Multiply numbers - Excel Excel Applies To: Excel 2010 More... Less Let's say you want to figure out how much bottled water that you need for a customer conference (total attendees × 4 days × 3 bottles per day) or the reimbursement travel cost for a business trip (total miles × 0.46). There are several ways to multiply numbers. What do you want to do? Multiply numbers in a cell To do this task, use the * (asterisk) arithmetic operator. For example, if you type =5*10 in a cell, the cell displays the result, 50. Top of Page Multiply a column of numbers by a constant number Suppose you want to multiply each cell in a column of seven numbers by a number that is contained in another cell. In this example, the number you want to multiply by is 3, contained in cell C2. 1 4306 =A8*$C$2 Type =A2*$C$2 in cell B2. Be sure to include a $ symbol before C and before 2 in the formula. Drag the formula in B2 down to the other cells in column B.

Note: Using $ symbols tells Excel that the reference to C2 is "absolute," which means that when you copy the formula to another cell, the reference will always be to cell C2. If you didn't use $ symbols in the formula and you dragged the formula down to cell B3, Excel would change the formula to =A3*C3, which wouldn't work, because there is no value in C3. Multiply numbers in different cells by using a formula To do this task, use the asterisk (*) operator or the PRODUCT function. Example The example may be easier to understand if you copy it to a blank worksheet. How to copy an example Create a blank workbook or worksheet. Select the example in the Help topic. Note: Do not select the row or column headers. Selecting an example from Help Press CTRL+C. In the worksheet, select cell A1, and press CTRL+V. To switch between viewing the results and viewing the formulas that return the results, press CTRL+` (grave accent), or on the Formulas tab, in the Formula Auditing group, click the Show Formulas button. 1 Exponents | Wyzant Resources Certain numbers as exponents cause the base to behave in a special way.

Zero Exponent As a general rule, any variable or non-zero number raised to the power zero is equal to one. This is always true regardless of what the base a is. The base can be a non-zero number or a variable, positive or negative and, as long as it is raised to the power zero, it will equal to one. The only exception is the number zero itself because when zero is raised to the power zero, the result is still zero. So it is important to remember that any other number, except zero, when raised to the power of zero is equal to one. For example: Positive One Exponent The number one is another interesting exponent. Any number raised to the power of positive one is equal to the number itself. This is regardless of whether the number is positive or negative. and Odd and Even Whole Number Exponents When a positive number is raised to an odd or even whole number, the result is always positive. Negative numbers, on the other hand, behave differently. When a negative number is raised to an even whole number power, the result will ALWAYS be a positive number.

For example: The above is also true if the exponent is a negative whole number: You can try out different negative numbers and raise them to even whole numbers and the result is always a positive number. Conversely, when a negative base is raised to an odd whole number exponent, the result is ALWAYS a negative number. For example: the same applies to negative exponents: Exponents of Ten Ten (10) is a special number because raising ten to any whole number exponent is the in effect adding a number of trailing zeros to 10, and these are as many as the value of the exponent, i.e. where the last 0 is the nth zero. For example: Fraction Exponents Exponents can be whole numbers or fractions. Whole number exponents have been discussed above. Fractional Exponents behave different, instead of having the effect of multiplication of the base by itself, they have the effect of finding the root of the base. i.e. is the same as finding the nth root of a, which is also written as: Thus to find the square root of a number, we can also write: n can be positive or negative, but in order to get real roots, a must be positive. Negative a would result in complex roots.

For example; Properties of Exponents Numbers or variables with exponents satisfy the following properties which makes them easy to manipulate. If the bases in a given exponential equation are equal, the exponents are also equal. i.e. Solving Exponential Equations Exponential equations are those with polynomials as exponents, for example: Ordinarily, such exponential equations would be solved by using logarithms (refer to section on logarithms), but some exponential equations can be solved using the property mentioned above applying this property to the equations above: would become which then implies that x = 2 since the bases are the same Similarly, is the same as from which you would proceed to solve for x. From the above example, you should have noticed that we had to put all the components of the exponential equation into the same base in order to apply that property. So this property will only apply if the bases can be expressed in one base. For example, while the property can be used to solve the equation below because 128 and 16 can be expressed as exponential functions of the same base, i.e.

which using another of the properties mentioned before becomes which then leads to: Other Number Properties | Purplemath They didn't change anything, so this is an identity. They multiplied, so this is: the multiplicative identity This is obvious: anything equals itself. They used: the reflexive property. If 10 = y, then y = 10. They only switched sides; they didn't actually change anything. All they did was swap across the "equals" sign. They used: the symmetric property. They didn't change anything, so this is an identity. They added, so they used: the additive identity. If 2(a + b) = 3c, and a + b = 9, then 2(9) = 3c. You might be torn here between the transitive property and the substitution property. If you look closely, they didn't say that 2(a + b) equalled something, which then equalled 3c. Instead, what they did was substitute the "9" from the second equation for the "a + b" in the first equation, so they used: the substitution property.

2 = x, so 2 + 5 = x + 5 Usually I'd do something to both sides of an equation in order to get the variable by itself. Here, they made the equation more complicated by adding to both sides, so now the variable is not by itself. But the point here is that all they did was add the same thing to both sides. That is, they used: the additive property of equality. Content Continues Below If x + 2 = 10, then x + 2 + (–2) equals what, and why? If they'd just given me the first equation and asked me to solve it, I'd have subtracted 2 from both sides, and simplified to get x = 8. Here, they've made things complicated. But all they've really done is show only the left-hand side of the subtraction (or, to be technical, the addition of a "minus" number), leaving me to figure out that they're wanting me to picture the right-hand side and given them their answer. Since all they really did was add the same thing to both sides, they got: x = 8 by the additive property of equality.

(x – 3)(x + 4) = 0, so x = 3 or x = –4. They set the quadratic equal to zero, factored, and then solved each factor. This is: the zero-product property. 4x = 8, so x = 2 They solved the equation by dividing both sides by 4, or, which is the same thing, multiplying both sides by . In other words, they changed the equation by doing the same multiplying to both sides. They used: the multiplicative property of equality. If x is not equal to y and is not less than y, what must be true of x, and why? By the trichotomy law, there are only three possible relationships between x and y (namely, to be greater than, to be equal to, or to be less than), and they've eliminated two of these options. Then: x > y, by the trichotomy law. x + (–x) = 0 They added, and they ended up with zero. Zero is the thing which, when added to other things, doesn't change anything. So zero is the additive identity. Since they ended up with the additive identity, then they used: the additive inverse.

They converted to a common denominator by multiplying both fractions by a useful form of 1; remember that and are just useful forms of 1! By multiplying by 1, they didn't actually change the numbers; they only changed how the numbers are being stated. Since they didn't really change anything, this is an identity. Since they multiplied, they used: the multiplicative identity. 4 Ways to Add 5 Consecutive Numbers Quickly - wikiHow Using the Middle Number 1 Mentally multiply the middle number by 5 ... finished!? That's all you do!. e.g. with your example 53 X 5 = 265. Here's how to multiply this mentally: First separate 53 into 50 and 3. Now 50 X 5 = 250. Also 3 X 5 = 15. Now add those separate answers. 250 + 15 = 265. 2 Learn how it works: Let the smallest number be (x - 2). Then the other 4 are (x - 1), (x), (x + 1) and (x + 2).

The sum: (x - 2) + (x - 1) + (x) + (x + 1) + (x + 2) = 5x Using the above method: 10x/2 = 5x Method If this question (or a similar one) is answered twice in this section, please click here to let us know. Tips You can add any sequence of consecutive numbers, even or odd, regardless of the number of integers the sequence contains, by adding the first and the last number in the sequence, dividing it by two and multiplying the answer by the number of integers in the sequence. In algebra , we can say ((a+b)/2)*n, or rearrange it to n*(a+b)/2 to remove a set of parenthesis. Method 2 can be used for any odd amount of consecutive numbers, but instead of using "5x" you must use "(amount of consecutive numbers)x" e.g. in 6 + 7 + 8, seven is x. (3)7 = 21, and 6 + 7 + 8 = 21 Advanced Usage They don't need to be consecutive numbers.

They just need to be a sequential subset of any linear equation. ( The examples above use the linear equation of x = c + 1 * n ) For example, let's use the linear equation x = 10 + 7y, therefore, {xϵN| 17,24,31,38,45,...} So if we use: 17, 24, 31, 38, 45 31 x 10 = 310 and 310/2 = 155 They also do not need to be integers. Let's use the linear equation x = 1 + y / 20, therefore, {xϵN| 1.05, 1.1, 1.15, 1.2, 1.25, ...} So if we use: 1.05, 1.1, 1.15, 1.2, 1.25 1.15 x 10 = 11.5 and 11.5/2 = 5.75 They also do not need to be positive values.

The set can contain negative numbers, positive numbers, or both. This method can be extended to be used (as above) for any ODD number of consecutive integers --> 5, 7, 13, 25, 99 consecutive as long as you can locate the median digit and then multiply it be the number of integers. (Example 12, 13, 14, 15, 16, 17, 18, 19, 20 = 144 = 16(median) x 9(integer count). This can be most impressive if coupled with the simple trick of multiplying by 11 . Video Prime numbers and prime factorization - A complete course in arithmetic It means that the larger number can be composed -- made up -- of the smaller number. 2 is a proper divisor of 10 because 10 can be additively composed of 2's: 5 and 1 are also proper divisors of 10, because 10 can be composed of 5's and 1's.

As for 10 itself, we are not interested that 10 is equal to one 10; we want to know what other numbers will compose it. Nevertheless, 10 is called a divisor of itself, but not a proper divisor. The proper divisors do not include the number itself. It is important to note that 1 is a proper divisor of every natural number (except itself), because every natural number is a sum of 1's. That is what a natural number is. 5 = 1 + 1 + 1 + 1 + 1. Problem 1. a) Which numbers are the divisors of 12? To see the answer, pass your mouse over the colored area. To cover the answer again, click "Refresh" ("Reload"). Do the problem yourself first! 1, 2, 3, 4, 6, 12. a) Which are the proper divisors of 12? 1, 2, 3, 4, 6. b) 16 can be composed of which other numbers? 1, 2, 4, 8. Those are the proper divisors of 16.

c) 17 can be composed of which other numbers. 1. 1 is its only proper divisor. d) Write all the proper divisors of 29. 1. A prime number is a number whose only proper divisor is 1. Thus a prime number can be composed only of 1's. 17 and 29 are prime numbers. Again, 1 is a proper divisor of every number (except itself), but a prime number has 1 as its only proper divisor. Problem 2. Is 1 itself a prime number? No. 1 has no proper divisors. 1 cannot be composed of other numbers. 1 in this case is the measure. It cannot be measured. Problem 3. Write the first ten prime numbers. 2, 3, 5, 7, 11, 13, 17, 19, 23, 29. With the exception of 2, then, -- which is the only even prime -- a prime number is a kind of odd number. 3. A composite number. 6 is a composite number, because it can be composed of other numbers besides 1; namely 2 and 3. 1 itself is neither prime nor composite.

It is the measure by which we decide whether a number is prime or composite. Square roots When we multiply a number by itself, we say that we have "squared" the number. Thus the square of 5 is 5 × 5 = 25. We then say that 5 is the square root of 25. The square numbers are the numbers we get by squaring a number: 1, 4, 9, 16, 25, and so on. 50, for example, is not a square number, therefore it does not have an exact square root. Its square root is closest to 7, however, because 7 × 7 = 49. Problem 4. The square root of 175 is closest to what number? 13. 13 × 13= 169. Divisors We can always find the divisors of a number in pairs. One member of the pair will be less than the square root of the number, and the other will be more.

(If the number is a square number, then its square root will be its own partner.) For example, here are the pairs of divisors of 24: 1 and 24. (Because 1 × 24 = 24.) 2 and 12. (Because 2 × 12 = 24.) 3 and 8. (Because 3 × 8 = 24.) 4 and 6. (Because 4 × 6 = 24.) Each number on the left is less than the square root of 24, and each number on the right is more. The point is: When we look for divisors of a number, it is necessary to look only up to its square root. Because for each divisor we find less than the square root, we will have found its partner, which is more. Problem 5. If we are looking for the divisors of 157, up to what number must we look? 12. Because 13 is more than the square root of 157. In this Lesson we will be looking for the prime divisors of a number, and to that we now turn.

Prime divisors Every number except 1 is either prime or composite. And if a number is composite, it can be composed of at least one prime number. It will have at least one prime divisor.. (Euclid, VII. 31) 28 is composite. It is composed of fourteen 2's, and it is composed of four 7's. 2 and 7 are prime divisors of 28. Example 1. 63 could be composed of which prime numbers? Answer. Here again are the first few prime numbers: 2, 3, 5, 7, 11, 13, 17. We must test whether 2 is a divisor, or 3, or 5, and so on. But we need test only up to 7, because 11 is more than the square root of 63. Is 2 a divisor of 63? No, it is not. How do we know? Because 63 is not an even number. And how do we know that? Because even numbers end in 0, 2, 4, 6, or 8. Next, is 3 a divisor of 63?

There is a test for divisibility by 3, and it is as follows: A number is divisible by 3 if the sum of the digits is divisible by 3. The sum of the digits of 63 is 6 + 3 = 9, which is divisible by 3. That tells us that 63 can be composed of 3's. Next, is 63 divisible by 5? There is a simple test for divisibility by 5, namely, the number ends in either 0 or 5. 63 does not end in 0 or 5. Therefore, 63 is not divisible by 5. Finally, is 63 divisible by 7? Yes, it is. 63 is equal to nine 7's. 63, then, has could be composed of the prime numbers 3 and 7. Example 2. Which prime numbers are divisors of 78? Answer. 2 is a prime divisor: 78 = 2 × 39. Now, 39 is composite. Therefore it has a prime divisor: 39 = 3 × 13. 39 is thus divisible by both 3 and 13.

And since 78 is divisible by 39, then 78 is also divisible by 3 and 13.. That is called the transitive property of " is divisible by." If a is divisible by b, and b is divisible by c, then a is divisible by c. 78 therefore has three prime divisors: 2, 3, and 13. Problem 6. What is the smallest prime that is a divisor of the following? a) 231. 3. The sum of the digits is divisible by 3. b) 3,165. 3. When numbers are multiplied, they are called factors. Since 30 = 2 × 15, we say that 2 and 15 are factors of 30. Now, 2 is a prime factor -- but 15 is not. However, 15 = 3 × 5. Therefore we can express 30 as a product of prime factors only: 30 = 2 × 3 × 5. "2 × 3 × 5" is called the prime factorization of 30. And it is unique.

That is, apart from the order of the factors: Every composite number can be uniquely factored as a product of prime numbers only. Note: We could have found those same factors by factoring 30 in any way. For example, 30 = 5 × 6. And since 6 = 2 × 3, 30 = 5 × 2 × 3. Apart from the order, we have found the same prime factors. Example 3. Find the prime factorization of 102. Solution. 2 is obviously a prime factor. Its partner will be half of 102 -- 51 ( Lesson 16 ): 102 = 2 × 51. Because the sum of the digits is divisible by 3, we know that 51 has a prime factor 3. Now, 3 times what is equal to 51? On mentally decomposing 51 into 30 + 21, 51 Problem 8. Find the square factors of each number by writing its prime factorization. a) 112 = 2 × 2 × 2 × 2 × 7 = 16 × 7.

b) 450 = 3 × 3 × 5 × 5 × 2 = 3 × 5 × 3 × 5 = 225 × 2. c) 153 = 3 × 51 = 3 × 3 × 17 = 9 × 17. d) 294 = 2 × 147 = 2 × 3 × 49 = 49 × 6. e) 1225 = 25 × 49. 1225 is itself a square number. It is the square of 5 × 7 = 35. Is there a last prime? As the numbers get larger, the greater the possibility that they will have a divisor, so that there might in fact be a last prime. Now, if there were a last prime, then we could imagine a list that contains every prime up to and including the last one. We will now prove that there is no such list, which is to say, There is no last prime. Here is the theorem: Given any list of prime numbers, there will always be a prime number that is not on the list.

Let the following, then, be any list of prime numbers: 2, 3, 5, 7, 11, . . . , P. Now construct the number N which will be the product of every prime on that list: N = 2 × 3 × 5 × 7 × 11 × . . . × P. Every prime on the list is thus a divisor of N. Add 1 to N: N + 1 = (2 × 3 × 5 × 7 × 11 × . . . × P) + 1. The first thing to note is that N + 1 is not on the list, because it is greater than every number on the list. Now, N + 1 is either prime or composite. If it is prime, then we have found a prime that is not on the list, and the theorem is proved. If N + 1 is composite, then it has a prime factor p. But p is not one of the primes on the list For if it were, then p would be a divisor of both N + 1 and N.

That would imply that p divides their difference ( Lesson 11 ), namely 1 -- which is absurd. Therefore, if N + 1 is composite, then there is a prime p that is a divisor of N + 1 but not a divisor of N , which is to say, p is not a prime on the list. Therefore, given any list of primes, there will always be a prime that is not on the list. Which is what we wanted to prove. * A modern enunciation of this theorem is: The number of primes is infinite. Euclid thus teaches us what we mean, or rather what we should mean, when we say that a collection of numbers is "infinite." It means: "No list of them is complete; there will always be more." That meaning of "infinite" refers to something that we could actually sense -- actually be aware of -- namely a list. It does not refer to something that we cannot be aware of: a list that has no end. * A famous problem in mathematics is the twin prime conjecture. It states that there are infinitely many pairs of primes that differ only by 2.

For example, 5 and 7, 17 and 19, 41 and 43. The conjecture has never been proved. What about prime triples, which are three primes that differ only by 2? For example, 3, 5, 7. What do you think? Are there many -- perhaps even an infinite number -- of such triples? Or is 3, 5, 7 the only one? 3, 5, 7 is the only such triple. Because in every sequence of three odd numbers, at least one of them is a multiple of 3. If the first is a multiple of 3, then the proposition is proved; for example, 21, 23, 25. 21 is not a prime. If the first is 1 more than a multiple of 3, then, on adding 2, the next will be a multiple of 3; for example, 25, 27, 29.

Finally, if the first is 1 less than a multiple of 3, then the next will be 1 more, and the third will be a multiple of 3; for example, 35, 37, 39. In the sequence, 3, 5, 7, 3 is the only multiple of 3 that is a prime. Next Lesson: Math Tricks - Behavioral Sciences Research Core - UMASS Medical School Behavioral Sciences Research Core Math Tricks for All Ages This web page is devoted to the incredibly boffo idea that math can be fun! Try these tricks: Here are some interesting links: A Reading list of Tricky Math Books, most of which I have used for this site. Learn about the original computer: The Abacus Play a Math-Chase Game -- for one or two players. (If you're using Netscape, Do Not Scroll down the page while this loads. Test your knowledge of the multiplication tables . Try your hand at estimation . Explore Geometry in a fun and interactive way. See what a Mandelbrot set is. If you want more math challenges try the new PBS MATHLINE MATH CHALLENGES site. Try it, you'll like it.

(But remember we were first.) Magic Addition Trick #1 Amaze the peons with this one. It's simple. It's effective. It gets them every time. Ask your mark to pick three (3) different numbers between 1 and 9. Tell him or her (or her or him) to write the three numbers down next to each other, largest first and smallest last, to form a single 3-digit number. Tell him/her not to tell you what the numbers are. Next have her or him form a new 3-digit number by reversing the digits, putting the smallest first and the largest last. And write this number right underneath the first number. Now have him or her subtract the lower (and smaller) 3-digit number from the upper (and larger) 3-digit number. Tell them not to tell you what the result is. Now you have a choice of wrap-ups: Ask your friend to add up the three digits of the number that results from subtracting the smaller from the larger 3-digit number. Then amaze him or her by teling them what the sum of those three numbers is. The sum of the three digit answer will always be 18!

Tell your friend that if she or he will tell you what the first OR last digit of the answer is, you will tell her or him what the other two digits are. This is possible because the middle digit will always be 9, and the other two digits will always sum to 9! So to get the digit other than the middle one (which is 9) and other than the digit that your friend tells you, just subtract the digit your friend tells you from 9, and that is the unknown digit. Back to Top A Recipe for Your Own 3 X 3 Magic Square Here's a recipe for making your own 3 X 3 magic number square. This recipe and both of the above two magic squares comes from one heck of a great book called, Mathematics for the Million, by Lancelot Hogben, published by Norton and Company. I highly recommend it. You don't need much math at all to get into the adventure of numbers told in this classic book. Some necessary rules and definitions: Let the letters a, b, and c stand for integers (that is, whole numbers). Always choose a so that it larger than the sum of b and c. That is, a > b + c.

This guarantees no entries in the magic square is a negative number. Do not let 2 X b = c. This quarantees you won't get the same number in different cells. Using the formulas in the table below, you can make magic squares where the sum of the rows, columns, and diagonals are equal to 3 X whatever a is. a - b + c a - c To create the first Magic Square #15 above, you let a be equal to 5, let b be equal to 3, and let c be equal to 1. Here are some others: a = 6, b = 3, c = 2 a = 6, b = 3, c = 1 a = 7, b = 3, c = 2 a = 7, b = 4, c = 2 a = 8, b = 6, c = 1 a = 8, b = 5, c = 2 a = 8, b = 4, c = 3 Try making up some of your own.

Back to Top Upside Down Magic Square Here's a magic square that not only adds up to 264 in all directions, but it does it even when it's upside down! If you don't believe me, look at it while you are standing on your head! (Or, just copy it out and turn it upside down.) Back to Top Win Bets with this Magic Square OK, here's a neat way to win bets with a magic square. Call a friend on the phone. Have him or her get a pencil and paper and bring it to the phone, so he or she can write down numbers from 1 to 9. Tell your friend that you will take turns calling out numbers from 1 to 9. Neither one of you can repeat a number that the other one calls out. Both of you then write down the numbers 1 to 9. Then when your friend says one of the numbers he or she draws a circle around that number, and so do you. When you say a number, you draw a square around that number, and so does your friend. The winner is whoever is the first one to get three numbers that add up exactly to 15. Say you go first and you call out 8.

Your friend might call out 6. You then call out 2. Your friend calls out 5, and you call out 4. Your friend calls out 7, and you call out 3. Then you tell your friend that you have just won because you called out 8, 3, and 4, which add up to 15. Your friend will want to play again. So this time you can bet him you'll win, with the condition that in case of a draw (where you use up the numbers 1 to 9 without either of you getting a 15 total) nobody owes anything. If you know the trick, you will never lose, and will probably will most times. The tricks Actually the trick is based on both tic-tac-toe and a magic square. The magic square look like this: 9 2 Because this is a magic square, every row and every column and every diagonal adds up to 15. So if you've got this square in front of you with your friend on the phone, you can put an X in the squares of the number you call out, and an O in the squares of the numbers your friend calls out.

Then, just like in tic-tac-toe, you try to get three X's in a row, because those will always add up to 15. So in the example above, when you call out 8, you put an X in the upper left corner. When your friend says 6, you put an ) in the upper right corner. And so on. Back to Top Mathematical Card Trick You need an ordinary pack of cards for this wower. No fancy shuffling is required. Just follow these easy steps: Shuffle the cards to mix them thoroughly. Deal out 36 cards into a pile. Ask a friend to pick one of the 36 cards, look at it and memorize it, and then put it back in the pile without letting you see it. Shuffle the 36 cards. Lay the 36 cards out in 6 rows of 6 cards each. Be sure to deal the top row from left to right. Then deal the second row beneath it from left to right. And so on with each succeeding row laid out underneath the one before. Ask your friend to look at the cards and tell you which row the chosen card is in. Remember what number the row is.

Carefully pick the cards up in the same order you laid them down. So the first card on the left of the top row is on top of the stack, and the last card on the right of the bottom row is on the bottom of the stack. Now lay the cards down in 6 rows of 6 cards each, but this time spread the card one column at a time. Instead of proceeding from one row to the next, proceed from one column to the next. Lay the first six cards in a column from top to bottom on the far left. Then lay out the next six cards in a second column of six cards just to the right of the first column of six cards. Continue doing this until you have 6 columns of 6 cards each (which looks the same as 6 rows of 6 cards each -- because it is the same). Once again ask your friend which row contains the chosen card. When you friend tells you which row the card is in, you can say what the exact chosen card is. How? If your friend said the card was in row 2 the first time, and in row 5 the second time, then the chosen card is the one in the second column of the fifth row.

This is because the way you arrange the cards, what were rows the first time around become columns the second time around. Back to Top Lightning Calculator Here's a trick to wow them everytime! Have someone write down their Social Security number. Then have them rewrite it so that it is all scrambled up. (If they don't have a Social Security number, have them write down any 9 digits between 1 and 9.) If there are any zeroes, have them change them to any other number between 1 and 9. Then have them copy their nine numbers, in the same order, right next to the orginal nine numbers. This will give them a number with 18 digits in it, with the first half the same as the second half. Next change the second digit to a 7, and change the eleventh digit (this will be the same number as the second digit but in the second nine digits) to a 7 also. Then bet them that you can tell them what is left after dividing the number by 7 faster than they can figure it out by hand. The answer is 0 -- 7 divides into this new number exactly with nothing left over!

Back to Top Fun Number Tables The following fun tables are from one of my favorite books of all time, Recreations in the Theory of Numbers, by Albert H. Beiler, published by Dover Publications. This book actually explains the mathematical reasons these tricks work.

3 x 37 = 111 and 1 + 1 + 1 = 3 6 x 37 = 222 and 2 + 2 + 2 = 6 9 x 37 = 333 and 3 + 3 + 3 = 9 12 x 37 = 444 and 4 + 4 + 4 = 12 15 x 37 = 555 and 5 + 5 + 5 = 15 18 x 37 = 666 and 6 + 6 + 6 = 18 21 x 37 = 777 and 7 + 7 + 7 = 21 24 x 37 = 888 and 8 + 8 + 8 = 24 27 x 37 = 999 and 9 + 9 + 9 = 27 1 x 9 + 2 = 11 12 x 9 + 3 = 111 123 x 9 + 4 = 

1111 1234 x 9 + 5 = 11111 12345 x 9 + 6 = 111111 123456 x 9 + 7 = 1111111 1234567 x 9 + 8 = 11111111 12345678 x 9 + 9 = 111111111 123456789 x 9 +10 = 1111111111 9 x 9 + 7 = 88 98 x 9 + 6 = 888 987 x 9 + 5 = 8888 9876 x 9 + 4 = 88888 98765 x 9 + 3 = 888888 987654 x 9 + 2 = 8888888 9876543 x 9 +

 1 = 88888888 98765432 x 9 + 0 = 888888888 1 x 8 + 1 = 9 12 x 8 + 2 = 98 123 x 8 + 3 = 987 1234 x 8 + 4 = 9876 12345 x 8 + 5 = 98765 123456 x 8 + 6 = 987654 1234567 x 8 + 7 = 9876543 12345678 x 8 + 8 = 98765432 123456789 x 8 + 9 = 987654321 Back to Top Did You Know...?

Each and every 2-digit number that ends with a 9 is the sum of the multiple of the two digits plus the sum of the 2 digits. Thus, for example, 29= (2 X 9) + (2 + 9). 2 X 9 = 18. 2 + 9 = 11. 18 + 11 = 29. 40 is a unique number because when written as "forty" it is the only number whose letters are in alphabetical order. A prime number is an integer greater than 1 that cannot be divided evenly by any other integer but itself (and 1). 2, 3, 5, 7, 11, 13, and 17 are examples of prime numbers. 139 and 149 are the first consecutive primes differing by 10. 69 is the only number whose square and cube between them use all of the digits 0 to 9 once each: 692 = 4761 and 693 = 328,509.

One pound of iron contains an estimated 4,891,500,000,000,000,000,000,000 atoms. There are some 318,979,564,000 possible ways of playing the first four moves on each side in a game of chess. The earth travels over one and a half million miles every day. There are 2,500,000 rivets in the Eiffel Tower. If all of the blood vessels in the human body were laid end to end, they would stretch for 100,000 miles. Back to Top A Math Trick for This Year This one will supposedly only work in 1998, but actually one change will let it work for any year. 1. Pick the number of days a week that you would like to go out (1-7). 2. Multiply this number by 2. 3. Add 5. 4. Multiply the new total by 50. 5. In 1998, if you have already had your birthday this year, add 1748.

If not, add 1747. In 1999, just add 1 to these two numbers (so add 1749 if you already had your birthday, and add 1748 if you haven't). In 2000, the number change to 1749 and 1748. And so on. 6. Subtract the four digit year that you were born (19XX). Results: You should have a three-digit number. The first digit of this number was the number of days you want to go out each week (1-7). The last two digits are your age. (Thanks for passing this one on to me, Judy.) Back to Top Where is the String? The next time you are with a group of people, and you want to impress them with your psychic powers, try this. Number everyone in the group from 1 to however many there are. Get a piece of string, and tell them to tie it on someone's finger while you leave the room or turn your back.

Then say you can tell them not only who has it, but which hand and which finger it is on, if they will just do some easy math for you and tell you the answers. Then ask one of them to answer the following questions: 1. Multiply the number of the person with the string by 2. 2. Add 3. 3. Multiply the result by 5. 4. If the string is on the right hand add 8. If the string is on the left hand add 9. 5. Multiply by 10. 6. Add the number of the finger (the thumb = 1). 7. Add 2. Ask them to tell you the answer. Then mentally subtract 222. The remainder gives the answer, beginning with the right-hand digit of the answer. For example, suppose the string is on the third finger of the left hand of Player #6: 1. Multiply by 2 = 12. 2. Add 3 = 15. 3. Multiply by 5 = 75. 4. Since the string is on the left hand, add 9 = 84. 5. Multiply by 10 = 840. 6.

Add the number of the finger (3) = 843. 7. Add 2 = 845. Now mentally subtract 222 = 623. The right-hand digit (3) tells you the string is on the third finger. The middle digit tells it is on the left hand (the right hand would = 1). The left-hand digit tells you it is Player #6 who has the string. By the way, when the number of the person is over 9, you will get a FOUR-digit number, and the TWO left-hand digits will be the number of the Player. What is the Secret? (This is from a great book called, Giant Book of Puzzles & Games, by Sheila Anne Barry. Published by Sterling Publishing Co., Inc., NY, 1978, recently reissued in paperback.) Stay tuned for more Math Tricks. They will be added from time to time, so be sure to check in again. Integers Integers Comparing integers Positive and Negative Integers Positive integers are all the whole numbers greater than zero: 1, 2, 3, 4, 5, ... .

Negative integers are all the opposites of these whole numbers: -1, -2, -3, -4, -5, …. We do not consider zero to be a positive or negative number. For each positive integer, there is a negative integer, and these integers are called opposites. For example, -3 is the opposite of 3, -21 is the opposite of 21, and 8 is the opposite of -8. If an integer is greater than zero, we say that its sign is positive. If an integer is less than zero, we say that its sign is negative. Example: Integers are useful in comparing a direction associated with certain events. Suppose I take five steps forwards: this could be viewed as a positive 5. If instead, I take 8 steps backwards, we might consider this a -8. Temperature is another way negative numbers are used. On a cold day, the temperature might be 10 degrees below zero Celsius, or -10°C.

The Number Line The number line is a line labeled with the integers in increasing order from left to right, that extends in both directions: For any two different places on the number line, the integer on the right is greater than the integer on the left. Examples: 9 > 4, 6 > -9, -2 > -8, and 0 > -5 Absolute Value of an Integer The number of units a number is from zero on the number line. The absolute value of a number is always a positive number (or zero). We specify the absolute value of a number n by writing n in between two vertical bars: |n|. Examples: |-1234| = 1234 Adding Integers 1) When adding integers of the same sign, we add their absolute values, and give the result the same sign. Examples: (-7) + (-2) = -(7 + 2) = -9 (-80) + (-34) = -(80 + 34) = -114 2) When adding integers of the opposite signs, we take their absolute values, subtract the smaller from the larger, and give the result the sign of the integer with the larger absolute value.

Example: 8 + (-3) = ? The absolute values of 8 and -3 are 8 and 3. Subtracting the smaller from the larger gives 8 - 3 = 5, and since the larger absolute value was 8, we give the result the same sign as 8, so 8 + (-3) = 5. Example: 8 + (-17) = ? The absolute values of 8 and -17 are 8 and 17. Subtracting the smaller from the larger gives 17 - 8 = 9, and since the larger absolute value was 17, we give the result the same sign as -17, so 8 + (-17) = -9. Example: -22 + 11 = ? The absolute values of -22 and 11 are 22 and 11. Subtracting the smaller from the larger gives 22 - 11 = 11, and since the larger absolute value was 22, we give the result the same sign as -22, so -22 + 11 = -11.

Example: 53 + (-53) = ? The absolute values of 53 and -53 are 53 and 53. Subtracting the smaller from the larger gives 53 - 53 =0. The sign in this case does not matter, since 0 and -0 are the same. Note that 53 and -53 are opposite integers. All opposite integers have this property that their sum is equal to zero. Two integers that add up to zero are also called additive inverses. Subtracting Integers Subtracting an integer is the same as adding its opposite. Examples: In the following examples, we convert the subtracted integer to its opposite, and add the two integers. 7 - 4 = 7 + (-4) = 3 12 - (-5) = 12 + (5) = 17 -8 - 7 = -8 + (-7) = -15 Note that the result of subtracting two integers could be positive or negative. Multiplying Integers To multiply a pair of integers if both numbers have the same sign, their product is the product of their absolute values (their product is positive).

If the numbers have opposite signs, their product is the opposite of the product of their absolute values (their product is negative). If one or both of the integers is 0, the product is 0. Examples: In the product below, both numbers are positive, so we just take their product. 4 × 3 = 12 In the product below, both numbers are negative, so we take the product of their absolute values. (-4) × (-5) = |-4| × |-5| = 4 × 5 = 20 In the product of (-7) × 6, the first number is negative and the second is positive, so we take the product of their absolute values, which is |-7| × |6| = 7 × 6 = 42, and give this result a negative sign: -42, so (-7) × 6 = -42.

In the product of 12 × (-2), the first number is positive and the second is negative, so we take the product of their absolute values, which is |12| × |-2| = 12 × 2 = 24, and give this result a negative sign: -24, so 12 × (-2) = -24. To multiply any number of integers: 1. Count the number of negative numbers in the product. 2. Take the product of their absolute values. 3. If the number of negative integers counted in step 1 is even, the product is just the product from step 2, if the number of negative integers is odd, the product is the opposite of the product in step 2 (give the product in step 2 a negative sign). If any of the integers in the product is 0, the product is 0. Example: 4 × (-2) × 3 × (-11) × (-5) = ? Counting the number of negative integers in the product, we see that there are 3 negative integers: -2, -11, and -5.

Next, we take the product of the absolute values of each number: 4 × |-2| × 3 × |-11| × |-5| = 1320. Since there were an odd number of integers, the product is the opposite of 1320, which is -1320, so 4 × (-2) × 3 × (-11) × (-5) = -1320. Dividing Integers To divide a pair of integers if both integers have the same sign, divide the absolute value of the first integer by the absolute value of the second integer. To divide a pair of integers if both integers have different signs, divide the absolute value of the first integer by the absolute value of the second integer, and give this result a negative sign. Examples: In the division below, both numbers are positive, so we just divide as usual. 4 ÷ 2 = 2. In the division below, both numbers are negative, so we divide the absolute value of the first by the absolute value of the second. (-24) ÷ (-3) = |-24| ÷ |-3| = 24 ÷ 3 = 8.

In the division (-100) ÷ 25, both number have different signs, so we divide the absolute value of the first number by the absolute value of the second, which is |-100| ÷ |25| = 100 ÷ 25 = 4, and give this result a negative sign: -4, so (-100) ÷ 25 = -4. In the division 98 ÷ (-7), both number have different signs, so we divide the absolute value of the first number by the absolute value of the second, which is |98| ÷ |-7| = 98 ÷ 7 = 14, and give this result a negative sign: -14, so 98 ÷ (-7) = -14. Integer coordinates Integer coordinates are pairs of integers that are used to determine points in a grid, relative to a special point called the origin. The origin has coordinates (0,0). We can think of the origin as the center of the grid or the starting point for finding all other points. Any other point in the grid has a pair of coordinates (x,y).

The x value or x-coordinate tells how many steps left or right the point is from the point (0,0), just like on the number line (negative is left of the origin, positive is right of the origin). The y value or y-coordinate tells how many steps up or down the point is from the point (0,0), (negative is down from the origin, positive is up from the origin). Using coordinates, we may give the location of any point in the grid we like by simply using a pair of numbers. Example: The origin below is where the x-axis and the y-axis meet. Point A has coordinates (2,3), since it is 2 units to the right and 3 units up from the origin. Point B has coordinates (3,1), since it is 3 units to the right, and 1 unit up from the origin. Point C has coordinates (8,-5), since it is 8 units to the right, and 5 units down from the origin. Point D has coordinates (9,-8); it is 9 units to the right, and 8 units down from the origin.

Point E has coordinates (-4,-3); it is 4 units to the left, and 3 units down from the origin. Point F has coordinates (-7,6); it is 7 units to the left, and 6 units up from the origin. Comparing Integers We can compare two different integers by looking at their positions on the number line. For any two different places on the number line, the integer on the right is greater than the integer on the left. Note that every positive integer is greater than any negative integer. Examples: 9 > 4, 6 > -9, -2 > -8, and 0 > -5 -2 < 1, 8 < 11, -7 < -5, and -10 < 0 Multiplying Negatives Makes A Positive Yes indeed, two negatives make a positive, and we will explain why, with examples! Signs "+" is the positive sign, "−" is the negative sign. When a number has no sign it usually means that it is positive. Example: 5 is really +5 And we can put () around the numbers to avoid confusion.

Example: 3 × −2 can be written as 3 × (−2) Two Signs: The Rules "Two like signs make a positive sign, two unlike signs make a negative sign" Example: (−2) × (+5) The signs are − and + (a negative sign and a positive sign), so they are unlike signs (they are different to each other) So the result must be negative: (−2) × (+5) = −10 Example: (−4) × (−3) The signs are − and − (they are both negative signs), so they are like signs (like each other) So the result must be positive: (−4) × (−3) = +12 Why does multiplying two negative numbers make a positive? Well, first there is the "common sense" explanation: When I say "Eat!" I am encouraging you to eat (positive) But when I say "Do not eat!" I am saying the opposite (negative). Now if I say "Do NOT not eat!", I am saying I don't want you to starve, so I am back to saying "Eat!" (positive).

So, two negatives make a positive, and if that satisfies you, then you don't need to read any more. Direction It is all about direction. Remember the Number Line ? Well here we have Baby Steven taking his first steps. He takes 2 paces at a time, and does this three times, so he moves 2 steps x 3 = 6 steps forward: Now, Baby Steven can also step backwards (he is a clever little guy). His Dad puts him back at the start and then Steven steps backwards 2 steps, and does this three times: Once again Steven's Dad puts him back at the start, but facing the other way. Steven takes 2 steps forward (for him!) but he is heading in the negative direction. He does this 3 times: Back at the start again (thanks Dad!), still facing in the negative direction, he tries his backwards walking, once again taking two steps at a time, and he does this three times: So, by walking backwards, while facing in the negative direction, he moves in the positive direction. Try it yourself! Try walking forwards and backwards, then again but facing the other direction.

More Examples 16 Same pattern: we can follow along a row (or column) and the values change consistently: Follow the "4" row along: it goes -16, -12, -8, -4, 0, 4, 8, 12, 16. Getting 4 larger each time. Follow the "-4" row along: it goes 16, 12, 8, 4, 0, -4, -8, -12, -16. Getting 4 smaller each time. etc... So it all follows a neat pattern! What About Multiplying 3 or More Numbers Together? Multiply two at a time and follow the rules. Example: What is (−2) × (−3) × (−4) ? First multiply (−2) × (−3). Two like signs make a positive sign, so: (−2) × (−3) = +6

Question: If the numbers from 1 to 10 are mutiplied by each other, what is the last number of the result?
Answer: 0