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Natural Numbers
Published in Nita H. Shah, Vishnuprasad D. Thakkar, Journey from Natural Numbers to Complex Numbers, 2020
Nita H. Shah, Vishnuprasad D. Thakkar
Multiplication is taking collections, each having as many as the first operand objects; the number of collections taken are as many as the second operand. The result of multiplication, called the product, is the total number of objects. For multiplication, we take the collection of bricks. We put as many bricks as the first operand in a row and make as many rows as the second operand. The total number of bricks is the result of the multiplication. The shape formed by the arrangement is a rectangle formed with rows as the length side and rows forming width. If we look from the width side, rows become columns and columns become rows. This can be interpreted as the second operand multiplied by the first operand. This establishes that multiplication is commutative. If we need to multiply three operands, multiplication of the first two operands forms a rectangle. We can stack such rectangles (as many as the third operand). Total bricks in this structure gives the product of the three numbers as the first operand multiplied by the second and then multiplied by the third operand. We can look at the cuboid structure from different faces, and it can be interpreted as a different grouping of operands. Thus, we can say that multiplication is associative.
Arithmetic and Logic Unit Organisation
Published in Pranabananda Chakraborty, Computer Organisation and Architecture, 2020
In multiplication, if either operand is 0, the result is automatically declared as 0. The next step is to add the exponents. If the exponents are stored in biased form, the sum of the exponents would then contain double the bias value. Hence, the bias value must be subtracted from the sum. The result may sometimes give rise to a situation of exponent overflow or underflow which must be intimated with the termination of the process. However, if the exponent of the product (result) lies within the specified range, the next step is to multiply the significands of the input operands, taking into account their signs, as is done for integer multiplication (already described earlier). The product (result) will be double the length of the multiplier or multiplicand, which one is larger. The extra bits may be lost due to rounding-off the result. After obtaining the product, the result as usual needs to be normalized, and rounded-off, if required. The action of normalization may sometimes lead to a situation of exponent underflow. Appropriate actions should then be taken to resolve the situation.
Fixed-Point Multiplication
Published in Joseph Cavanagh, Computer Arithmetic and Verilog HDL Fundamentals, 2017
Multiplication is a process of multiplying the multiplicand by the multiplier to produce a product. The general procedure consists of scanning the multiplier from the low-order bit to the high-order bit. If the multiplier bit is a 1, the multiplicand becomes the partial product; if the multiplier bit is a 0, then 0s are entered as the partial product. Each partial product is then shifted left one bit position relative to the previous partial product.
Understanding the properties of operations: a cross-cultural analysis
Published in International Journal of Mathematical Education in Science and Technology, 2021
Meixia Ding, Xiaobao Li, Ryan Hassler, Eli Barnett
The commutative, associative, and distributive properties undergird the arithmetic operations (NRC, 2001). Considering a, b, and c as any arbitrary numbers in a given number system, the commutative property of addition (CP+) states that a + b = b + a while the commutative property of multiplication (CP×) states that a × b = b × a. In other words, CP deals with the changing of order of numbers with the results invariant. Distinct from CP, the associative property of addition (AP+) states that (a + b)+ c = a +(b + c), while the associative property of multiplication (AP×) states that (a × b) × c = a × (b × c). As such, instead of changing the order of numbers, AP deals with changing the order of operations. Finally, the distributive property of multiplication over addition (DP) states that a × (b + c) = a × b + a × c, which involves the interaction between the two different operations.
An MDPSK homodyne receiver with adaptive phase-diversity
Published in Journal of Modern Optics, 2020
Changqing Cao, Zengyan Wu, Wenrui Zhang, Xiaodong Zeng, Xu Yan, Zhejun Feng, Yutao Liu, Bo Wang
Here, denotes multiplication. The recovered base band signals, shown in Equations (19) and (20), are independent of the phase fluctuations of the LO without phase locking. The phase difference associated with the asymmetry of the two branches has also been eliminated simultaneously.
How well prepared are the teachers of tomorrow? An examination of prospective mathematics teachers' pedagogical content knowledge
Published in International Journal of Mathematical Education in Science and Technology, 2019
As indicated in Table 3, many of the prospective teachers responded correctly to items Q-2 and Q-3. In contrast, the incorrect response rates for the remaining three questions were remarkable; while the participants performed at a high level with respect to the knowledge of learner component, they performed poorly with respect to presenting content. In other words, the prospective teachers were successful at identifying student difficulties or misconceptions in certain scenarios. However, when the scenario related to a presentation of content in terms of how to overcome or eliminate student difficulties or misconceptions, many of them performed at quite a low level. For instance, the first item in the test involved an algorithm of the division with fractions (see Appendix Q-1). As can be seen in Table 3, only a few of the participants (20%) responded correctly to this question. When the incorrect responses were analysed, it was noted that some of the prospective teachers asserted that inverting and multiplying should be taught as a rule without questioning. This response, which was also frequently given for the following questions, does not address why the operation is conceptually valid for the situation. Therefore, these responses were coded as incorrect. One example of such a response is as follows:PT7: Addition is the inverse of subtraction. Then, I say that multiplication is the inverse of division. Just as for we directly multiply two fractions, for the division of we invert and then multiply. This is a rule.Another of the common responses given by the prospective teachers is illustrated below: PT51: If , then we can write . Now it is .