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Simple System Design Techniques
Published in A. Arockia Bazil Raj, FPGA-Based Embedded System Developer's Guide, 2018
Subtraction is a mathematical operation in which one number is deducted from another to obtain the equivalent quantity. The number from which the other number is to be deducted is called the minuend, and the number subtracted from the minuend is called the subtrahend. In all the operations, each subtrahend bit is deducted from the minuend bit. Similarly to binary addition, a single-bit binary subtraction also has four possible operations, such as (0–1 = 0), (1–0 = 1), (1–1 = 0), and (0–1 = 10–1 = 11). This means that, like decimal subtraction, we can simply subtract a smaller number from a bigger number without borrowing anything as the first three subtractions, whereas in the last case, that is, (0–1), subtraction of “1” from “0” is not possible; hence, we need to get a borrow. Then the “0” becomes “10” (“10”–“1” = 1); thus, the difference is “1” and the borrow is also “1”. Similarly to adder circuits, subtraction circuits are also classified as half and full subtractors. A half subtractor is a combinational circuit that does the subtraction of two bits of binary data. It has two input variables and two output variables that correspond to difference and borrow bits. Binary subtraction is performed by the xor gate and a not with an and gate, as shown in Figure 3.3a, which produces the difference and borrow, respectively. Thus, a half subtractor is designed by an xor gate including an and gate with input (A) complemented before being fed to the and gate [21–25].
Algebra
Published in John Bird, Bird's Engineering Mathematics, 2021
The order of precedence is brackets, multiplication, then subtraction. Hence (a+5a)×2a−3a=6a×2a−3a=12a2−3a or 3a(4a−1)
Vector Algebra and Calculus
Published in William F. Ames, George Cain, Y.L. Tong, W. Glenn Steele, Hugh W. Coleman, Richard L. Kautz, Dan M. Frangopol, Paul Norton, Mathematics for Mechanical Engineers, 2022
A vector is a directed line segment, with two vectors being equal if they have the same length and the same direction. More precisely, a vector is an equivalence class of directed line segments, where two directed segments are equivalent if they have the same length and the same direction. The length of a vector is the common length of its directed segments, and the angle between vectors is the angle between any of their segments. The length of a vector u is denoted ∣u∣. There is defined a distinguished vector having zero length, which is usually denoted 0. It is frequently useful to visualize a directed segment as an arrow; we then speak of the nose and the tail of the segment. The sumu + v of two vectors u and v is defined by taking directed segments from u and v and placing the tail of the segment representing v at the nose of the segment representing u and defining u + v to be the vector determined by the segment from the tail of the u representative to the nose of the v representative. It is easy to see that u + v is well defined and that u + v = v + u. Subtraction is the inverse operation of addition. Thus the differenceu − v of two vectors is defined to be the vector that when added to v gives u. In other words, if we take a segment from u and a segment from v and place their tails together, the difference is the segment from the nose of v to the nose of u. The zero vector behaves as one might expect; u + 0 = u, and u − u = 0. Addition is associative: u + (v + w) = (u + v) + w.
Pre-service elementary teachers’ reasoning in introducing multidigit subtraction
Published in International Journal of Mathematical Education in Science and Technology, 2020
Only 5 out of the 230 responses included the word ‘regrouping’ but each of these responses only used regrouping as a term, and not as an underlying concept.The number sentence or the subtraction sentence above is solved with regrouping. 4–9 cannot be so we borrow to the next number which is 5. So 5 becomes 4 and 4 will become 14. 14–9 is 5. 4–1 is 3 and 2–1 is 1. The answer is 135. (Participant #165)[translated to English] The process to get the answer is subtraction. To subtract, if the minuend is smaller than the subtrahend, we need to borrow to subtract. This is called subtraction with regrouping. (Participant #166)