12 Questions
Which type of number consists of all rational numbers and irrational numbers?
Real numbers
Which property states that integer addition follows the equation a + b = b + a?
Commutative property
What type of number is pi (3.141...)?
Real number
What was the primary reason for inventing complex numbers?
To solve cubic equations with negative coefficients
Which of the following is not a type of number discussed in the text?
Fractional
What is the imaginary part in a complex number indicated by?
$i$
What does algebra involve manipulating?
Variables and unknowns
In which field are financial calculations heavily reliant on integers and decimals?
Business & Finance
What unique property do prime numbers have in relation to perfect squares?
There are infinitely more primes than perfect squares
In computer science, what tasks do computers perform that require numerical values?
Sorting and searching
What are the common bases used for expressing numbers in today's systems?
Binary and hexadecimal
What is a characteristic of perfect squares in terms of their ending digits?
Always end in 0, 1, 4, 5, 6, or 9
Study Notes
Number Systems
Numbers play a fundamental role in mathematics and our daily lives. They allow us to quantify quantities, measure distances, count objects, and communicate complex ideas using simple symbols. In this article, we will explore various aspects of number systems, including their types, operations, representations, applications, and conversions.
Types of Numbers
There are several distinct categories of numbers:
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Integer: These are whole numbers such as -87, 963, or zero. Integer addition follows the commutative property, meaning
a + b = b + a
, while multiplication is associative and distributive. -
Rational Numbers: Rational numbers can be expressed as fractions, like (\frac{2}{3}) or (-\frac{2}{\pi}), where the fraction is a ratio of two integers.
-
Real Numbers: Real numbers consist of all rational numbers and irrational numbers. Irrational numbers cannot be expressed exactly by a repeating decimal, such as pi ((3.141\ldots)) or the square root of 2.
-
Complex Numbers: A complex number consists of real and imaginary parts, represented as (a+bi), where (i=\sqrt{-1}). Complex numbers were invented to solve cubic equations with negative coefficients.
Each type of number has its own set of rules regarding how they behave under common mathematical operations like addition, subtraction, multiplication, division, exponentiation, and comparison.
Operations on Numbers
Mathematical operations performed on numbers can take different forms depending on the context. For example, arithmetic operations involve manipulating numerals according to basic principles like carrying and borrowing when adding and subtracting digits. Algebra involves manipulating expressions involving variables, unknowns, and other numerical quantities.
In some cases, the order of performing certain operations does not matter due to associative properties, as seen in integer arithmetic where (a+(b+c)=(a+b)+c) or multiplying monomials and polynomials. However, even though (x^2=xx), it does not imply that ((xy)^2 = xyx y).
Applications of Number Systems
Number systems have numerous practical uses across diverse fields:
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Business & Finance: Financial calculations rely heavily on integers and decimals, often dealing with dollars, cents, percentages, and fractions.
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Science: Scientists use numbers to represent physical measurements, molecular structures, and many other phenomena. Some scientists also use nonstandard units to describe dimensions beyond traditional ones.
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Computer Science & Engineering: Computers need numerical values to perform tasks such as addition, subtraction, multiplication, division, sorting, searching, counting, and probability analysis.
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Government & Politics: Governments and political organizations frequently require statistical data, demographics, budgetary needs, and poll results which often involve numbers.
Properties of Numbers
Certain unique characteristics define each kind of number:
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Prime Numbers: Prime numbers only differ from perfect squares by one; therefore, there are infinitely more primes than perfect squares. Each prime has an infinite number of divisors.
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Perfect Square: Every odd square is one less than a multiple of four. Perfect squares always end in 0, 1, 4, 5, 6, or 9. Squaring any number twice gives you back that original number.
These properties can help determine if a number belongs to a specific category or aid in rearranging terms inside parentheses during algebraic simplification.
Conversions Between Number Systems
Different bases, known as positional systems, exist for expressing numbers. Common bases used today include binary (base 2), octal (base 8), decimal (base 10), and hexadecimal (base 16). To convert between these bases, people employ algorithms designed specifically for that task.
Explore the types, operations, applications, and conversions of number systems. Learn about integers, rational numbers, real numbers, complex numbers, and the unique properties of prime numbers and perfect squares. Discover how number systems are used in business, science, computer science, engineering, government, and politics.
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