Lesson History of mathematics

Vocabulary : 

Research  

Izlanish  

Rigorous  

Mukammal  

Discipline  

Tartib intizom  

Enormous  

Ulkan  

Observation  

Kuzatish  

Practical  

Amaliy  

Formulate  

Ta’riflamoq  

Appropriate  

Muvofiq  

Phenomenon  

Hodisa, voqea  

Application  

Ariza  

 

 

 

 

 

 

Lesson 3. Arithmetic.  

Arithmetic (a term derived from the Greek word arithmos, “number”) refers generally 

to the elementary aspects of the theory of numbers, arts of mensuration 

(measurement), and numerical computation (that is, the processes of addition, 

subtraction, multiplication, division, raising to powers, and extraction of roots). Its 

meaning, however, has not been uniform in mathematical usage. An eminent German 

mathematician, Carl Friedrich Gauss, in Disquisitiones Arithmeticae (1801), and 

certain modern-day mathematicians have used the term to include more advanced 

topics. The reader interested in the latter is referred to the article number theory.  

Fundamental Definitions And Laws  

Natural numbers  

In a collection (or set) of objects (or elements), the act of determining the number of 

objects present is called counting. The numbers thus obtained are called the counting 

numbers or natural numbers (1, 2, 3, …). For an empty set, no object is present, and 

the count yields the number 0, which, appended to the natural numbers, produces 

what are known as the whole numbers.  

 

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If objects from two sets can be matched in such a way that every element from each 

set is uniquely paired with an element from the other set, the sets are said to be equal 

or equivalent. The concept of equivalent sets is basic to the foundations of modern 

mathematics and has been introduced into primary education, notably as part of the 

“new math” (see the figure) that has been alternately acclaimed and decried since it 

appeared in the 1960s.  

Addition and multiplication  

Combining two sets of objects together, which contain a and b elements, a new set is 

formed that contains a + b = c objects. The number c is called the sum of a and b; and 

each of the latter is called a summand. The operation of forming the sum is called 

addition, the symbol + being read as “plus.” This is the simplest binary operation, 

where binary refers to the process of combining two objects.  

---

From the definition of counting it is evident that the order of the summands can be 

changed and the order of the operation of addition can be changed, when applied to 

three summands, without affecting the sum. These are called the commutative law of 

addition and the associative law of addition, respectively.  

If there exists a natural number k such that a = b + k, it is said that a is greater than b 

(written a > b) and that b is less than a (written b < a). If a and b are any two natural 

numbers, then it is the case that either a = b or a > b or a < b (the tracheotomy law).  

From the above laws, it is evident that a repeated sum such as 5 + 5 + 5 is 

independent of the way in which the summands are grouped; it can be written 3 × 5. 

Thus, a second binary operation called multiplication is defined. The number 5 is 

called the multiplicand; the number 3, which denotes the number of summands, is 

called the multiplier; and the result 3 × 5 is called the product. The symbol × of this 

operation is read “times.” If such letters as a and b are used to denote the numbers, 

the product a × b is often written a∙b or simply ab.  

If three rows of five dots each are written, as illustrated below, it is clear that the total 

number of dots in the array is 3 × 5, or 15. This same number of dots can evidently be 

written in five rows of three dots each, whence 5 × 3 = 15. The argument is general, 

leading to the law that the order of the multiplicands does not affect the product, 

called the commutative law of multiplication. But it is notable that this law does not 

apply to all mathematical entities. Indeed, much of the mathematical formulation of 

modern physics, for example, depends crucially on the fact that some entities do not 

commute.  

By the use of a three-dimensional array of dots, it becomes evident that the order of 

multiplication when applied to three numbers does not affect the product. Such a law 

is called the associative law of multiplication. If the 15 dots written above are 

separated into two sets, as shown, then the first set consists of three columns of three 

dots each, or 3 × 3 dots; the second set consists of two columns of three dots each, or 

2 × 3 dots; the sum (3 × 3) + (2 × 3) consists of 3 + 2 = 5 columns of three dots each, 

or (3 + 2) × 3 dots. In general, one may prove that the multiplication of a sum by a 

number is the same as the sum of two appropriate products. Such a law is called the 

distributive law.  

Integers  

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Subtraction has not been introduced for the simple reason that it can be defined as the 

inverse of addition. Thus, the difference a − b of two numbers a and b is defined as a 

solution x of the equation b + x = a. If a number system is restricted to the natural 

numbers, differences need not always exist, but, if they do, the five basic laws of 

arithmetic, as already discussed, can be used to prove that they are unique. 

Furthermore, the laws of operations of addition and multiplication can be extended to 

apply to differences. The whole numbers (including zero) can be extended to include 

the solution of 1 + x = 0, that is, the number −1, as well as all products of the form −1 

× n, in which n is a whole number. The extended collection of numbers is called the 

integers, of which the positive integers are the same as the natural numbers. The 

numbers that are newly introduced in this way are called negative integers.  

Exponents  

Just as a repeated sum a + a + 

⋯ + a of k summands is written ka, so a repeated 

product a × a × 

⋯ × a of k factors is written a

k

. The number k is called the exponent, 

and a the base of the power a

k

.