The History of Imaginary Numbers
The origin of imaginary numbers dates back to the ancient Greeks. Although, at
one time they believed that all numbers were rational numbers. Through the years
mathematicians would not accept the fact that equations could have solutions that were
less than zero. Those type of numbers are what we refer to today as negative numbers.
Unfortunately, because of the lack of knowledge of negative numbers, many equations
over the centuries seemed to be unsolvable. So, from the new found knowledge of
negative numbers mathematicians discovered imaginary numbers.
Around 1545 Girolamo Cardano, an Italian mathematician, solved what seemed to
be an impossible cubic equation. By solving this equation he attributed to the acceptance
of imaginary numbers. Imaginary numbers were known by the early mathematicians in
such forms as the simple equation used today x = +/- ^-1. However, they were seen as
useless. By 1572 Rafael Bombeli showed in his dissertation “Algebra,” that roots of
negative numbers can be utilized.
To solve for certain types of equations such as, the square root of a negative
number ( ^-5), a new number needed to be invented. They called this number “i.” The
square of “i” is -1. These early mathematicians learned that multiplying positive and
negative numbers by “i” a new set of numbers can be formed. These numbers were then
called imaginary numbers. They were called this, because mathematicians still were
unsure of the legitimacy. So, for lack of a better word they temporarily called them
imaginary. Over the centuries the letter “i” was still used in equations therefore, the name
stuck. The original positive and negative numbers were then aptly named real numbers.
What are Imaginary Numbers?
An imaginary number is a number that can be shown as a real number times “i.”
Real numbers are all positive numbers, negative numbers and zero. The square of any
imaginary number is a negative number, except for zero. The most accepted use of
imaginary numbers is to represent the roots of a polynomial equation (the adding and
subtracting of many variables) in one variable. Imaginary numbers belong to the complex
number system. All numbers of the equation a + bi, where a and b are real numbers are a
part of the complex number system.
Imaginary Numbers at Work
Imaginary numbers are used in a variety of fields and holds many uses. Without
imaginary numbers you wouldn’t be able to listen to the radio or talk on your cellular
phone. These type of devices work by receiving and transmitting radio waves. Capacitors
and inductors are used to make circuits that are used to make radio waves.
For example,"(-x+√(x^2-1))” plus “(x+√(x^2-1))” cancel because they equal to zero. This leaves one with x^2 – (x^2-1), so the minus sign is to be distribute to “(x^2-1)”. This equals to x^2 – x^2+1, and makes x^2 and negative x^2 cancel out because it will also equal to zero. As a result, log(a) is the same as saying log(1), which equals to zero. This is because the “10” to the power of “0” equals to “1”.
This means that although the specific numbers systems may vary from culture to culture, the basic concept that one thing plus another thing means you have two things is seemingly
After Gretchen is given the problem, she approaches it with her first method: the standard algorithm. To start off, she sets up the problem by writing out “70” and writing “- 23” directly below it, finishing out with a line underneath. This setup indicates several things about Gretchen’s basic mathematical understanding. First, it shows that she understands a connection between the words and the actual written symbols for each number. Also, since she writes “70,” Gretchen probably has knowledge of numbers up to ninety-nine. Last, her arrangement of the numbers indicates that she has knowledge of the minus being a symbol for “take away” and the second number be placed underneath the first. As she works the problem and su...
The Golden ratio is an infinite number that is rounded approximately to 1.618. Euclid referred to the decimal form of the golden ratio, which is 0.61803…, in his book The Elements. The golden ratio is a very special number with many properties. One of its properties is that to square the golden ratio, you could just add one to it. The formula for squaring the golden ratio would be phi²= Phi + 1. Another property of the golden ratio is that to get the reciprocal you can just subtract one. The reciprocal of Phi would be Phi-1. The golden ratio is often written as a/b
People use numbers whenever they do math. Yet, do they know that each number in the number system has its own unique trait? Numbers such as 4 and 9 are considered square numbers because 2 times 2 is 4, and 3 times 3 is 9. There also prime numbers. Prime numbers are numbers that have exactly two divisors. The number one is not included because it only has one divisor, itself. The smallest prime number is two, then three, then five, and so on. This list goes on forever and the largest known primes are called Mersenne primes. A Mersenne prime is written in the form of 2p-1. So far, the largest known Mersenne prime is 225,964,951-1, which is the 42nd Mersenne prime. This prime number has 7,816,230 digits!
Cantor created M, where M is a real number that does not correspond with any natural number. Taking the first digit in the first real number, write down any other number for the tenth’s place of M. Then, take the second digit for the second real number and write down any other number for the hundredth’s place of M.
Countless time teachers encounter students that struggle with mathematical concepts trough elementary grades. Often, the struggle stems from the inability to comprehend the mathematical concept of place value. “Understanding our place value system is an essential foundation for all computations with whole numbers” (Burns, 2010, p. 20). Students that recognize the composition of the numbers have more flexibility in mathematical computation. “Not only does the base-ten system allow us to express arbitrarily large numbers and arbitrarily small numbers, but it also enables us to quickly compare numbers and assess the ballpark size of a number” (Beckmann, 2014a, p. 1). Addressing student misconceptions should be part of every lesson. If a student perpetuates place value misconceptions they will not be able to fully recognize and explain other mathematical ideas. In this paper, I will analyze some misconceptions relating place value and suggest some strategies to help students understand the concept of place value.
...letter “f” and parentheses for a function; the use of the symbol π for the ratio of circumference to diameter in a circle; and i for √ (-1) (Leonhard Euler).
Prior to the 15th century, Italy was still using roman numerals. Solving mathematical problems with roman numerals was problematic to the Venetian merchants of the time. Sometime during the 15th century, Venetian merchants began using Arabic numbers. Arabic numbers made mathematics much easier. (Kestenbaum, 2012)
Tubbs, Robert. What is a Number? Mathematical Concepts and Their Origins. Baltimore, Md: The Johns Hopkins
Present day zero is quite different from its previous forms. Many concepts have been passed down, and many have been forgotten. Zero is the only number that is neither positive of negative. It has no effect on any quantity. Zero is a number lower than one. It is considered an item that is empty. There are two common uses of zero: 1. an empty place indicator in a number system, 2. the number itself, zero. Zero exist everywhere; although it took many civilizations to establish it.
Irrational numbers are real numbers that cannot be written as a simple fraction or a whole number. For example, irrational numbers can be included in the category of √2, e, Π, Φ, and many more. The √2 is equal to 1.4142. e is equal to 2.718. Π is equal to 3.1415. Φ is equal to 1.6180. None of these numbers are “pretty” numbers. Their decimal places keep going and do not end. There is no pattern to the numbers of the decimal places. They are all random numbers that make up the one irrational number. The concept of irrational numbers took many years and many people to discover and prove (I.P., 1997).
in exponential form. For instance, in a base 2 system, 4 can be written as 2
The history of math has become an important study, from ancient to modern times it has been fundamental to advances in science, engineering, and philosophy. Mathematics started with counting. In Babylonia mathematics developed from 2000B.C. A place value notation system had evolved over a lengthy time with a number base of 60. Number problems were studied from at least 1700B.C. Systems of linear equations were studied in the context of solving number problems.