Abstract
The invention is directed to an improved abacus and related calculating method. The improved abacus and method are adapted to perform mathematical operations including addition, subtraction, multiplication, division and factoring. Additional mathematical operations are performable using described techniques. The improved abacus has a frame divided into an upper and lower section by a horizontal bar. The abacus includes a plurality of rods in which each rod contains four movable beads on the upper section of each rod and one movable bead on the lower section of each rod. The beads in the upper section have a value of two and the bead in the lower section has a value of one. Beads moved toward the bar are counted while those beads away from the bar are not counted. The improved abacus also includes a unit indicator, mathematical operational signs and positive and negative sliding indicators signs on the frame.
Claims
1. An improved abacus for use by an individual to make mathematical calculations, the improved abacus comprising: a frame; a plurality of vertical rods positioned parallel to each other across the frame, wherein each vertical rod represents a column of numbers; a plurality of movable beads positioned on each rod, wherein the beads are divided into two unequal sections by a horizontal bar that is perpendicular to the rods; and a mathematical operation indicator disposed in the frame, wherein the mathematical operation indicator is adapted to expose and cover more than one mathematical operation sign.
2. The improved abacus of claim 1, wherein the mathematical operation sign includes at least two from the group consisting of a plus sign, a negative sign, a multiplication sign and a division sign.
3. The improved abacus of claim 1, further comprising a comma positioned after every third rod on the horizontal bar, wherein the comma allows easier organization of the numbers.
4. The improved abacus of claim 1, wherein the plurality of vertical rods form a right section, a middle section, and a left section of the frame, wherein mathematical calculations with the beads are separately performable on the right section, the middle section and the left section of the frame.
5. The improved abacus of claim 1, further comprising a decimal point indicator slidably disposed on the horizontal bar.
6. The improved abacus of claim 1, further compromising an operational sign adapted to indicate the type of the operation to be performed.
7. The improved abacus of claim 1, further compromising a positive/negative indicator disposed on the frame, wherein the positive/negative indicator is adapted to represent the positive or negative sign of numbers used in mathematical operations.
8. The improved abacus of claim 1, further compromising a plurality of positive/negative indicators disposed on the frame, wherein the positive/negative indicators are adapted to represent the positive or negative sign of numbers used in mathematical operations.
9. The improved abacus of claim 1, wherein one row of the beads is below the horizontal bar and four rows of beads are above the horizontal bar, wherein each of the beads below the horizontal bar are assigned a value of 1 and each of the beads above the horizontal bar are assigned a value of 2.
10. A method of using an improved abacus for performing calculations, the method comprising the steps of: providing an abacus having a plurality of movable beads positioned on a plurality of rods, wherein the beads are divided into two sections by a horizontal bar that is perpendicular to the rods, wherein one row of the beads is below the horizontal bar and four rows of beads are above the horizontal bar; assigning a value of 1 to each of the beads below the horizontal bar that are moved upward towards the horizontal bar and assigning a value of 2 to each of the beads above the horizontal bar that are moved downward towards the horizontal bar; performing a mathematical calculation by setting a first number by moving one or more of the beads on one or more of the rods, and setting a second number by moving one or more of the beads on one of more of the rods; and reading the result of the calculation by summing the value of the beads on the one or more rods.
11. The method as set forth in claim 10, further comprising: before the step of performing a mathematical calculation, clearing the abacus by moving all of the beads positioned above the horizontal bar away from the horizontal bar, and moving all of the beads positioned below the horizontal bar away from the horizontal bar.
12. The method as set forth in claim 10, further comprising: wherein the abacus comprises a mathematical operation indicator disposed in a frame of the abacus, wherein the mathematical operation indicator is adapted to expose and cover more than one mathematical operation sign, wherein the mathematical operation sign includes at least two from the group consisting of a plus sign, a negative sign, a multiplication sign and a division sign; manipulating the mathematical operation indicator to identify the mathematical operation being performed.
13. The method as set forth in claim 10, wherein the step of performing a mathematical calculation further comprises setting the first number on right-hand portion of the abacus, setting the second number on a left-hand portion of the abacus and calculating the result by moving the beads in a middle portion of the abacus.
14. The method as set forth in claim 10, wherein the abacus further comprises a decimal point indicator slidably disposed on the horizontal bar and movable to set a decimal point in the mathematical calculation.
15. The method as set forth in claim 10, wherein the step of moving one or more of the beads on one or more of the rods is performed by moving the beads below the horizontal bar using a thumb and moving the beads above the horizontal bar using a forefinger.
16. The method as set forth in claim 10, wherein the abacus is a constructed image in a user's mind and not a physical apparatus.
17. A method of using an improved abacus for performing a multiplication operation, the method comprising the steps of: providing an abacus having a plurality of movable beads positioned on a plurality of rods, wherein the beads are divided into two sections by a horizontal bar that is perpendicular to the rods, wherein one row of the beads is below the horizontal bar and four rows of beads are above the horizontal bar; assigning a value of 1 to each of the beads below the horizontal bar that are moved upward towards the horizontal bar and assigning a value of 2 to each of the beads above the horizontal bar that are moved downward towards the horizontal bar; setting a first number to be multiplied by moving one or more of the beads on one or more of the rods on a right portion of the abacus; setting a second number to be multiplied by moving one or more of the beads on one of more of the rods on a left portion of the abacus; performing the multiplication operation in a middle portion of the abacus by copying the positions of the one or more beads representing the second number or the easier number to calculate in the middle portion of the abacus; duplicating the uncopied number in the middle portion by the positioned value of the beads representing the uncopied number; and adding the total value of the duplicated numbers of the uncopied number in the middle portion to obtain the result.
18. The method as set forth in claim 17, wherein the abacus is a constructed image in a user's mind and not a physical apparatus.
19. A method using an improved abacus for performing a division calculation, the method comprising the steps of: providing an abacus having a plurality of movable beads positioned on a plurality of rods, wherein the beads are divided into two sections by a horizontal bar that is perpendicular to the rods, wherein one row of the beads is below the horizontal bar and four rows of beads are above the horizontal bar; assigning a value of 1 to each of the beads below the horizontal bar that are moved upward towards the horizontal bar and assigning a value of 2 to each of the beads above the horizontal bar that are moved downward towards the horizontal bar; setting the first number on a first side of the abacus by moving the beads up or down towards the horizontal bar; setting the second number on a second side of the abacus by moving the beads up or down towards on the horizontal bar; performing the calculation in a middle section of the abacus by positioning a unit indicator on the horizontal bar to the middle section of the abacus to indicate the unit's rod position of the numbers; marking the indicator first with the same digit as the divisor on the middle rod in the middle section of the abacus and then duplicating the divisor's left digit by the value of the bead on a middle rod to check if it is at least equal to the dividend; duplicating the divisor starting with the bead position of the left digit on a rod that is marked by the unit indicator, based on the digits of the divisor, until the value of the duplication equals or is closest to the value of the dividend; moving the right bead on the rod until an equal or close enough value of the dividend is reached; and reading the final result with the values of the beads based on their position on the rods.
20. The method as set forth in claim 19, wherein the abacus is a constructed image in a user's mind and not a physical apparatus.
Description
BRIEF DESCRIPTION OF THE DRAWINGS
[0055] Advantages of the present invention will be readily appreciated as the same becomes better understood by reference to the following detailed description when considered in connection with the accompanying drawings wherein:
[0056] FIG. 1 shows a front view of an improved abacus in accordance with a nonlimiting embodiment of the invention;
[0057] FIG. 2 shows an example of an addition operation using the beads of the improved abacus in accordance with a nonlimiting embodiment of the invention;
[0058] FIG. 3 shows another example of a mathematical operation using the beads of the improved abacus in accordance with a nonlimiting embodiment of the invention;
[0059] FIG. 4 shows another example of a mathematical operation using the beads of the improved abacus in accordance with a nonlimiting embodiment of the invention and depicting the steps used in an addition operation of an even and odd number of 4 and 5;
[0060] FIG. 5 shows another example of a mathematical operation using the beads of the improved abacus in accordance with a nonlimiting embodiment of the invention and depicting the steps used in an addition operation of an odd and odd number of 1 and 5;
[0061] FIG. 6 shows another example of a mathematical operation using the beads of the improved abacus in accordance with a nonlimiting embodiment of the invention showing a negative result from adding a positive and negative number of 5 and −10;
[0062] FIG. 7 shows another example of a mathematical operation using the beads of the improved abacus in accordance with a nonlimiting embodiment of the invention and depicting the steps used in the subtraction of even numbers 56 and 4;
[0063] FIG. 8 shows another example of a mathematical operation using the beads of the improved abacus in accordance with a nonlimiting embodiment of the invention and depicting the steps used in the subtraction of an odd and even numbers 9 and 4;
[0064] FIG. 9 shows another example of a mathematical operation using the beads of the improved abacus in accordance with a nonlimiting embodiment of the invention and depicting the steps used in the subtraction of an even and odd numbers 16 and 3;
[0065] FIG. 10 shows another example of a mathematical operation using the beads of the improved abacus in accordance with a nonlimiting embodiment of the invention and depicting the steps used in the subtraction of an odd and odd numbers 9 and 5;
[0066] FIG. 11 shows another example of a mathematical operation using the beads of the improved abacus in accordance with a nonlimiting embodiment of the invention and depicting the steps used in the multiplication of the numbers 3 and 5;
[0067] FIG. 12 shows another example of a mathematical operation using the beads of the improved abacus in accordance with a nonlimiting embodiment of the invention and depicting the steps used in the multiplication of the numbers 50 and 4;
[0068] FIG. 13 shows another example of a mathematical operation using the beads of the improved abacus in accordance with a nonlimiting embodiment of the invention and depicting the steps used in the multiplication of the numbers 162 and 24;
[0069] FIG. 14A shows another example of a mathematical operation using the beads of the improved abacus in accordance with a nonlimiting embodiment of the invention and depicting the steps used in the multiplication of the numbers 350 and 250;
[0070] FIG. 14B shows another example of a mathematical operation using the beads of the improved abacus in accordance with a nonlimiting embodiment of the invention and depicting the steps used in the multiplication of the numbers 250 and 350;
[0071] FIG. 15 shows another example of a mathematical operation using the beads of the improved abacus in accordance with a nonlimiting embodiment of the invention and depicting the steps used in the division of the numbers 24 and 4;
[0072] FIG. 16 shows another example of a mathematical operation using the beads of the improved abacus in accordance with a nonlimiting embodiment of the invention and depicting the steps used in the division of the numbers 550 and 25;
[0073] FIG. 17 shows another example of a mathematical operation using the beads of the improved abacus in accordance with a nonlimiting embodiment of the invention and depicting the steps used in the division of the numbers 23 and 2;
[0074] FIG. 18 shows another example of a mathematical operation using the beads of the improved abacus in accordance with a nonlimiting embodiment of the invention and depicting the steps used in the division of the numbers 55 and 25;
[0075] FIG. 19 is a flowchart of the steps of mathematical operations using the beads of the improved abacus in accordance with a nonlimiting embodiment of the invention including in an addition and abstraction operation when two numbers have the same and different signs;
[0076] FIG. 20 is a flowchart of the steps of mathematical operations using the beads of the improved abacus in accordance with a nonlimiting embodiment of the invention including multiplication and division operations when two numbers have the same and different signs;
[0077] FIG. 21 shows another example of a mathematical operation using the beads of the improved abacus in accordance with a nonlimiting embodiment of the invention and depicting the steps used in finding the simplest version of fractions;
[0078] FIG. 22A shows another example of a mathematical operation using the beads of the improved abacus in accordance with a nonlimiting embodiment of the invention and depicting the steps used in simplifying fractions into decimals;
[0079] FIG. 22B shows another example of a mathematical operation using the beads of the improved abacus in accordance with a nonlimiting embodiment of the invention and depicting the steps used in simplifying fractions into decimals; and
[0080] FIG. 22C shows another example of a mathematical operation using the beads of the improved abacus in accordance with a nonlimiting embodiment of the invention and depicting the steps used in simplifying fractions into decimals.
DETAILED DESCRIPTION OF THE DRAWINGS
[0081] Reference will now be made to the drawings in which the various elements of the present invention will be given numeral designations and in which the invention will be discussed to enable one skilled in the art to make and use the invention.
[0082] Referring to FIG. 1 an improved abacus 10 of the invention is intended for use by an individual to make mathematical calculations. The improved abacus 10 is also, a teaching tool to help students learn math. In a nonlimiting embodiment, the improved abacus 10 has a frame 12 with twelve vertical rods 14 (columns). The frame 12 of the abacus 10 has a left section 4, a right section 6, and a middle section 8. Each of the left section 4, right section 6 and middle section 8 include a certain number of the vertical rods 14. The number of vertical rods in each of the left section 4, the right section 6 and the middle section 8 can vary and is not limited to what is depicted in FIG. 1, or any of the other figures. Further, the abacus 10 is expandable 9 in its width so that the total number of vertical rods 14 can be more than twelve. It is the express intent of the invention to allow for expandability of the number of columns so that mathematical operations of large numbers can be performed. Each of the vertical rods 14 has five (5) movable beads 16. The improved abacus 10 operates using a system where each rod 14 represents a column of numbers. For example, starting from the right and moving left, the first rod 14a represents a one's rod, the second rod 14b represents a ten's rod, the third rod 14c represents a hundred's rod, and so forth. A horizontal bar 22 is perpendicular to the vertical rods 14, separating the abacus into two unequal sections including a lower section 24 and an upper section 26. The movable beads 16 are located above and below the horizontal bar 22. Each rod 14 of the improved abacus 10 has four upper beads 20 on the upper section 26 and one bead 18 on the lower section 24. Each upper bead 20 in the upper section 26 has a value of two (2) and the bead 18 in the lower section 24 has a value of one (1). The improved abacus also includes a unit indicator 28 (decimal point marker) on the right-hand side of the frame 12. The unit indicator 28 is adapted to move along the bar 22 in order to indicate the position of the unit (one's units) to the left of the unit indicator 28. When the unit indicator 28 is set to a position on the bar 22, the first bead in the lower section 24 where the unit indicator is marked represents a one's unit bead 18a, the second bead 18b to the left of the one's bead represents the ten's bead, the third bead 18c to the left of the ten's bead represents a hundred's bead, and so forth. The first bead in the upper section 26 represents a one's (or single unit) bead 20a, the second bead 20b to the left of the one's bead represents ten's bead, the third bead 20c to the left of the ten's bead represents a hundred's bead, and so forth. The abacus 10 further includes comma separators 48 located across the bar 22 of the abacus after every three columns. The commas located after every third column are used to allow for easier recognition of the numbers.
[0083] To indicate the number one (1) on the abacus, the lower bead 18 would be moved up to the bar 22 with the thumb. For the number two (2), the upper bead 20 is moved down to the bar 22 with the forefinger. Representing the number 3 requires moving the bottom bead 18 up, which represents the number 1 and one upper bead 20, which represents the number 2 is moved down to the bar 22. Representing the number 22 requires moving one upper bead 20 on the first rod 14a and one upper bead 20a on the second rod 14b down to the bar. Representing the number 222 would require moving down one upper bead 20 on the first rod 14a, one upper bead 20a on the second rod 14b, and one upper bead 20b on the third rod 14c. The unit indicator 28 is movable to determine the location of the unit (single digit).
[0084] The improved abacus 10 includes operational signs 30 represented by mathematical symbols for addition 30a, subtraction 30b, multiplication 30c and, division 30d. The operational signs 30 are placed on the upper left corner 40 of the abacus 10 in the form of a slider or sliders 38 which are adapted to be moved to indicate the type of the operation to be performed by the user. The improved abacus 10 includes on the left side a positive/negative sliding sign indicator 32 to represent a positive sign 32a or a negative sign 32b of a second number positioned on the left side 44 of the abacus 10. On the right side 46 of the abacus 10 a positive/negative sliding sign indicator 34 represents a positive sign 34a or a negative sign 34b of a first number. Near the middle of the abacus, a positive/negative sliding sign indicator 36 represents a positive sign 36a or a negative sign 36b of the calculated result of the first number and second number on the abacus 10.
[0085] With added reference to FIGS. 2 to 6, examples of how addition operations are to be performed on the abacus 10 in forms including, addition of: two even numbers; an odd number and an even number; two even numbers; and two odd numbers. In using the improved abacus 10 it is set on a rigid medium (not shown), for example, a table. Prior to use, all the beads in the lower section are moved to the lower base and all the beads in the upper section are moved to the upper base. An addition calculation is performed by moving the appropriate beads towards the bar. To illustrate the use of the invention through an example, with particular reference to FIG. 2, to add two (2) and two (2); first, assign the addition sign 30a on the upper left side corner 40 of the abacus 10 as shown in FIG. 1; next, perform the addition operation by setting the number 4 by using the forefinger to move two upper beads 20 on the one's rod 14a of the upper section 26 down to the bar 22. Then, read the final result by adding the values of the moved beads based on their position on the rods, which equals four (4).
[0086] Now with emphasis on FIG. 3, the addition operation of an odd and even number is described. To add the numbers five (5) and six (6), first, assign the addition operation sign 30a on the upper left side corner 40 of the abacus 10 and set the number 5 by moving two upper beads 20 on the upper section 26 on the one's rod 14a down to the bar 22 and move up the bead 18a on the lower section 24 on the one's rod 14a to the bar 22. Next, to add the number 6, use the rod on the left side of the abacus to set the second number 6 by moving three beads 20a on the upper section 26 on the one's rod 14a down to the bar 22. Next, read the final result by adding the values of the moved beads based on their position on the rods, which equals eleven (11).
[0087] With emphasis on FIG. 4, the addition operation of an even and odd number is described. To add the numbers four (4) and five (5), first, assign the addition operation sign 30a on the upper left side corner 40 of the abacus 10 and set the number 4 by moving two beads 20 on the upper section 26 on the one's rod 14a down to the bar 22. To add 5, move up the lower bead 18a on the one's rod 14a to the bar 22 and two beads 20 on the upper section 26 on the one's rod 14a down to the bar 22. Next, read the final result by adding the values of the beads based on their position on the rods, which equals nine (9).
[0088] With emphasis on F1G. 5, the addition operation of odd numbers is described. To add the numbers one (1) and five (5), first, assign the addition operation sign 30a on the upper left side corner 40 of the abacus 10 and set the number 1 by moving up the lower bead 18a on the one's rod 14a to the bar 22. To add the number 5, move the lower bead 18a back to its normal position to the base of the frame 12 by using the thumb and move three beads 20 on the upper section 26 on the one's rod 14a down to the bar 22. Read the final result by adding the values of the moved beads based on their position on the rods, which equals to six (6).
[0089] Now with emphasis on FIG. 6 the addition operation of a positive and a negative number is described. To add the numbers five (5) and negative ten (−10), first, assign the addition operation sign 30a on the upper left side corner 40 of the abacus 10 and assign the positive sign 32a on the left side 44 of the abacus 10 to the second number five (5) and the second sign 34b on the right side 46 of the abacus 10 to the first number negative ten (−10). For the calculation, begin with the bigger number by setting the number ten (10) by moving up the tower bead 18 on the ten's rod 14b to the bar 22 on the right-hand side of the abacus. Next, setting five (5) on the right-hand side of the abacus 4 by moving up the lower bead 18a on the one's rod 14a to the bar 22 and moving down the upper bead 20a down to the bar 22. To add the number five (5) and negative ten (−10), move down the lower bead 18b on the ten's rod 14b back to its normal position to the base of the frame 12, calculate the result in the middle section 8 by using the thumb to move up the lower bead 18a on the one's rod 14a to the bar 22. Next, use the forefinger to move down two beads 20 on the upper section 26 to the bar 22. Read the final result by adding the values of the moved beads based on their position on the rods, which equals five (5).
[0090] With additional reference to FIGS. 7 to 10, an approach for using a subtraction operation in the form of two even numbers, an odd and even numbers, two even numbers and two odd numbers is described.
[0091] With emphasis on FIG. 7, the subtraction operation of two even numbers is described. To subtract the numbers fifty-six (56) and four (4), first, assign the subtraction operation sign 30b on the upper left side corner 40 of the abacus 10. Second, set the number fifty-six (56) by moving three beads 20b on the upper section 26 on the ten's rod 14b down to the bar 22 and moving up a bead 18a on the lower section 24 on the one's rod 14a to the bar 22 to represent the number fifty-six (56). To subtract the numbers fifty-six (56) and four (4), move back two beads 20 on the one's rod 14a to its normal position. The result, as represented by the values of the beads is fifty-two (52) as shown in the middle section 8 of the abacus.
[0092] Now with emphasis on FIG. 8, the subtraction operation of an odd and even number is described. To subtract the numbers nine (9) and four (4), first, assign the subtraction operation sign 30b on the upper left side corner 40 of the abacus 10. Second, set number nine (9) by moving four beads 20 on the upper section 26 on the one's rod 14a down to the bar 22 and moving up the bead 18a on the lower section 24 on the one's rod 14a to the bar 22. To subtract number four (4) from nine (9), remove the last two beads 20a on the upper section of the one's rod back to its normal position. The result as representing by the remining beads will read as number five (5) as shown in the middle section 8.
[0093] With emphasis on FIG. 9, the subtraction operation of an even and odd number is described. To subtract the numbers sixteen (16) and three (3), first assign the subtraction operation sign 30b on the upper left side corner 40 of the abacus 10. Second, set the number sixteen (16) by moving up the bead 18b on the lower section 24 on the ten's rod 14b to the bar 22, and moving three beads 20a on the upper section 26 on the one's rod 14a down to the bar 22. To subtract the numbers three (3) from sixteen remove two beads 20a on the upper section 26 on the one's rod 14a back to its normal position and move up one bead 18a on the lower section 24 of the abacus to the bar 22. The result as represented by the remining beads is thirteen (13) as displayed in the middle section 8.
[0094] With emphasis on FIG. 10, the subtraction operation of two odd numbers is described. To subtract the numbers nine (9) and five (5), first, assign the subtraction operation sign 30b on the upper left side corner 40 of the abacus 10. Second, set the number nine (9) by moving up the bead 18a on the lower section 24 on the one's rod 14a to the bar 22, and moving 4 beads 20 on the upper section 26 on the one's rod 14a down to the bar 22. To subtract the number five (5) from nine (9), move back the bottom bead 18a to its original position to the base of the frame 12 and move back the last two beads 20a on the upper section 26 to their original position to the top base of the frame 12. The result as represented by the remining beads is four (4) as displayed in the middle section 8.
[0095] With added reference to FIGS. 11 to 15, simple and more complicated multiplication calculations are described. FIG. 11 illustrates the multiplication of a simple number. To multiply the numbers three (3) and five (5), first, assign the multiplication operation sign 30c on the upper left side corner 40 of the abacus 10. Second, set the number three (3) as shown in FIG. 11 by beginning with the beads on the rods in the left section 4 of the abacus 10 and moving the lower bead 18a to the bar 22 and by moving one bead 20a in the upper section 26 down to the bar 22. Next, to set the number five (5), move up the lower bead 18a to the bar 22 and move two beads 20 in the upper section 26 down to the bar 22 in the right section 6 of the abacus 10. To reach the result, copy the position of the beads of the second number five (5) in the middle section 8 of the abacus and duplicate the first number (3) by the value of the bead's position in the second number and assign the bead in the middle section the value of the calculated numbers, (3×1), (3×2) and (3×2), Lastly, adding the final value of the duplicated numbers 3+6+6=15.
[0096] Alternatively, the result can be calculated and shown in the middle section 8 of the improved abacus 10 instead of assigning the beads in the middle section 8 a value based on one of the numbers in the calculation. Showing the calculated result in the middle section 8 of the improved abacus 10 can be done by an iterative process of duplicating the value of the first number by the value of the bead's position in the second number or vice versa and moving the beads in the middle section 8 of the abacus 10 based on the result of each duplicated number and then adding the next duplicated number to the numbers already calculated until reaching the fubak result. Lastly, the final calculated value of the moved beads in the middle section of the abacus is read as the result of the calculation. For example, 3×5=((3×1)=3)+((3×2)=6)+((3×2)=6)=15. The result of this example will be shown in the middle section of the abacus equaling fifteen (15). Since the first column has 5 beads equaling 9, receiving a number above 9 after some of the calculation requires clearing up the beads on the first column 14a in the middle section 8 and moving the beads on the next column from the left of the first column 14a in the middle section 8 depending on the number of the calculation. Thus, the result value of number 15 would require moving the lower bead 18b on the ten's bar 14b up to the bar 22 and moving down two upper beads 20a on the one's rod 14a to the bar 22 and moving the lower head 18a on the one's bar 14a to the bar 22. This same alternative method, and nonlimiting embodiment of the invention, can be used for all multiplication and division operations on the improved abacus 10.
[0097] It should be appreciated that operations on the improved abacus can, and advantageously should be performed on a mentally imaged copy of the abacus. However, it requires more than mental steps to accomplish the result because it is not natural to carry a mental image of the improved abacus in the user's head. It is only with prolonged training and use of the teachings explained herein that an experienced operator of the improved abacus can perform the operations as mental steps aided with a mentally derived image of the improved abacus.
[0098] With emphasis on FIG. 12, is another example of multiplication. To multiply the numbers fifty (50) and four (4), first, assign the multiplication operation sign 30c on the upper left side corner 40 of the abacus 10. Second, set number fifty (50) by beginning with the beads on the rods positioned on the left section 4 of the abacus 10 and moving two upper beads 20b on the ten's rod 14a to the bar 22 and moving the lower bead 18b on the ten's rods 14a to the bar 22. The beads on the one's rod are unmoved since it is a zero. Third, set the second number four (4) by move two beads 20a in the upper section 26 down to the bar 22 in the right section 6 of the abacus 10. To reach the result, copy the positions of the beads of the second number (4) in the middle section 8 and duplicate the first number fifty (50) by the value of the beads position in the second number and assign the numbers, (50×2), (50×2), and then add the final numbers 100 +100=200.
[0099] With emphasis on FIG. 13, the multiplication of a complicated number is described. To multiply the numbers one hundred sixty-two (162) and twenty-four (24), first, assign the multiplication operation sign 30c on the upper left side corner 40 of the abacus 10. Second, set the number one hundred sixty-two (162) by beginning with the beads on the rods in the left section 4 of the abacus and marking the one's rod with the unit indicator 28 on the bar. Move the upper bead 20a to the bar 22 on the one's rod 14a that is indicated by the unit indicator 28 and move three beads 20b on the upper section 26 on the ten's rod 14b towards to the bar 22 and move up the lower bar 18c on the hundred's rod 14c to the bar 22. For the number 24, begin with the rods on the right section 6 of the abacus 10 and set it by moving one bead 20b in the upper section of the ten's rod 14b towards to the bar 22 and two beads 20a in the upper section 26 of the one's rod 14a towards the bar. To reach the result, copy the positions of the beads of the second number twenty-four (24) in the middle section 8 of the abacus 10 and duplicate the number one hundred sixty-two (162) by the original values of the beads, (162×20), (162×2) and (162×2) then add the final results 3240+324+324=3888.
[0100] With emphasis on FIG. 14A, the multiplication of a complicated number is described. To multiply the numbers three hundred fifty (350) and two hundred fifty (250), first, assign the multiplication operation sign 30c on the upper left side corner 40 of the abacus 10. Second set the number three hundred fifty (350) by beginning with the beads on the rods in the left section 4 of the abacus 10. Move two upper beads 20c on the hundred's rod 14c to the bar 22, and move the lower bead 18c on the hundred's rod 14c to the bar 22, move two upper beads 20b on the ten's rod 14b to the bar 22 and the lower bead 18b on the ten's rod 14b to the bar 22. No beads are to be moved on the one's rod since it is a zero. For the number two hundred fifty (250), begin with the column on the right section 6 of the abacus 10 and set it by moving two upper beads 20c on the hundred's rod 14c, move two upper beads 20b on the ten's rod 14b to the bar 22 and move the lower bead 18b on the ten's rod 14b to the bar 22. No beads are to be moved in the one's rod since it is a zero. To reach the result, copy the position of the beads of the second number two hundred fifty (250) in to the middle section 8 of the abacus by first marking the unit indicator 28 for the position of the digits, then duplicate the second number three hundred fifty (350) by the original values of the beads that are based on the position of two hundred fifty (250) in the middle section starting from the hundred's rod and moving right to the one' rod, (350×200)+(350×100)+(350×20)+(350×20). Lastly, add the final numbers 70,000+7,000+7,000+3,500=87,200.
[0101] Referring to FIG. 14B, the multiplication of a complicated number is described. To multiply the numbers three hundred fifty (350) and two hundred fifty (250), first, assign the multiplication operation sign 30c on the upper left side corner 40 of the abacus 10. Second, set the first number three hundred fifty (350) by beginning with the beads on the rods in the left section. Move two upper bead 20c on the hundred's rod 14c to the bar 22 and move the lower bead 18c on the hundred's rod 14c to the bar 22. Next, move two upper beads 20b on the ten's rod 14b to the bar 22 and the lower bead 18b on the ten's rod 14b to the bar 22. No beads are to be moved on the one's rod since it is zero. For the number two hundred fifty (250), begin with the column on the right side 6 of the abacus 10 and set the second number by moving two upper beads 20c on the hundred's rod 14c, move two upper beads 20b on the ten's rod 14b to the bar 22 and move the lower bead 18b on the ten's rod 14b to the bar 22. No beads are to be moved on the one's rod since it is zero. To reach the result, copy the position of the beads of the first number three hundred fifty (350) because the second number is easier to add after duplication. Therefore, first copy the position of the beads of the firm number three hundred fifty (350) in to the middle section 8 of the abacus 10 by first marking the unit indicator 28 for the position of the one's digit, then duplicate second number two hundred fifty (250) by the original values of the beads that are based on the positions of first number three hundred fifty (350) in the middle section 8 starting from the hundred's rod and moving right to the one's rode, (250×200)+(250×100)+(250×20)+(250×20)+(250×20)+(250×10). Lastly, add the final numbers 50,000+25,000+5,000+2,500=87,200.
[0102] Now with emphasis on FIG. 15, the division of a simple number is described. To divide the number twenty-four (24) with four (4). First, assign the division operation sign 30d on the upper left side corner 40 of the abacus 10. Second, set the number twenty-four (24) in the left section of the abacus 10. Move the upper bead 20b on the ten's rod 14b to the bar 22 and move two beads 20a on the one's rod 14a to the bar 22. For number four (4), in the right section 6 of the abacus 10, move two beads 20a on the one's rod 14a towards the bar 22. In the middle section 8 of the abacus 10, begin with the beads 20a on the one's rod 14a on the upper section 26 of the abacus 10 since the operation involves dividing by an even number meaning the result should be even. Duplicate the second number four (4) by the value of the upper bead 20a on the one's rod 14a and continue duplicating 4×2 on the one's rod until the result equals twenty-four (24) is reached. Therefore, it is (4×2)+(4×2)+(4×2)=(8+8+8)=24. The result is to be read by the resultant position of the beads in its original value on the rod, which equals number six (6). This approach is used when dividing by a single digit.
[0103] With emphasis on FIG. 16, the division of a more complex number is described. To divide the number five hundred fifty (550) by twenty-five (25) first, assign the division operation sign 30d on the upper left side corner 40 of the abacus 10. Second set the first number five hundred fifty (550) in the left section 4 of the abacus 10, by moving the upper bead 20c on the hundred's rod 14c to the bar 22 and move two beads 20b on the ten's rod 14b to the bar 22 and the lower bead 18b on the ten's rod 14b to the bar 22. Lastly, no beads are to be moved on the one's rod 14a since it is zero. For the second number twenty-five (25), in the right section 6 of the abacus 10, move one bead 20b on the ten's rod 14b towards the bar 22, move two upper beads 20a on the one's rod 14a to the bar 22 and a lower bead 18a on the one's rod 14a to the bar 22. Move the unit indicator 28 to the middle section 8 of the abacus and assign the digits of the divisor. Begin with duplicating the divisor by the value of the beads 20a in the ten's rod 14b, which is 20, Therefore, 25×20=500. Since the duplicated result is close to the value of the dividend, move right to the one's rod and duplicate the devisor by the value of the beads 20 on the one's rod 14a until reaching the result (25×2)=50. Lastly, read the result by the original value of the beads as positioned on the rods in the middle section of the abacus, which equals number twenty-two (22).
[0104] Now with emphasis on FIG. 17, the division of a more complex number is described. To divide the number twenty-three (23) by two (2). First, assign the division operation sign 30d on the upper left side corner 40 of the abacus 10. Second, set the first number twenty-three (23) in the left section 4 of the abacus 10. Move the upper bead 20b on the ten's rod 14b to the bar 22 and move two beads 20a on the one's rod 14b to the bar 22 and the lower bead 18a on the one's rod 14a to the bar 22. For the second number two (2), set it in the right section 6 of the abacus 10 by moving the upper bead 20a on the one's rod 14a towards the bar 22. Move the unit indicator 28 to the middle section 8 of the abacus 10 and assign the digits of the divisor. Begin with duplicating the divisor by the value of the beads 20a on the one's rod 14a, which is the number 2. Since duplicating the divisor by itself would lead to a result that is far off from the dividend, move the digit indicator to the left of the abacus and duplicate the divisor by the value of the lower bead 18b on the ten's rod 14b, which is 2×10=20. Since duplicating the divisor by the value of the beads 18b results in a number that is close to the dividend, move right to the one's rod and duplicate the divisor by the value of the lower beads 18a on the one's rod 14a, (2×1)=2. If the calculated valve of the duplicated divisor equals the value of the dividend, read the result by the original value of the beads as positioned on the rods in the middle section 8 of the abacus, If there is a reminder, positioned the reminder by a decimal point. Therefore, the result is the number 11 with a reminder of 1.
[0105] With emphasis on FIG. 18, the division of a complex number is described. To divide the number fifty-five (55) by twenty-five (25), first, assign the division operation sign 30d on the upper left side corner 40 of the abacus 10. Second, set the first number fifty (50) in the left section 4 of the abacus 10 by moving two upper beads 20b on the ten's rod 14b to the bar 22 and moving the lower beads 18b on the ten's rod 14b to the bar 22. No beads are to be moved on the one's rod 14a since it is zero. For the second number twenty-five (25), in the right section 6 of the abacus 10, move the upper bead 20b on the ten's rod 14b towards the bar 22. Move two upper beads 20a on the one's rod 14a to the bar 22 and move the lower beads 18a on the one's rod 14a to the bar 22. Move the unit indicator to the middle section 8 of the abacus 10 and assign the digits of the divisor to the result. Begin with duplicating the divisor by the value of the beads 20b on the ten's rod 14b. Since duplicating the divisor by itself would lead to a result that is far bigger from the dividend, move the digit indicator to the right on the abacus and duplicate the divisor by the value of the upper bead 20a on the one's rod 14a, which is 25×2=50. Duplicating the divisor by the value of the lower bead 18a would result after calculating the value of the duplicated divisor in a number that is bigger than the dividend, which is the number 75. Read the result by the original value of the beads as positioned on the rods in the middle section of the abacus and position the reminder by a decimal point. Therefore, the result is the two (2) with a remainder of five (5).
[0106] Referring now to FIG. 19 with continuing reference to FIG. 1, the processing steps performed on the abacus 10 in an addition 30a and subtraction 30b operations when two numbers have the same and different signs can be demonstrated. The process starts 100 with reading the numbers 102 and then reading the first number with its positive/negative sign 104 and assigning it its positive/negative sign 34 on the abacus. The next step involves reading the second number with its sign 106 and assigning it its positive/negative sign 32 on the abacus. The next step involves placing the numbers in the absolute value form 108. The next step involves reading whether the first number has the same positive/negative sign as the second number 110. If yes, select the appropriate operation 112. For performing an additional operation 114, select the addition sign 30a from the selector 38 on the abacus and add both numbers together with placing the sign of the result 36 in front of the operation 116.
[0107] For performing a subtraction operation 118, determine whether the absolute value of the first number is greater than or equal to the absolute value of the second number 122. If the absolute value of the first number is greater than or equal to the absolute value of the second number, the sign of the result is the sign of the first number 124. If the absolute value of the first number is less than the absolute value of the second number, the sign of the result is the sign of the second number 126. The final step for the numbers that have the same positive/negative sign is determining whether other numbers are remaining in the calculation 130. If yes, the result is assigned as the first number 134 of a next operation and the next operation will proceed with the fourth step 106 from the start 100. If there are no other numbers remaining in the calculation, the operation ends 132.
[0108] For numbers that do not have the same positive/negative sign, read the operation of the mathematical calculation 114. For performing a subtraction operation 118, select the subtraction sign 30b from the selector 38 and determine whether the absolute value of the first number is greater than or equal to the absolute value of the second number 122. If the absolute value of the first number is greater than or equal to the absolute value of the second number, the sign of the result is the sign of the first number 124. If the absolute value of the first number is less than the absolute value of the second number, the positive/negative sign of the result is the sign of the second number 126. For performing an additional operation 120, select the addition sign 30a from the selector 38 on the abacus, and the result will be the sign of the first number 128. The final step for the numbers that have different positive/negative signs is determining whether other numbers are remaining in the calculation 130. If yes, the result is assigned as the first number 134 of a next operation, and the next operation will proceed with the fourth step 106 from the start 100. If there are no other numbers remaining in the calculation, the operation ends 132.
[0109] Referring now to FIG. 20 with continuing reference to FIG. 1, the processing steps performed on the abacus 10 in multiplication and division operations when two numbers have the same and different positive/negative signs will be described.
[0110] The process starts 200 with reading the numbers 202 and then reading the first number with its positive/negative sign 204 and assigning it its positive/negative sign 34 on the abacus. The next step involves reading the second number with its positive/negative sign 206 and assigning it its positive/negative sign 32 on the abacus. The next step involves placing the numbers in their absolute value form 208. The next step involves reading whether the first number has the same positive/negative sign as the second number 210. If yes, read the operation 212 and select the appropriate operation sign 30 on the abacus. For performing a multiplication operation 214, select the appropriate sign 30c from the selector 38 on the abacus, calculate the numbers together, and place the positive sign on the result 216. For performing a division operation 218, select the appropriate sign 30d from the selector on the abacus, calculate the numbers together, and place the positive sign on the result 220.
[0111] For numbers that do not have the same positive/negative sign, read the operation 222 of the mathematical calculation 30 on the abacus. For performing a multiplication operation 224, select the appropriate sign 30c from the selector on the abacus, calculate the numbers together, and place the negative sign on the result 226. For performing a division operation 228, select the appropriate sign 30d from the selector on the abacus, calculate the numbers together, and place the negative sign on the result 230.
[0112] The final step for the numbers that have the same and different positive/negative signs is determining whether there are other numbers remaining in the calculation 232. If yes, the result is assigned as the first number 236 of a next operation and the next operation will proceed with the fourth step 206 from the start 200. If there are no other numbers remaining in the calculation, the operation ends 234.
[0113] Referring now to FIG. 21, finding the simplest version of a fraction of the numbers twenty-four (24) and thirty (30) are demonstrated. Since the nominator and the dominator are even numbers, divide both numbers by two (2) to reduce them to the numbers twelve (12) and fifteen (15). That shows that the nominator is an even number, and the dominator is an odd number with its one's digit beginning with the number five (5), meaning that fifteen (15) is dividable by five (5). Dividing fifteen (15) by five (5) would give the result of three (3). Therefore, to reduce the number of the numerator or the denominator further, divide both numbers by three (3) to obtain the prime number for the numerator or the denominator that is no longer possible to simplify anymore. The method used of knowing that the number can be divided by three (3) is by the addition of the digits of the number is to show that can divided by three (3) such as adding the digits of the number twelve (12) together as 1+2=3, thus it can be divided by three (3) to simplify further the numbers.
[0114] Referring now to FIGS. 22A to 22C, demonstration of simplifying a fraction into a decimal is shown. For a multiplication or division operation, when a denominator is a number that can be multiplied by another number to become 10 or its multiples, the numerator is multiplied by that number. Therefore, since it is known that 100 is the 4.sup.th factor of 25, convert the fraction 13/25 to a decimal fraction by multiplying the nominator thirteen (13) by four (4) and the denominator twenty-five (25) by four (4). The result being 52/100, which is 0.52 in decimals. FIG. 22A demonstrates multiplying the nominator thirteen (13) by four (4) equaling 52. FIG. 22B demonstrates multiplying the denominator twenty-five (25) by four (4) equaling 100. FIG. 22C demonstrates resolving the fraction 13/25 into a decimal 0.52.
[0115] Therefore, a student or accomplished user can use the improved abacus 10 and associated improved calculating methods to make both simple and complex mathematical calculations. The above examples and nonlimiting embodiments are intended to help explain how to use the abacus but are not meant as a limitation on its form or use. There are an unlimited number of variations of calculations that can be performed on the improved abacus and by using the improved method. What is common among all calculation methods using the improved abacus is the ability to more easily represent the numbers being operated upon and the result of the operation. The improved abacus 10 and method are not only an improvement in the function and type of an abacus but also will make learning fun while offering educational benefits unavailable with other current methods of teaching mathematics. Accomplished users will have a lifelong tool in a physical or mental form that allows for more easily understanding and resolving mathematical calculations.
