniques from the book – The Trachtenberg Speed System of Basic Mathematics

I would like to explain this method of squaring any 2 digits number with the help of an illustration.

What is the square of 32?

Step 1. In finding the last two digits of the answer, we shall find the square of the last digit of the number. Square the right-digit digit, which is 2 in this case. Hence we get 04

_ _04

Step 2. We shall now need to use the cross product. This is what we get when we multiply the two digits of the given number together. Multiply the two digits of the number together and double it: 3 times 2 is 6, doubled is 12: We write 12 as 2 and carry over 1 to the next step.

_ 24

Step 3: In finding the first two digits of the answer we shall still need to square the first digit of the number. That means we square the left hand figure of the number. Here square of 3 will be 9. Add 1 which is carried over from last step. Hence we get 9 + 1 =10

1024

That’s the answer.

This method can also be compared with another shortcut to find the square of any number posted by me on Quickermaths.com in the past.

Let’s try another example by squaring 64.

Square of 64 =     Square of 6 | double of cross product of both given digits 4 & 6| square of 4

Square of 64 =      36 |2x6x4 | 16

=      36 | 48 | 16

Collapsing the numbers

=      36 | 48 + 1 | 6

=      36 + 4 | 9 | 6

=      40 | 9 | 6

Hence the answer is 4096.

Have you come across any other squaring method like this?

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Multiplication Trick for Multiples of 11

If you know how to quickly multiply any number by 11 (click on the link to read further), the short cut multiplication method for 22, 33, etc. becomes easy to grasp. It’s an extension to the earlier method and you’ve seen earlier that there is no need to remember multiplication tables.

Multiplication by 11 is easy. Start from the right, add the two adjacent digits and keep on moving left. Since you can write only one digit in each step, if there is a carryover add it to the number obtained in the next step. So let’s begin to learn multiplication by multiples of 11.

Multiplication Trick for 22

For multiplication with 22, the rule is (number +next number)*2

Let us look at it step by step –
Step 1: For sake of simplicity, assume that there are two invisible 0 (zeroes) on both ends of the given number.
Say if the number is 786, assume it to be 0 7 8 6 0

Step 2:Start from the right, add the two adjacent digits and multiply by 2. Keep on moving left.
07860
Add the last zero to the digit in the ones column (6), and multiply by 2. Write the answer below the ones column.
Then add this 6 with digit on the left i.e. 8 and multiply by 2.
Next add 8 with 7 and multiply with 2.
Next add 7 with 0 and multiply by 2.
(0+7)*2     |    (7+8)*2  |    (8+6)*2   |   (6+0)*2
=   14   |   30   |  28   |  12

Step 3: Start from right most digit. Keep only the unit’s digit. Carryover and add the ten’s digit to the next number to the left. Doing this we get the answer as 17292.
Yes, job done. Quite simple, isn’t it?

Multiplication with Other Multiples Of 11

For multiplication with 33, the rule is (number +next number)*3
For multiplication with 44, the rule is (number +next number)*4
and so on….till
For multiplication with 99, the rule is (number +next number)*9. However, their is a simpler way of multiplication trick for 99

Multiplication Examples:
56789*22 = 0567890*22= (0+5)*2  |   (5+6)*2  |   (6+7)*2  |   (7+8)*2  |   (8+9)*2  |   (9+0)*2=1249358
123678*88 = 01236780*88 = (0+1)*8  |   (1+2)*8  |   (2+3)*8  |   (3+6)*8  |   (6+7)*8  |   (7+8)*8  |   (8+0)*8=10883664

Try it yourself. Share your experience with all by posting a comment below.

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Shortcut to Find Square of 2-Digit Numbers with Unit Digit as 1

Special shortcut methods of squaring 2 digit numbers

In previous articles we’ve discussed special shortcut Vedic Math Techniques to find the square of any number ending in 5 and square of 2 digit numbers ending in 9.  In this article we will discuss –

Square of 2-digit numbers whose unit digit is 1

Let us take a 2 digit number in its generic form. Any two digit number whose unit digit is 1, say a1 can be expressed as 10a+1, where a is the digit in ten’s place

Square of a1= a2 | 2xa | 1

Here, ‘|’ is used as separator.

That means for the left most part of the answer, a is squared, hence first part will be a2. The middle part will be twice of a and the last or the right most part will always be 1.

Let us see a few examples.

(21)2= 2 squared | 2 . 2 |1 = 441
(31)2= 3 squared | 2 . 3 |1 = 961
(41)2= 4 squared | 2 . 4 |1 = 1681
(51)2= 5 squared | 2 . 5 |1 = 2601 (Here the square of 5 is 25 but since the product of 2.5 is 10 we write down 0 and add 1 to 25). Similarly,
(91)2= 9 squared | 2 . 9 |1 = 8281

Square of 3-digit numbers ending in 1

Now let us try to extend the above shortcut method to 3 digit numbers as well. Let us straight away start with an example 131.

Like earlier separate the given number in 2 halves, left hand side will have digits other than 1 and right hand side will have 1 as usual.

Hence, the answer is

(131)2= 13 squared | 2 . 13 | 1 =

= 169 | 26 | 1

= 17161

Let’s take another example of squaring a three digit number ending in 1.

261 = 26 squared | 2×26 | 1

= 676 | 52 | 1

= 68121

From the above illustrations you must have noticed that higher the initial number to be squared higher is the square in first part. Hence to extend this method to large 3 digit numbers one should be proficient in squaring any 2 digit number.

Try it yourself

21, 71, 91, 161, 231 and 321

Answers: 441, 5041, 8281, 25921, 53361 and 103041

This article is based on a suggestion given by a Quickermaths.com reader Vishal Mishra. If you also have some suggestions you can email it to me or post it here as comment. 

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Find Value of Sin and Cos using Fingers

Today I am going to share with you a special memory trick for trigonometry, mailed to me by Debasis Basak – a young Class IX follower of QuickerMaths.com

By this method we can find out Sines and Cosines of different angles. It just requires your left hand. Let’s understand this trick step by step –

Step 1:-
First mark the angles of 0, 30, 45, 60, and 90 on little, ring, middle and pointer finger and thumb of your left hand.

Step 2:-
On the palm of your left hand write the equation √x /2

Find Value of Sin and Cos

Step 3:-
Suppose, you want to find Cos30. 

Fold the finger representing 30. i.e. ring finger of your left hand palm.  Count the numbers of fingers on the left of the ring finger. So since there are 3 fingers, x=3; put the value in the equation given in step 2

Step 4:-
Hence,
Cos30 = √3/2

Finding Sine of the same angle is just a simple step away. Of the same left hand, if we count the fingers to the right of the folded finger, we will get value of sine.

Sin30 = √1 /2 = 1/2

Let’s take another example, sin60

60 is represented by pointer finger and there are 3 fingers to the right of this folded finger.

Hence, sin60 = √3/2

I appreciate the efforts of Debasis in sharing this trick with all of us. In case you want to share some quick calculation trick or technique, mail it to me at vineetpatawari [at] gmail [dot] com.

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