One of my favourite mathematical results is the famous formula
How quickly do these series converge to a value involving π?
This formula has a splendid history. It was derived in the West in 1671 by James Gregory from the formula for arctan(x) and slightly later and independently
by Gottfried Leibniz.
However, the same formula (along with many other results involving infinite series) was discovered long before in the
1300s by the great Indian mathematician Madhava.
Similar results of equal beauty are the convergent series given by
and
However, in one sense, these formulae are disappointing. If you want to actually calculate or then you probably would not reach for one of these formulae. The reason is that they converge very slowly. If you take formula (1), add up 100 terms and multiply by 4, you get 3.146567747182956, which whilst fairly close to is not a particularly accurate estimate given the effort involved in adding up 100 terms. If you wanted to calculate or to an accuracy of six decimal places, you would have to take on the order of terms of either series (1) or (2), and long before you have added up all of the terms in the series, the rounding errors associated with computer calculations will have accumulated to the point where the accuracy of the answer is severely degraded.
The world needs pi
So what, you (and many pure mathematicians) might say. Surely you don’t need to know the value of that accurately, after all the Bible was content to give it to just one significant figure. However, is not any number.
It lies at the heart of any technology that involves rotation or waves, and that is much of mechanical and electrical engineering. If rotating parts in, say, a typical jet engine are not manufactured to high tolerance, then the parts simply won’t rotate.
This typically involves measurements correct to one part in and, as these measurements involve , we require a value of to at least this order of accuracy to prevent errors.
In medical imaging using CAT or MRI scanners, the scanning devices move on a ring which has to be manufactured to a tolerance of one part in , requiring an even more precise value of .
However, even this level of accuracy pales into insignificance when we look at modern electrical devices.
In high frequency electronics, with frequencies in the order of 1GHz (typical for mobile phones or GPS applications), electrical engineers have to work with functions of the form where and is a number close to one.
To get the accuracy in the function needed for GPS to work requires a precision in the value used for in the order of one part in .
So, to live in the modern world we really do need to know very accurately.
So, what can we do? One possibility is to take a vast number of terms of the series for etc. above, book lots of time on a very expensive computer, sit back and wait (and wait, and wait). Or we can try and accelerate their convergence.
So that with only a small number of terms (say 10) we can get 10 significant figures for . The nice thing about this method is that the derivation of the formulae is very transparent (well within the reach of a first year undergraduate or even a good Alevel student).
In principle this method can also be used to find the sum of other slowly convergent series.
Accelerating the convergence of a series
Let’s suppose that we have a series
and we define the sum by
We will assume that this series converges. This means there is a limiting sum so that
How to Solve Math Problems Faster: 15 Techniques to Show Students
“Test time. No calculators.”
You’ll intimidate many students by saying this, but teaching techniques to solve math problems with ease and speed can make it less daunting.
This can also make math more rewarding. Instead of relying on calculators, students learn strategies that can improve their concentration and estimation skills while building number sense. And, while there are educators who oppose math “tricks” for valid reasons, proponents point to benefits such as increased confidence to handle difficult problems.
Here are 15 techniques to show students, helping them solve math problems faster:
Addition and Subtraction
1. TwoStep Addition
Many students struggle when learning to add integers of three digits or higher together, but changing the process’s steps can make it easier.
The first step is to add what’s easy. The second step is to add the rest.
Let’s say students must find the sum of 393 and 89. They should quickly see that adding 7 onto 393 will equal 400 — an easier number to work with. To balance the equation, they can then subtract 7 from 89.
Broken down, the process is:
 393 + 89
 (393 + 7) + (89 – 7)
 400 + 82
 482
With this fast technique, big numbers won’t look as scary now.
2. TwoStep Subtraction
There’s a similar method for subtraction.
Remove what’s easy. Then remove what’s left.
Suppose students must find the difference of 567 and 153. Most will feel that 500 is a simpler number than 567. So, they just have to take away 67 from the minuend — 567 — and the subtrahend — 153 — before solving the equation.
Here’s the process:
 567 – 153
 (567 – 67) – (153 – 67)
 500 – 86
 414
Instead of two complex numbers, students will only have to tackle one.
By making math engaging, students that use Prodigy consistently outperform those that don’t on standardized assessments
3. Subtracting from 1,000
You can give students confidence to handle fourdigit integers with this fast technique.
To subtract a number from 1,000, subtract that number’s first two digits from 9. Then, subtract the final digit from 10.
Let’s say students must solve 1,000 – 438. Here are the steps:
 9 – 4 = 5
 9 – 3 = 6
 10 – 8 = 2
 562
This also applies to 10,000, 100,000 and other integers that follow this pattern.
Multiplication and Division
4. Doubling and Halving
When students have to multiply two integers, they can speed up the process when one is an even number. They just need to halve the even number and double the other number.
Students can stop the process when they can no longer halve the even integer, or when the equation becomes manageable.
Using 33 x 48 as an example, here’s the process:
 66 x 24
 132 x 12
 264 x 6
 528 x 3
 1,584
The only prerequisite is understanding the 2 times table.
5. Multiplying by Powers of 2
 This tactic is a speedy variation of doubling and halving.
 It simplifies multiplication if a number in the equation is a power of 2, meaning it works for 2, 4, 8, 16 and so on.
 Here’s what to do: For each power of 2 that makes up that number, double the other number.
 For example, 9 x 16 is the same thing as 9 x (2 x 2 x 2 x 2) or 9 x 24. Students can therefore double 9 four times to reach the answer:
 9 x 24
 18 x 23
 36 x 22
 72 x 2
 144
10 Tricks for Doing Math Quickly in Your Head  Resilient Educator
You don’t have to be a math teacher to know that a lot of students—and likely a lot of parents (it’s been awhile!)—are intimidated by math problems, especially if they involve large numbers. Learning techniques on how to do math quickly can help students develop greater confidence in math, improve math skills and understanding, and excel in advanced courses.
If it’s your job to teach those, here’s a great refresher.
10 tricks for doing fast math
Here are 10 fast math strategies students (and adults!) can use to do math in their heads. Once these strategies are mastered, students should be able to accurately and confidently solve math problems that they once feared solving.
1. Adding large numbers
Adding large numbers just in your head can be difficult. This method shows how to simplify this process by making all the numbers a multiple of 10. Here is an example:
644 + 238
While these numbers are hard to contend with, rounding them up will make them more manageable. So, 644 becomes 650 and 238 becomes 240.
Now, add 650 and 240 together. The total is 890. To find the answer to the original equation, it must be determined how much we added to the numbers to round them up.
 650 – 644 = 6 and 240 – 238 = 2
 Now, add 6 and 2 together for a total of 8
 To find the answer to the original equation, 8 must be subtracted from the 890.
 890 – 8 = 882
 So the answer to 644 +238 is 882.
2. Subtracting from 1,000
 Here’s a basic rule to subtract a large number from 1,000: Subtract every number except the last from 9 and subtract the final number from 10
 For example:
 1,000 – 556
 Step 1: Subtract 5 from 9 = 4
 Step 2: Subtract 5 from 9 = 4
 Step 3: Subtract 6 from 10 = 4
 The answer is 444.
3. Multiplying 5 times any number
When multiplying the number 5 by an even number, there is a quick way to find the answer.
For example, 5 x 4 =
 Step 1: Take the number being multiplied by 5 and cut it in half, this makes the number 4 become the number 2.
 Step 2: Add a zero to the number to find the answer. In this case, the answer is 20.
 5 x 4 = 20
 When multiplying an odd number times 5, the formula is a bit different.
 For instance, consider 5 x 3.
 Step 1: Subtract one from the number being multiplied by 5, in this instance the number 3 becomes the number 2.
 Step 2: Now halve the number 2, which makes it the number 1. Make 5 the last digit. The number produced is 15, which is the answer.
5 x 3 = 15
4. Division tricks
Here’s a quick way to know when a number can be evenly divided by these certain numbers:
 10 if the number ends in 0
 9 when the digits are added together and the total is evenly divisible by 9
 8 if the last three digits are evenly divisible by 8 or are 000
 6 if it is an even number and when the digits are added together the answer is evenly divisible by 3
 5 if it ends in a 0 or 5
 4 if it ends in 00 or a two digit number that is evenly divisible by 4
 3 when the digits are added together and the result is evenly divisible by the number 3
 2 if it ends in 0, 2, 4, 6, or 8
5. Multiplying by 9
 This is an easy method that is helpful for multiplying any number by 9. Here is how it works:
 Let’s use the example of 9 x 3.
 Step 1: Subtract 1 from the number that is being multiplied by 9.
 3 – 1 = 2
 The number 2 is the first number in the answer to the equation.
 Step 2: Subtract that number from the number 9.
 9 – 2 = 7
 The number 7 is the second number in the answer to the equation.
 So, 9 x 3 = 27
6. 10 and 11 times tricks
The trick to multiplying any number by 10 is to add a zero to the end of the number. For example, 62 x 10 = 620.
There is also an easy trick for multiplying any twodigit number by 11. Here it is:
11 x 25
Take the original twodigit number and put a space between the digits. In this example, that number is 25.
 2_5
 Now add those two numbers together and put the result in the center:
 2_(2 + 5)_5
 2_7_5
 The answer to 11 x 25 is 275.
 If the numbers in the center add up to a number with two digits, insert the second number and add 1 to the first one. Here is an example for the equation 11 x 88
 8_(8 +8)_8
 (8 + 1)_6_8
 9_6_8
 There is the answer to 11 x 88: 968
7. Percentage
Finding a percentage of a number can be somewhat tricky, but thinking about it in the right terms makes it much easier to understand. For instance, to find out what 5% of 235 is, follow this method:
 Step 1: Move the decimal point over by one place, 235 becomes 23.5.
 Step 2: Divide 23.5 by the number 2, the answer is 11.75. That is also the answer to the original equation.
8. Quickly square a twodigit number that ends in 5
Let’s use the number 35 as an example.
 Step 1: Multiply the first digit by itself plus 1.
 Step 2: Put a 25 at the end.
 35 squared = [3 x (3 + 1)] & 25
 [3 x (3 + 1)] = 12
 12 & 25 = 1225
 35 squared = 1225
9. Tough multiplication
When multiplying large numbers, if one of the numbers is even, divide the first number in half, and then double the second number. This method will solve the problem quickly. For instance, consider
20 x 120
Step 1: Divide the 20 by 2, which equals 10. Double 120, which equals 240.
 Then multiply your two answers together.
 10 x 240 = 2400
 The answer to 20 x 120 is 2,400.
10. Multiplying numbers that end in zero
Multiplying numbers that end in zero is actually quite simple. It involves multiplying the other numbers together and then adding the zeros at the end. For instance, consider:
 200 x 400
 Step 1: Multiply the 2 times the 4
 2 x 4 = 8
 Step 2: Put all four of the zeros after the 8
 80,000
 200 x 400= 80,000
 Practicing these fast math tricks can help both students and teachers improve their math skills and become secure in their knowledge of mathematics—and unafraid to work with numbers in the future.
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Addition Tips and Tricks
Here are lots of “thinking tricks” you can use to make addition easier.
Use the ones that make sense to you!
Hint: start from the larger number.
2 + 6 is Harder: “2 … 3, 4, 5, 6, 7, 8”
6 + 2 is Easier: “6 … 7, 8”
So do 6 + 2 instead (you get the same answer).
Jump Strategy
We can also count by 2s or 10s, or make any “jumps” we want to help us solve a calculation.
Think “4 … 14 … 15, 16”
Adding Up To Ten
See if any numbers add to 10. They don't have to be next to each other.
 7+3 is 10,
 8+2 is another 10, which makes 20,
 Plus 5 is 25
Do The Tens Last
Break big numbers into Tens and Units, add the Units, then add on the Tens.
Example: 14+5
 Break the “14” into Tens and Units: 10 + 4
 Add the Units: 4 + 5 = 9
 Now add the Tens: 10 + 9 = 19
 Think “4 plus 5 is 9, plus 10 is 19”
 Break into Tens and Units: 10 + 4 + 10 + 2
 Add the Units: 4 + 2 = 6
 Now add on the Tens: 6 + 10 + 10 = 26
Aim for Ten
When a number is close to ten we can “borrow” from the other number so it reaches ten.
9 is only 1 away from 10 
so take 1 from the 7: 9 + 1 + 6 and give it to the 9: 10 + 6 = 16Think “9 plus 1 is 10 … 7 less 1 is 6 … together that is 16” 
8+2=10, so lets take 2 from the 5: 8 + 2 + 3 and give it to the 8: 10 + 3 = 13
We can also move backwards to ten, by making the other number bigger as needed:
Reduce 12 by 2: 12 − 2 = 10 Increase 7 by 2: 7 + 2 = 9
12 + 7 = 10 + 9 = 19
Compensation Method
“Compensation” is where you round up a number (to make adding easier) and then take away the extra after you have added.
It is easier to do 20 + 16 = 36
Then take away the extra 1 (that made 19 into 20) to get: 35
It is easier to do 400 + 126 = 526
Then take away the extra 5 (that made 395 into 400) to get: 521
Double when the numbers are the same
Example 5 + 5 = 2 x 5 = 10
5 + 6 = two 5s + 1 = 10 + 1 = 11
7 + 9 = “8 less 1” + “8 add 1” = two 8s = 16
We can also use the Addition Table to help us.
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Ways to add values in a spreadsheet
Excel for Microsoft 365 Excel for the web Excel 2019 Excel 2016 Excel 2013 Excel 2010 Excel 2007 More… Less
One quick and easy way to add values in Excel is to use AutoSum.
Just select an empty cell directly below a column of data. Then on the Formula tab, click AutoSum > Sum. Excel will automatically sense the range to be summed.
(AutoSum can also work horizontally if you select an empty cell to the right of the cells to be summed.)
AutoSum creates the formula for you, so that you don't have to do the typing. However, if you prefer typing the formula yourself, see the SUM function.
 Use the SUMIF function when you want to sum values with one condition. For example, when you need to add up the total sales of a certain product.
 Use the SUMIFS function when you want to sum values with more than one condition. For instance, you might want to add up the total sales of a certain product, within a certain sales region.
For an overview of how to add or subtract dates, see Add or subtract dates. For more complex date calculations, see Date and time functions.
For an overview of how to add or subtract time, see Add or subtract time. For other time calculations, see Date and time functions.
You can always ask an expert in the Excel Tech Community, get support in the Answers community, or suggest a new feature or improvement on Excel User Voice.
7 practical tips for mental math (that ANYONE can use!)
You have most likely heard about mental math — the ability to do calculations in one's head — and how important it is for children to learn it.
But why is it important? Because mental math relates to NUMBER SENSE: the ability to manipulate numbers in one's head in various ways in order to do calculations. And number sense, in return, has been proven to predict a student's success in algebra.
Essentially, what we do with variables in algebra is the same as what students can learn do with numbers in the lower grades.
People with number sense use numbers flexibly. They are able to take them apart and put them together in various ways in order to do calculations. It is quite similar to being able to “PLAY” with words in order to make interesting sentences, or being able to play with chords and melodies in order to make songs.
But mental math/number sense isn't just for “math whizzes” — quite the contrary! EVERYONE can learn the basics of it, and it will make learning math and algebra so much easier! We expect our children to learn lots of English words and to be able to put those words together in many different ways to form sentences, so why not expect them to do the same with numbers? And they can, as long as they are shown the basics and shown examples of how it happens. So let's get on to the practical part of this writing: mental math strategies for EVERYONE.
 The “9trick”.
To add 9 to any number, first add 10, and then subtract 1. In my Math Mammoth books, I give children this storyline where nine really badly wants to be 10… so, it asks this other number for “one”. The other number then becomes one less. For example, we change the addition 9 + 7 to 10 + 6, which is much easier to solve.
But this “trick” expands. Can you think of an easy way to add 76 + 99? Change it to 75 + 100. How about 385 + 999?
How would you add 39 + 28 in your head? Let 39 become 40… which reduces 28 to 27. The addition is now 40 + 27. Yet another way is by thinking of compensation: 39 is one less than 40, and 28 is two less than 30. So, their sum is three less than 70.
 Doubles + 1.
Encourage children to memorize the doubles from 1 + 1 through 9 + 9. After that, a whole lot of other addition facts are at their fingertips: the ones we can term “doubles plus one more”. For example, 5 + 6 is just one more than 5 + 5, or 9 + 8 is just one more than 8 + 8.
 Use addition facts when adding bigger numbers.
Once you know that 7 + 8 = 15, then you will also be able to do all these additions in your head:
 70 + 80 is 15 tens, or 150
 700 + 800 is 15 hundreds, or 1500
 27 + 8 is 20 and 15, which is 35. Or, think this way: since 7 + 8 is five more than ten, then 27 + 8 is five more than the next ten.
 Subtract by adding.
How to Add
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1
Use a number line if you’re just learning to add. Draw a line, then write numbers along the line from 015. Circle the first number you want to add. Start at that number. Then, count down your number line, moving the same number of spaces as the second number you’re adding. You’ll land on your answer.[1]
 Let’s say you want to add 4+5. Circle 4 on your number line, then count 5 spaces down the line. You will land on 9, which is your answer.
 Use your number line to add these numbers:

2
Add using beans to get more practice. Start with a basic math problem, such as 4+6. Create two piles of beans to represent your problem, including one group of 4 beans and one group of 6 beans. Next, you can combine your two sets of beans to get the answer to your addition problem. Count your beans to see that you now have a total of 10 beans.[2]
 If you don’t have beans, you can use any small item that fits in your hand to practice addition! For example, you can use blocks, candies, coins, or legos.
 Use your beans to add these numbers:

3
Draw oranges for each number to help you see how adding works. If you were adding 7+4, you’d draw one group of 7 oranges and one group of 4 oranges. Then, count all of the oranges to see how many you have when you add them together, which is 11 oranges. This is your answer.[3]
 You can draw any object to represent your numbers, but it’s best to choose something easy. As another option, you can use stickers!
 Use drawing to add these numbers:

1
Learn the number places. Each digit in a number has a name, and knowing them will help you add. Number places go from left to right:[4]
 The spot on the left is the “ones” place.
 The second spot from the left is the “tens” place.
 The third spot from the left is the “hundreds” place.
 Here’s an example: In the number 583, the 3 is in the ones place, the 8 is in the tens place, and the 5 is in the hundreds place.

2
Write out your problem vertically. Line up your numbers so that each digit place is in a row. This makes it easier to add each column of numbers to get a final sum.[5]
 If your problem is presented horizontally on a worksheet, it helps to rewrite it vertically to make it easier to solve.

3
Line up the numbers. Each number place should be in its own line with each number stacked vertically. If one number uses fewer number spaces than another, leave the left spot blank. Here's an example:[6]
 Here’s how you’d write 16+4+342:
 342
 _16
 +_4

4
Add the ones column first. The ones column is on the right. Once you have the sum of these numbers, write the ones digit of the sum in the ones place of your answer spot. If you have a tens digit in your sum, write it above the tens column in your problem.[7]
 For 342+16+4, you’d add 2+6+4=12. Write the 2 in your ones place in your answer. Carry the other 1 into your tens column.

5
Carry the tens from the ones sum into the column of your problem. If you have a number in the tens place, write the tens place number at the top of the tens column. This is the column located to the left of your ones column. You will include this number in the tens total.[8]
 In the above example, you’d write the 1 from the tens place in 12 on top of the tens column.

6
Count the next column. Go to the tens column, which is the next column to the left. Add up the numbers in this column, including your carry over number, if you had one. Write the ones place in this sum in the tens spot on your answer, then carry over the tens spot in your sum, if you had one.[9]
 For 342+16+4, you’d add 4+1+1=6. Remember, the second 1 is the carry over from your ones sum. You’d write 6 down in the tens spot of your answer. You don’t have any carry over from this sum.

7
Continue working these steps until you get a final sum. For longer problems, you’ll need to add each column, moving from right to left. For each column, write the ones place of that column’s sum in the corresponding number place in the answer. Then, carry over the tens place of the sum in the next column to the left.[10]
 When you complete the last column on the right, you’ll have your final sum.
 In our example, you only have one number in the hundreds column, so you’d carry down the 3 to your answer. Your final sum for 342+16+4=362.

8
Follow the same steps when adding decimals. Although it seems challenging, you use the same process to add decimals as you do with whole numbers. Just make sure you line up each numbers place properly, including your decimals. If a number in the problem doesn’t have a decimal, add a .0 to make it easier to workout the problem. Here’s an example:[11]

1
Round all of your numbers to multiples of ten or one hundred. Multiples of ten and one hundred are much easier to add together! Multiples of ten are easier to manage, but using hundreds can be helpful for larger numbers.[12]
 Always round up because it’s easier to keep track of how much you’re adding to your original numbers as you round.
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