machine arithmétique
What is machine arithmetic and why is it important?
Machine arithmetic refers to the operations that a computer's CPU can perform: addition, subtraction, multiplication, division, amongst others. They are fundamental because they form the basis of all computations performed by a computer such as executing software, running games, and processing data.
What do you mean by CPU?
CPU stands for Central Processing Unit. It is the primary component of a computer that performs most of the processing inside the computer. The CPU interprets and carries out the instructions of a computer program.
What are the basic operations in machine arithmetic?
The four fundamental operations of arithmetic - addition, subtraction, multiplication, and division are the basic operations. These are performed using binary, the number system that computers use because of their digital nature.
What is meant by binary numbers and why do computers use them?
A binary number is a number expressed in the base-2 numeral system, a method which uses only two symbols: typically 0 (zero) and 1 (one). Computers use binary because they operate electrically and electronically, with two states - on and off, which can be represented by 1 and 0 respectively.
How does addition work in machine arithmetic?
In machine arithmetic, addition is done using what's known as a binary adder. Binary addition follows the same rules as mathematical arithmetic but only uses two numbers, 0 and 1. When both numbers are 1, it results in a binary 10 (2 in decimal). The '1' is carried over to the next higher bit, similar to carrying in decimal arithmetic.
What is a binary adder?
A binary adder is a hardware circuit that is used in computers to perform binary addition of two numbers. Simplest one is a half-adder, which adds two single binary digits and accounts for values carried in and out.
How does subtraction work in machine arithmetic?
Subtraction in machine arithmetic uses a process called two's complement method. This method transforms the number to be subtracted into a form that when added to the first number gives the desired result.
What is the two's complement method?
Two's complement is a method for representing signed integers in binary number systems. For subtraction, it is used as it allows the subtraction of more significant numbers from less significant ones by adding the two's complement of the second number to the first.
How does multiplication work in machine arithmetic?
In machine arithmetic, multiplication of two binary numbers is performed by a sequence of shifting and adding operations. Each bit in the multiplier is used to determine whether a shifted version of the multiplicand should be added to the product.
What does shifting mean in this context?
In the context of binary arithmetic, shifting a number means to move its digits left or right. Each move to the left increases the number by a power of two (similar to a tenfold increase in decimal); a shift to the right decreases the number by a power of two.
How does division work in machine arithmetic?
Division in machine arithmetic works by using a similar process to long division in decimal arithmetic. The divisor is repeatedly subtracted from the dividend and the quotient is incremented each time until the dividend is less than the divisor. The remainder is then the final adjusted dividend.
Is the process faster in machine arithmetic than manual?
Yes, because computers can perform these operations very quickly, machine division is much faster than manual long division. However, depending on the complexity of the numbers involved, it can still be a relatively slow operation in computational terms.
What are boolean operations in machine arithmetic and how are they used?
Boolean operations are simple comparisons of binary digits, or bits, and are named after George Boole, the 19th-century mathematician who introduced them. They include AND, OR, NOT, XOR (exclusive OR), and others. These operations are used extensively in computer programming and digital electronics.
Can you elaborate on what AND, OR and NOT operations are?
In Boolean logic, an AND operation results in true if both operands are true; an OR operation results in true if at least one operand is true; a NOT operation inverts the truth value, true becomes false and vice versa.
How does a computer handle large numbers and floating point arithmetic in machine arithmetic?
Computers handle large numbers and floating-point arithmetic using specific storage formats. Large numbers are broken down into smaller parts, each of which is processed separately using the computer's basic arithmetic capabilities. Floating-point numbers are represented as a binary fraction and an exponent, allowing them to accommodate an enormous range of scales.
What is the significance of floating-point numbers?
Floating-point representation is used for real numbers, that is, numbers that have a decimal point. They allow for very large and very small numbers and are utilized when precision is necessary, such as in scientific calculations.
What is overflow and underflow in the context of machine arithmetic?
Overflow and underflow refer to a situation where a calculation exceeds the maximum or minimum limit that a computer can represent. Overflow occurs when a result is too large for the assigned space, and underflow occurs when a result is too close to zero for the computer to store it accurately.
How do computers handle situations of overflow and underflow?
Most computers handle these situations by returning an error or a special value that represents infinity or zero, and the software running on the computer should be designed to handle these values appropriately.
What are some real-world examples where machine arithmetic is used?
Machine arithmetic is integral to almost all digital technology. It's used in everything from scientific computations like modelling climate change or the spread of diseases, to the basic functioning of websites, applications and games. In robotics, it controls motor movements and sensor data analysis, it's also central to image and video processing in digital cameras and film production.
Can you give an example of how machine arithmetic is used in games?
In games, machine arithmetic might be used to calculate trajectories in physics-based games or handle scoring systems. The coordinates and movement of an object in 3D space for example are often calculated using floating-point arithmetic.