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”.
natural numbers are labeled 1, 2, 10, 11, 12, 20, 21, 22, 100, and so forth.
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!
Numbers do not exist. They are creations of the mind, existing only in the realm of understanding. No one has ever touched a number, nor would it be possible to do so. You may sketch a symbol on a paper that represents a number, but that symbol is not the number itself. A number is just understood. Nevertheless, numbers hold symbolic meaning. Have you ever asked yourself serious questions about the significance, implications, and roles of numbers? For example, “Why does the number ten denote a change to double digits?” “Is zero a number or a non-number?” Or, the matter this paper will address: “Why does the number three hold an understood and symbolic importance?”
If you are asked to give an exact solution for a quadratic equation that does not have x-intercepts, then you will answer that question using the variable i. Say you need to find the square root of negative sixty-four. We know that the perfect square of positive sixty-four is eight. What we are going to do is take out i, resulting in the square root of positive sixty-four, which we know is eight, therefore the answer is plus or minus eight i. It is plus or minus because square roots can be positive or negative because a negative times a negative is also equal to a positive. You can also simplify an equation if there is a constant before the negative under the radical. You do the same thing in terms of simplifying as you would do without the constant. After you get your imaginary number you put the constant in the correct position, and then you are left with a complex number, such as a+bi. You can also simplify i when it is involved in a polynomial. If you multiply out two polynomials that have imaginary numbers in them you may end up with i with an exponent attached. You can use your previous knowledge of patterns to simplify the equation. Say you end up with a term along the lines of thirty-six i squared. We know that i to the second power is the same as negative one. From there we can multiply negative one and thirty-six to result in the product of negative thirty-six. As you have
How far does imaginary numbers go back in history? First must know that an imaginary number is a number that is expressed in terms of the square root of a negative number. This fact took several centuries of convincing for certain mathematicians to believe, but imaginary numbers have been used all the back to the first century, and is now being widely used by people all around the world to this day. It is thanks to people like Heron of Alexandria, Girolamo Cardano, Rafael Bombelli, and other mathematician’s for making imaginary numbers as impactful as they are being used for signal processing, control theory, electromagnetism, quantum mechanics, cartography, vibration analysis, and many others.
The Ancient Indians had some mathematical achievements. One of their mathematical achievements, which was shown in the Vedic texts, is that they had names for every number up to one billion. The Vedic texts also show that they managed to calculate irrational numbers, such as√3, very accurately (Whitfield, Traditions 42)....
There was no discussion in regard to the use of computers and technology in this article, however by assumption there must have been high mathematical equations which must have needed the use of advanced technology like we have today.
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
structures he had never seen before. The type of numbers he was used to had
The simplest forms of equations in algebra were actually discovered 2,200 years before Mohamed was born. Ahmes wrote the Rhind Papyrus that described the Egyptian mathematic system of division and multiplication. Pythagoras, Euclid, Archimedes, Erasasth, and other great mathematicians followed Ahmes (“Letters”). Although not very important to the development of algebra, Archimedes (212BC – 281BC), a Greek mathematician, worked on calculus equations and used geometric proofs to prove the theories of mathematics (“Archimedes”).
The 17th Century saw Napier, Briggs and others greatly extend the power of mathematics as a calculator science with his discovery of logarithms. Cavalieri made progress towards the calculus with his infinitesimal methods and Descartes added the power of algebraic methods to geometry. Euclid, who lived around 300 BC in Alexandria, first stated his five postulates in his book The Elements that forms the base for all of his later Abu Abd-Allah ibn Musa al’Khwarizmi, was born abo...
in exponential form. For instance, in a base 2 system, 4 can be written as 2
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.