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The decimal multiplication table was traditionally taught as an essential part of elementary arithmetic around the world, as it lays the foundation for arithmetic operations with our base-ten numbers. Many educators believe it is necessary to memorize the table up to 9 Ã— 9.
In his 1820 book The Philosophy of Arithmetic, mathematician John Leslie published a multiplication table up to 99 Ã— 99, which allows numbers to be multiplied in pairs of digits at a time. Leslie also recommended that young pupils memorize the multiplication table up to 25 Ã— 25.
In 493 A.D., Victorius of Aquitaine wrote a 98-column multiplication table which gave (in Roman numerals) the product of every number from 2 to 50 times and the rows were "a list of numbers starting with one thousand, descending by hundreds to one hundred, then descending by tens to ten, then by ones to one, and then the fractions down to 1/144" (Maher & Makowski 2001, p.383)
The traditional rote learning of multiplication was based on memorization of columns in the table, in a form like
1 Ã— 10 = 10 2 Ã— 10 = 20 3 Ã— 10 = 30 4 Ã— 10 = 40 5 Ã— 10 = 50 6 Ã— 10 = 60 7 Ã— 10 = 70 8 Ã— 10 = 80 9 Ã— 10 = 90
10 x 10 = 100 11 x 10 = 110 12 x 10 = 120 13 x 10 = 130 14 x 10 = 140 15 x 10 = 150 16 x 10 = 160 17 x 10 = 170 18 x 10 = 180 19 x 10 = 190 100 x 10 = 1000
This form of writing the multiplication table in columns with complete number sentences is still used in some countries instead of the modern grid above.
Patterns in the tables
There is a pattern in the multiplication table that can help people to memorize the table more easily. It uses the figures below:
â†’ â†’ 1 2 3 2 4 â†‘ 4 5 6 â†“ â†‘ â†“ 7 8 9 6 8 â†� â†� 0 0 Fig. 1 Fig. 2
For example, to memorize all the multiples of 7:
- Look at the 7 in the first picture and follow the arrow.
- The next number in the direction of the arrow is 4. So think of the next number after 7 that ends with 4, which is 14.
- The next number in the direction of the arrow is 1. So think of the next number after 14 that ends with 1, which is 21.
- After coming to the top of this column, start with the bottom of the next column, and travel in the same direction. The number is 8. So think of the next number after 21 that ends with 8, which is 28.
- Proceed in the same way until the last number, 3, which corresponds to 63.
- Next, use the 0 at the bottom. It corresponds to 70.
- Then, start again with the 7. This time it will correspond to 77.
- Continue like this.
Figure 1 is used for multiples of 1, 3, 7, and 9. Figure 2 is used for the multiples of 2, 4, 6, and 8. These patterns can be used to memorize the multiples of any number from 1 to 9, except 5.
In abstract algebra
Standards-based mathematics reform in the USA
In 1989, the National Council of Teachers of Mathematics (NCTM) developed new standards which were based on the belief that all students should learn higher-order thinking skills, and which recommended reduced emphasis on the teaching of traditional methods that relied on rote memorization, such as multiplication tables. Widely adopted texts such as Investigations in Numbers, Data, and Space (widely known as TERC after its producer, Technical Education Research Centers) omitted aids such as multiplication tables in early editions. It is thought by many that electronic calculators have made it unnecessary or counter-productive to invest time in memorizing the multiplication table. NCTM made it clear in their 2006 Focal Points that basic mathematics facts must be learned, though there is no consensus on whether rote memorization is the best method.
Mental calculation comprises arithmetical calculations using only the human brain, with no help from calculators, computers, or pen and paper. People use mental calculation when computing tools are not available, when it is faster than other means of calculation (for example, conventional methods as taught in educational institutions), or in a competition context. Mental calculation often involves the use of specific techniques devised for specific types of problems.
Many of these techniques take advantage of or rely on the decimal numeral system. Usually, the choice of radix determines what methods to use and also which calculations are easier to perform mentally. For example, multiplying or dividing by ten is an easy task when working in decimal (just move the decimal point), whereas multiplying or dividing by sixteen is not; however, the opposite is true when working in hexadecimal.
Methods and Techniques
Casting out nines
- Main article:Casting out nines
After applying an arithmetic operation to two operands and getting a result, you can use this procedure to improve your confidence that the result is correct.
- # Sum the digits of the first operand; any 9s (or sets of digits that add to 9) can be counted as 0.
- # If the resulting sum has two or more digits, sum those digits as in step one; repeat this step until the resulting sum has only one digit.
- # Repeat steps one and two with the second operand. You now have two one-digit numbers, one condensed from the first operand and the other condensed from the second operand. (These one-digit numbers are also the remainders you would end up with if you divided the original operands by 9; mathematically speaking, they're the original operands modulo 9.)
- # Apply the originally specified operation to the two condensed operands, and then apply the summing-of-digits procedure to the result of the operation.
- # Sum the digits of the result you originally obtained for the original calculation.
- # If the result of step 4 does not equal the result of step 5, then the original answer is wrong. If the two results match, then the original answer may be right, though it isn't guaranteed to be.
- * Say we've calculated that 6338 × 79 equals 500702
- # Sum the digits of 6338: (6 + 3 = 9, so count that as 0) + 3 + 8 = 11
- # Iterate as needed: 1 + 1 = 2
- # Sum the digits of 79: 7 + (9 counted as 0) = 7
- # Perform the original operation on the condensed operands, and sum digits: 2 × 7 = 14; 1 + 4 = 5
- # Sum the digits of 500702: 5 + 0 + 0 + (7 + 0 + 2 = 9, which counts as 0) = 5
- # 5 = 5, so there's a good chance that we were right that 6338 × 79 equals 500702.
You can use the same procedure with multiple operands; just repeat steps 1 and 2 for each operand.
When checking the mental calculation, it is useful to think of it in terms of scaling. For example, when dealing with large numbers, say 1531 × 19625, estimation instructs you to be aware of the number of digits expected for the final value. A useful way of checking is to estimate. 1531 is around 1500, and 19625 is around 20000, so therefore a result of around 20000 Ã— 1500 (30000000) would be a good estimate for the actual answer (30045875). So if the answer has too many digits, you know you've made a mistake.
When multiplying, a useful thing to remember is that the factors of the operands still remain. For example, to say that 14 × 15 was 211 would be unreasonable. Since 15 was a multiple of 5, so should the product. The correct answer is 210.
Calculating differences: ''a'' − ''b''
When the digits of b are all smaller than the corresponding digits of a, the calculation can be done digit by digit. For example, evaluate 872 − 41 simply by subtracting 1 from 2 in the units place, and 4 from 7 in the tens place: 831.
When the above situation does not apply, the problem can sometimes be modified:
- If only one digit in b is larger than its corresponding digit in a, diminish the offending digit in b until it is equal to its corresponding digit in a. Then subtract further the amount b was diminished by from a. For example, to calculate 872 − 92, turn the problem into 872 − 72 = 800. Then subtract 20 from 800: 780.
- If more than one digit in b is larger than its corresponding digit in a, it may be easier to find how much must be added to b to get a. For example, to calculate 8192 − 732, we can add 8 to 732 (resulting in 740), then add 60 (to get 800), then 200 (for 1000). Next, add 192 to arrive at 1192, and, finally, add 7000 to get 8192. Our final answer is 7460.
- It might be easier to start from the left (the big numbers) first.
You may guess what is needed, and accumulate your guesses. Your guess is good as long as you haven't gone beyond the "target" number. 8192 − 732, mentally, you want to add 8000 but that would be too much, so we add 7000, then 700 to 1100, is 400 (so far we have 7400), and 32 to 92 can easily be recognized as 60. The result is 7460.
Look-ahead borrow method
This method can be used to subtract numbers left to right, and if all that is required is to read the result aloud, it requires little of the user's memory even to subtract numbers of arbitrary size.
One place at a time is handled, left to right.
4075 âˆ’ 1844 ------
Thousands: 4 âˆ’ 1 = 3, look to right, 075 < 844, need to borrow. 3 âˆ’ 1 = 2, say "Two thousand"
Hundreds: 0 âˆ’ 8 = negative numbers not allowed here, 10 âˆ’ 8 = 2, 75 > 44 so no need to borrow, say "two hundred"
Tens: 7 âˆ’ 4 = 3, 5 > 4 so no need to borrow, say "thirty"
Ones: 5 âˆ’ 4 = 1, say "one"
Calculating products: ''a'' Ã— ''b''
Many of these methods work because of the distributive property.
Multiplying by 2 or other small numbers
Where one number being multiplied is sufficiently small to be multiplied with ease by any single digit, the product can be calculated easily digit by digit from right to left. This is particularly easy for multiplication by 2 since the carry digit cannot be more than 1.
For example, to calculate 2 Ã— 167: 2Ã—7=14, so the final digit is 4, with a 1 carried and added to the 2Ã—6 = 12 to give 13, so the next digit is 3 with a 1 carried and added to the 2Ã—1=2 to give 3. Thus, the product is 334.
Multiplying by 5
To multiply a number by 5,
1. First multiply that number by 10, then divide it by 2.
The following algorithm is a
A formula calculator is a software calculator that can perform a calculation in two steps:
1. Enter the calculation by typing it in from the keyboard.
2. Press a single button or key to see the final result.
This is unlike button-operated calculators, such as the Windows Calculator or the Mac os calculator, which require the user to perform one step for each operation, by pressing buttons to calculate all the intermediate values, before the final result is shown.
In this context, a formula is also known as an expression, and so formula calculators may be called expression calculators. Also in this context, calculation is known as evaluation, and so they may be called formula evaluators, rather than calculators.
How they work
- Parentheses to enclose parts of a formula that must be calculated first.
- In the absence of parentheses, operator precedence, so that higher precedence operators, such as multiplication, must be applied before lower precedence operators, such as addition. For example, in 2 + 3*4, the multiplication, 3*4, is done first.
- Among operators with the same precedence, associativity, so that the left-most operator must be applied first. For example, in 2 - 3 + 4, the subtraction, 2 - 3, is done first.
Also, formulas may contain:
- Non-commutative operators that must be applied to numbers in the correct order, such as subtraction and division.
- The same symbol used for more than one purpose, such as - for negative numbers and subtraction.
Once a formula is entered, a formula calculator follows the above rules to produce the final result by automatically:
- Analysing the formula and breaking it down into its constituent parts, such as operators, numbers and parentheses.
- Finding both operands of each binary operator.
- Working out the values of these operands.
- Applying the operator to these values, in the correct order so as to allow for non-commutative operators.
- Evaluating the parts of a formula in parentheses first.
- Taking operator precedence and associativity into account.
- Distinguishing between different uses of the same symbol.
How to enter formulas
Formulas printed in many text books use juxtaposition, underline and superscripts for multiplication, division and exponentiation respectively. Also, some operations, such as square root, are represented by special symbols that are not usually available on a computer keyboard. For example, see the formulas in Amortization calculator, Heron's formula and Law of cosines.
means 2 multiplied by cos(3).
In calculators that donâ€™t allow juxtaposition, the asterisk (and possibly x rather than, or as well as, the asterisk), is used, and the calculation should be entered as:
Also, a period is sometimes used for multiplication, as in:
Because the period is also used as the decimal point in numbers, so that the â€œ2.â€� in the above would be interpreted as 2.0, the period is not used for multiplication in a formula calculator, and this calculation should be entered using a different symbol, as above.
Printed formulas often use a horizontal line for division, but in a formula calculator that uses only keyboard symbols, division is entered using the forward slash, /. When there is a calculation above or below the line, this should be done first, and so it should be enclosed in parentheses when typed in. For example,
2 + 3 â€”â€”â€”â€”â€” 4 - 5
should be entered as
(2 + 3)/(4 - 5)
Also, the symbol Ã· is often used for division, as in
2 Ã· 3
This symbol is not available on most computer keyboards, so this division operation is entered using the forward slash, as above.
Exponentiation, or raising to a power, is often represented using a superscript. For example:
means 2.45 squared.
With the limitations of a computer keyboard, in some software packages, such as Microsoft Excel, this is entered using the caret, ^:
but two asterisks are also used:
Exponentiation and functions
When using functions, the superscript is sometimes placed immediately after the function name. For example, it is common to write the trigonometric version of Pythagorasâ€™ Theorem (List of trigonometric identities) as:
sin2(x) + cos2(x) = 1
In this identity,
means the square of sin(x), and is the same as:
So, for x = 3.25, it could be entered into a formula calculator that uses the caret for exponentiation as:
Square roots are often specified using the âˆš symbol, but with the limitations of a keyboard this is can be entered by using exponentiation. For example, the square root of 2:
could be entered as:
but two asterisks are also used:
The parentheses specify that the division should be done first.
All roots can be specified in this way. For example, the cube root of 2 can be entered as:
Heronâ€™s formula example
An example that illustrates these features is Heronâ€™s formula.
One version of the formula, for a triangle with sides of length a, b and c, is equivalent to, using symbols that are available on a keyboard:
1/2*(a^2*c^2 - (a^2 + c^2 - b^2)/2)^0.5
For a = 2.5, b = 3.6 and c = 1.9, it could be entered into a formula calculator as:
1/2*(2.5^2*1.9^2 - (2.5^2 + 1.9^2 - 3.6^2)/2)^0.5
Types of calculator
The formula calculator concept can be applied to all types of calculator, including arithmetic,
From Yahoo Answers
Answers:I'll do nothing except follow the standard order of operations and the two instructions you quoted. Because of the parentheses, I need to perform the subtraction first. ( 7.87g ) / ( 16.1g - 8.14g ) For addition problems, the answer has the same number of decimal places as there are in the measurement with the fewest decimal places. The subtraction is 1 and 2 decimal places, so the result is 1 decimal place. ( 7.87g ) / ( 16.1g - 8.14 ) ( 7.87g ) / ( 8.0 ) For multiplication problems, the answer contains the same number of significant figures as in the measurement with the fewest significant figures. The division has 3 and 2 significant digits, so the result has 2 significant digits. 7.87g / 8.0g 0.98g You may want to check my math, and check my counting of digits.
Answers:/ No , there is no other way
Answers:Hint: If the denominator has factors of only 2 or 5 then you will get a terminating decimal. Notice: 1/2 = 0.5 (good) 1/3 = 0.3333.... (bad) 1/4 = 0.25 (good) 1/5 = 0.2 (good) 1/6 = 0.16666... (bad) 1/7 = 0.142857142857... (bad) 1/8 = 0.125 (good) 1/9 = 0.11111... (bad) 1/10 = 0.1 (good) 1/11 = 0.0909... (bad) etc. 189 --> 1 + 8 + 9 = 18 (multiple of 3) 196 --> 2 x 2 x 7 x 7 (multiple of 7) 225 --> 2 + 2 + 5 = 9 (multiple of 3) 144 --> 1 + 1 + 4 = 6 (multiple of 3) 128 --> 2 x 2 x 2 x 2 x 2 x 2 x 2 128 has only factors of 2 (or 5). Answer: E. 39/128
Answers:You can give the cell that holds your result, the property to have no decimal places. For example, if you cell's property is set to have several decimal places, the result would display 3.5257 or whatever. With one decimal place, it would be 3.6, and if you have no decimal places, then the result, regardless of the formula, would have to display 3. I hope that makes sense.