Calculus 1: Computing Limits II Case 0/0

Calculus 1: Computing Limits I

Calculus: One of the Most Successful Technologies

How to create a secure, memorable PIN number

Usually people will use their existing numbers, such as telephone number, house number, birth date or social security number as their PIN number for banking. These numbers are easy to remember, nevertheless easy to be broken.

This article will give an idea on how to create a secure, but memorable PIN number.

How to create the PIN number

The technique is so simple. You need to create your own simple algorithm to generate the PIN number, based on the number that you already memorized. In case you forget the PIN number, it can always be generate easily using the algorithm.

An example of algorithm

Let say you want to generate a PIN that consists of six digits.

1. First create the secret number. Think of a 3 digits number that you won’t forget. For example, if you are born in 1965, you can use 965 as the secret number.


2. Use your algorithm to create the PIN number.
   Example: Add the secret number in step 1 with the number one, and multiply it back to itself.
   965 * (965 +  1) = 932190


3. Therefore, your PIN number is 932190.

Different PIN number for different bank

Using this technique, you will have a PIN number that cannot easily be guessed by others. But, what if you want different PIN number for different bank?

You can also create different PIN number for different bank, using the same algorithm and secret number.

Just associate bank1 with 1, bank2 with 2, and bank3 with 3. Then, you can replace the number one in your algorithm with two for bank2 and three for bank3.

Conclusion

This article shows on how to create PIN numbers from one secret number and an algorithm. Using this technique you will only need to remember the secret number, the algorithm and the bank association number.

List of 6000+ high impact journal title

A very huge list of high impact journals.

Click here to download.

How to add two numbers less than hundred rapidly

Adding two numbers less than 100 is easy. For instance, 29 + 45 = 74. However, how fast can you do it. Can you do at least sixty additions within one minute. Actually, you can do it, but, with some practices, using the simple technique that will be discuss in this article.

In order to add two numbers less than 100, add the tens first, instead of the ones. For example, to add 29 to 45, you do (29 + 40) + 5 = 69 + 5 = 74. As an addition, you should not say, either verbally or mentally, such as 29 and 40 are 69, 69 and 5 are 74. Instead, just say 69, 74.

Try the following yourself. Please close up the answers.

23 + 48 = 63, 71
47 + 72 = 117, 119
79 + 69 = 139, 148

After some practices, the first reading should be left out. Thus the answer is gathered at once.

The only caveat to this technique is that, the calculation speed is proportional to the frequency of practices made. Without enough practices, you may not master this technique.

Multiplication of 2 numbers that are near to 100 or 1000

This tip is suitable for multiplying two numbers that are near to 100. For example, 103 x 104 and 97 x 98.

Example 1:

To calculate 103 x 104:

1. Add the the ones from the multiplicand to the multiplier.
   103 + 4 = 107.


2. Multiply the ones from the multiplier with the ones from the multiplicand.
   3 x 4 = 12.


3. Concatenate the results from step 1 and 2. Therefore, 103 x 104 = 10712.

Example 2:

To calculate 1005 x 1002:

1. Add the the ones from the multiplicand to the multiplier.
   1005 + 2 = 1007.


2. Multiply the ones from the multiplier with the ones from the multiplicand. Prepend a zero to the result.
   5 x 1 = 010.


3. Concatenate the results from step 1 and 2. Therefore, 1005 x 1002 = 1007010.

Example 3:

To calculate 97 x 98:

1. Subtract from 10, the ones from multiplicand.
   10 - 8 = 2.


2. Subtract from multiplier, the result from step 1.
   97 - 2 = 95.


3. Subtract from 10, the ones from multiplier.
   10 - 7 = 3.


4. Multiply the result from step 3 with the result from 1. Prepend a zero.
   2 x 3 = 06.


5. Concatenate the results from step 2 and step 4. Therefore, 97 x 98 = 9506.

Example 4:

To calculate 993 x 995:

1. Subtract from 10, the ones from multiplicand.
   10 - 5 = 5.


2. Subtract from multiplier, the result from step 1.
   993 - 5 = 988.


3. Subtract from 10, the ones from multiplier.
   10 - 3 = 7.


4. Multiply the result from step 3 with the result from 1. If you get a one digit number, prepend two zeroes, otherwise prepend just one zero.
   5 x 7 = 035.


5. Concatenate the results from step 2 and step 4. Therefore, 993 x 995 = 988035.

Exercise:

If domain name is cost $9.90 each, what is the cost if you want to buy 92 domain?

Multiplication of any number by a number near 100

It is easy to multiply any number by 100. We just need to append two zeroes to the end of that multiplicand number. For example, 100 x 23 = 2300. But, what if the multiplier is a number that is near to 100, such as 101, or 99?

To multiply a number, a, by a number, b, that is near to 100, here are the steps.

1. Multiply the multiplicand by 100.
   c = 100 x b


2. Subtract 100 from the multiplier.
   d = b - 100


3. Multiply multiplicand, a with the result from step 3, d.
   e = d x a


4. Finally, add the result from steps 1 and 3.

In simple words, we could write it as

( 100 x multiplicand ) + ( (multiplier - 100) x multiplicand )

Example:

1. 101 x 23 = (100 x 23) + ( (101 - 100) x 23 ) = 2300 + (1 x 23) = 2300 + 23 = 2323


2. 99 x 23 = (100 x 23) + ( (99 - 100) x 23 ) = 2300 + (-1 x 23) = 2300 - 23 = 2277

Now, you can try to do it on larger numbers such as 101 x 297, or even slightly larger or smaller than 100 multiplier, such as 103 x 74.

Simple division technique

This division technique can only be used if the divisor could be factorized.

Example 1:

272 ÷ 16 = 17

1. Find the factors of the divisor. Choose one of the factors. 16 = 4 × 4 or 16 = 8 × 2 For example, I will use 8 × 2


2. Divide the dividend by one of the factor. 272 ÷ 8 = 34


3. Divide the quotient from 2. by the second factor. 34 ÷ 2 = 17


4. Therefore, the quotient is 17

Note: Try with 4 × 4. What will happened if the remainder is not zero? Please look at the next example.

Example 2:

275 ÷ 16 = 17 with remainder 3

1. 16 = 4 × 4


2. 275 ÷ 4 = 68, r. 3


3. 68 ÷ 4 = 17


4. The quotient is 17, with remainder 3

Note: Try with 2 × 8

Remainder when a number is divided by a nine

Do you know that there is an easy way to find the remainder when a number is divided by 9?

Example 1:

43 ÷ 9 = 4, remainder = 7

quick remainder: 4 + 3 = 7

Example 2:

28 ÷ 9 = 3, remainder = 1

quick remainder: 2 + 8 = 10 ?

If the addition is ? 9, cast out the 9s, until it is lest than 9.

10 – 9 = 1

Example 3:

2345 ÷ 9 = 260, remainder = 5

quick remainder: 2 + 3 + 4 + 5 = 14

14 – 9 = 5

Example 4:

4489 ÷ 9 = 498, remainder = 7

quick remainder: 4 + 4 + 8 + 9 = 25

25 – 9 = 16

16 – 9 = 7

Squaring numbers more than 100

This technique is suitable for finding the square of number that is more than 100.

Example 1:

103 x 103 = 10609



Steps:

Find the difference between 100 and the number. 103 - 100 = 3

1. 103 + 3 = 106(00) <- Add the difference from the number being squared.


2. 3 x 3 = 9 <- Square the difference.


3. 10600 + 9 = 10609.

Example 2:

112 x 112 = 12544

Steps:
Find the difference between 100 and the number. 112 - 100 = 12

1. 112 + 12 = 124(00) <- Add the difference from the number being squared.


2. 12 x 12 = 144 <- Square the difference.


3. 12400 + 144 = 12544.

If you have better method to share, please don’t hesitate to leave your comment.

Squaring numbers between 51 and 100

This technique is suitable for finding the square of number that falls in range of 51 and 100.

Example 1:

97 x 97 = 9409



Steps:

Find the difference between 100 and the number. 100 - 97 = 3

1. 97 - 3 = 94(00) <- Subtract the difference from the number being squared.

2. 3 x 3 = 9 <- Square the difference.

3. 9400 + 9 = 9409.

Example 2:

76 x 76 = 5776

Steps:

Find the difference between 100 and the number. 100 - 76 = 24

1. 76 - 24 = 52(00) <- Subtract the difference from the number being squared.

2. 24 x 24 = 576 <- Square the difference.

3. 5200 + 576 = 5776.

If you have better method to share, please don’t hesitate to leave your comment.

Squaring numbers that end in 5

You know that 5 x 5 = 25. However, do you realize that the square of any number that ends with 5, will also contains 25 in the last 2 digits.



Example 1:

25 x 25 = 625

Steps:

Remove the last digit. We get 2.

1 + 2 = 3 <-- Add 1.

3 x 2 = 6 <-- Multiply 2 times the result.

6 and 25 give 625 <-- Concatenate the result with 25.

Example 2:

205 x 205 = 42025

Steps:

Remove the last digit. We get 20.

1 + 20 = 21 <-- Add 1.

21 x 20 = 420 <-- Multiply 20 times the result.

420 and 25 give 42025 <-- Concatenate the result with 25.

Example 3:

315 x 315 = 99225

Steps:

Remove the last digit. We get 31.

1 + 31 = 32 <-- Add 1.

32 x 31 = 992<-- Multiply 31 times the result.

992 and 25 give 99225 <-- Concatenate the result with 25.

If you have better method to share, please don't hesitate to leave your comment.

Multiply a number by 11

Multiplying a one digit number by 11, is easy. For example, 2 x 11 = 22. We just repeat the number back twice.

What if we have 1334 x 11? This strategy could help us to multiply any greater than 2 digits number by 11.



The best way to explain this strategy is by using examples.

Example 1: 243 x 11.

1. 2 _ _ 3   (3 digits - put extra 2 blanks in the middle)


2. 2 2+4 _ 3   (add the 1st 2 numbers)


3. 2 6 4+3 3   (add the last 2 numbers)


4. 2 6 7 3

Therefore, 243 x 11 = 2673

Example 2: 1435 x 11.

1. 1 _ _ _ 5  (4 digits - put extra 3 blanks in the middle)


2. 1 1+4 4+3 3+5 5 = 15785

Therefore, 1435 x 11 = 15785

Note:

If the addition of the 2 numbers would gives 2 digits number, add that number's left digit to the left digit of the original number.

Example 3: 99x 11.

Wrong: 99 x 11 = 9 9+9 9 = 9189

Note that 9 + 9 = 18 (a 2 digits number).

1. 9 _ 9


2. 9 9+9 9


3. 9 18 9 ---> 9 + 1 8 9


4. 1089

Example 4: 46 x 11.

46 x 11 = 4 4+6 6 = 4 10 6 = 4 +1 0 6 = 506

As a conclusion, to multiply a number by 11, add the digits of a 2 digits number, and place the sum between them. If the addition produce a more than 1 digit number, carry the tens column to the left.

Learn Calculus in 20 minutes

Two videos by Professor Edward Burger of Williams College teaching Calculus in 20 minutes.

The bottom line is, Calculus is not that hard, even though based on my experience, many students dislike it.

Calculus in 20 Minutes Part 1

Calculus in 20 Minutes Part 2

Product of a single digit number with 9

This algorithm is useful for kids that are lazy enough to memorize the multiplication table of 9.

The algorithm works like this:

Suppose you want a product of n and 9, you can do

1. n - 1


2. 10 -n


3. The answer is (n-1) as the first digit and (10 - n) as the second digit.

Example:

4 x 9 =36

First digit: 4 - 1 = 3

Second digit: 10 - 4 = 6

Note: This trick only works on multiplication of any single digit number with 9.

Box multiplication

The box multiplication technique is suitable for people that has problem with numbers, since it will organized all the number in place.

Example 1:

13 × 23 = 299 First, draw a box for each number. Divide each box diagonally, with a straight line. Then multiply diagonally to the right. For example, 3 × 2 = 6. Put 0 inside the left triangle and 6 inside the right triangle, and so on.

After we have fill in all the numbers inside the triangle, add all the numbers diagonally to the left. For example 6 + 0 + 3 = 9, 0 + 2 + 0 = 2. Write the addition at the end. Follow the arrow.

Example 2:

79 × 85 = 6715 In this example, when we add 2, 4 and 5, we get 11. So, we write 1 and carry 1 (just write it in bracket). Next, when we want to add 7, 6 and 3, add the carry number as well. So, it will become 7 + 6 + 3 + 1 = 17. The same thing goes here. Write 7 and carry 1. The last one is 5 + 1 = 6.

note: The box multiplication technique could also be applied to the numbers that are more than two digits, such as 234 × 346 and 342 × 34.

Multiply two 3 digits numbers in your head

This article will show how to do multiplication of two three digits numbers from right to left. Consider 234 × 456.

Step 1: Multiply the last digits. Write 2, carry 4.

Step 2: Cross multiply the one and the tenth, and add them together. We get 53. Add the 4 that we carry to 53. We get 57. Write 7, carry 5.

Step 3: Multiply all digits, like in the figure below. We get 55. Add the 5 that we carry to 55. We get 60. Write 0, carry 6.

Step 4: Finally, multiply the two left digits. We get 22. Add the 6 that we carry to 22. Write 2, carry 2.

Step 5: Finally, multiply the left digits. Add the 2 that we carry to 8. We get 10. Write both numbers.

Therefore, the answer is 108072.

Easy Multiplication I

This technique of multiplication will do the multiplication from right to left. Let’s look at an example. 23 x 12 = 276

Step 1: Multiply the last digits.

Step 2: Cross multiply both sets of numbers and add them together.

Step 3: Finally, multiply the left digits.

Add all the natural numbers below 1000 that are multiples of 3 or 5

Problem #1 of Project Euler

Add all the natural numbers below 1000 that are multiples of 3 or 5.

Analysis:

Sequence of multiple of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, ..., 999

Sequence of multiple of 5: 5, 10, 15, 20, 25, 30, ..., 995 So, the sum is 3 + 5 + 6 + 9 + 10 + 12 + 15 + 18 + ... + 999.

Take note that we have to remove the duplicate of 15, 30, 45, ..., 990

Method # 1: using Python

All the natural numbers below 1000 that are multiples of 3 or 5, are divisible by either 3 or 5. Hence, we need to add the numbers that are only divisible by 3 or 5. That is very easy in Python. Using the % operator will give use the remainder of the division.

For example, 4%2 = 0, 4%3 = 1. Therefore, in order to ad the number that is divisible by 3 or 5, we have to check for the remainder. We will only add the numbers that give zero for the remainder.

total = 0
for i in range(1000):
    if not (i % 3 and i % 5):
    total += i

Method # 2: using Arithmetic sequence

To add all natural numbers from 1 to 10.

Notice that the sum will always gives 11, if we write it this way. 11 occurs 10 times, so, 11 x 10 = 110. However, that is the sum of 1 to 10 twice. So, we divide 110 by 2, we get the answer, which is 55.

The formula: Sum = n (a1 + a2) / 2 where n is the number of occurrence of the sequence from a1 to a2.

Back to our problem, the sum of 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, ..., 999. Adding the first and last numbers gives us 3 + 999 = 1002. 1002 occurs 333 times, so 1002 * 333 = 333666. Divide it by 2, we get 166883.

Repeat the same procedure for the second sequence of 5s. Then, add the answers. However, as I mentioned previously, remember that we also have to remove the duplicates of 15, 30, 45, ..., 990.

The formula would be like this:

sum of multiple of 3 + sum of multiple of 5 - sum of multiple of 15