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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.
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
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
.
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