Unit VIII “The link of computer science with mathematics. Programming languages.”

1. Read and translate the text. Learn the words in bold type: Mathematical Language

There are many symbols in mathematics and most are used as a precise form of shorthand. We need to be confident when using these symbols, and to gain that confidence we need to understand their meaning. To understand their meaning there are two things to help us: context - this is the context in which we are working, or the particular topics being studied, and convention - where mathematicians and scientists have decided that particular symbols will have particular meaning.

Words associated with the symbol “+” are ‘plus’, ‘add’, ‘increase’ and ‘positive’. Words associated with the symbol “−” are ‘minus’, ‘subtract’, ‘take away’, ‘negative’ and ‘decrease’. Generally we use these two common signs to add or subtract something. The symbols “++” and “- -“ are used as an increment and a decrement in imperativeprogramming languages.

Multiplication is the mathematical operationofscaling one number by another. Words associated with the symbol "×" are ‘multiply’, ‘lots of’, and ‘times’. Because the result of scaling by whole numberscan be thought of as consisting of some number of copies of the original, whole-number products greater than 1 can be computed byrepeated addition; for example, 3 multiplied by 4 (often said as "3 times 4") can be calculated by adding 4 copies of 3 together:

Here 3 and 4 are the "factors" and 12 is the "product".

In mathematics, especially inelementary arithmetic, division (÷) is an arithmetic operation. In the expression a ÷ b = c, a is called thedividend or numerator, b the divisor or denominator and the result c is called the quotient. Dividing two integers may result in a remainder. To complete the division of the remainder, the number systemis extended to includefractions or rational numbersas they are more generally called.

Another symbol used frequently is the equals sign “=”. The “=” sign does not mean anything on its own - we need a context. For example, in the sum 1+2 = 3, what we are saying is that whatever we have on the left-hand side is exactly equal to whatever we have on the right-hand side.

Variations of the equals sign are:

• which means ‘is not equal to’;

≈ which means ‘is approximately equal to’;

≥ which means ‘is greater than or equal to’ ;

≤ which means ‘is less than or equal to’.

In computer science, the mentioned above signs belong torelational operators that test or define some kind of relationbetweentwo entities. In programming languages that include a distinctboolean data typein theirtype system, likeJava, these operators return true or false, depending on whether theconditional relationship between the two operandsholds or not. Inother languages such as C, relational operators return the integers 0 or 1.

In mathematics, avariable is a valuethat may change within the scope of a given problem or set of operations. In contrast, aconstantis a value that remains unchanged, though often unknown or undetermined. The concepts of constants and variables are fundamental to many areas of mathematics and itsapplications. Incomputer programming, a variable is astorage location and an associated symbolic name(anidentifier) which contains some known or unknown quantity or information, avalue. The variable name is the usual way to reference the stored value; this separation of name and content allows the name to be used independently of the exact information it represents. The identifier in computer source codecan beboundto avalueduringrun time, and the value of the variable may thus change during the course ofprogram execution. Variables in programming may not directly correspond to the concept ofvariables in mathematics. The value of a computing variable is not necessarily part of anequationorformulas in mathematics. In computing, a variable may be employed in a repetitive process: assigned a value in one place, then used elsewhere, thenreassigned a new value and used again in the same way (iteration). Variables in computer programming are frequently given long names to make them relatively descriptive of their use, whereas variables in mathematics often have one- or two-character names forbrevity in transcription and manipulation.

The positioning of numbers and symbols in relation to each other also gives meaning. For example, you will come across use of superscripts. These are small numbers or symbols written at the top right of another, as in 4². In this context 4² is a shorthand for ‘four squared’ or 4 x 4. Similarly 4³ is shorthand for ‘four cubed’ or 4 x 4 x 4. On the other hand 32° can mean different things in different contexts. It might mean an angle of 32 degrees. It could mean 32 to the power zero, which is actually 1.