Algebra is one of the most cardinal topics in mathematics that explains geometry, number theory, and analysis. Many of us find algebra very daunting and challenging to understand, as it is the study of complex mathematical symbols, rules, and language. It is enough to write “do my homework“, but it is a simple way. Though it may sound impossible initially, with thorough practice and apprehension, you can eliminate this dilemma.
So let us try to comprehend algebra’s meaning before delving deep into the subject matter of ‘i” in algebra. Algebra is a branch of mathematics that deals with symbols and rules associated with manipulating and handling mathematical symbols. While solving mathematical equations following problems arise-
How to substitute value in maths?
How to establish a relationship by using letters or symbols?
How to manipulate the equation to get the desired outcomes?
Algebra is a solution to all the problems. It allows you to substitute values in an equation to get favorable results. It helps you use numbers of the alphabet to show the value of any object, equation, or expressions. E.g. x = 1
Important Branches of Algebra
Elementary Algebra This branch of algebra talks about rudimentary topics like arithmetic and mathematical operations like +, -, x, ÷. In algebra, numbers are constituted as a, x, y. It helps in the systematic exploration of an equation like 2 = x-y.
- Advanced Algebra
The immediate level algebra deals with complex and high-level equations that include metrics, conic sections, trigonometry, sequence, and series. Moreover, it helps to solve multi-step equations, complex algebraic expressions, and problems.
- Abstract Algebra
One of the most crucial branches of algebra is abstract algebra. It is related to symbols associated with algebraic equations and systems as they are independent due to specific nature. For example – Fields, groups, modules, rings, lattices, and vector spaces are studied under the domain of abstract algebra. For this, the base of elementary algebra should be strong.
- Linear Algebra
It is used in both pure and applied mathematics. It studies linear mapping and vector spaces. It is concerned with linear equations and their representation in vector spaces with transformation properties.
- Commutative Algebra
It is that branch of algebra that studies commutative rings and their properties. Various theories in mathematics like number theory, invariant theory, order theory, general topology, and algebraic geometry depend on commutative algebra. Thus, it holds a significant role in modern mathematics.
It is just the gist about Algebra. For a more detailed explanation about algebra, try to understand Algebra the Cuemath way. But what is Cuemath? Well, for that, keep reading till the end. So now, let us learn about i in algebra.
Background of i in Algebra
i is an imaginary number and denoted as i. Greek mathematician Heron of Alexandria first coined these numbers. In 1952, Rafael Bombeli designed the rules for the multiplication of complex numbers. Earlier, imaginary numbers were overlooked by many mathematicians who considered it useless and illogical.
In 1843, William used imaginary numbers in-plane as four-dimensional space. Substantially, the imaginary number became popular and became widely accepted. In 1848, the idea of imaginary was first surfaced by James in his article.
Now imaginary numbers are used in pop culture also. Famous Protagonist Robert called Sophie’s brief in imaginary numbers in The Da Vinci Code. Many favorite writers like Issav Asimov used i in his various short stories.
Meaning
When any number is squared, and its outcome comes negative, it is called imaginary numbers denoted as i. In other words, when the number is a square root of a negative number and doesn’t have any tangible value and at the same time cannot be quantified.
Imaginary numbers are also called complex numbers as they are used in real-life applications like electricity and have real existence. It is highly helpful and useful in advanced calculus. Usually, in the case of electronics, imaginary numbers are represented by j instead of i because i already resonates with the current. An imaginary number is mostly associated with AC ( Alternating Current) as it changes from positive to negative in a sine wave. Yes, we use AC based on imagery numbers as well. In a nutshell, the letter i is a number, and when multiplied, gives – 1. The real numbers and imaginary current help in the calculation and protect against electrocution.
Graphic representation of i is –
√-1 = i
For Example – 3i
Where
3 – Real number
i:- Imaginary unit
-
- Now suppose, 3i is squared, the result will be negative as -9 as the value of i2 is always – 1. Thus – √-1 = i
Complex Number
The number is said to a complex number when it contains both imaginary units and real numbers. It is represented as a + bi
a and b are real numbers
i:- Imaginary unit
Unit Imaginary Number
In simple words, the square root of minus one is called the imaginary unit number equal to 1 in real numbers. Mathematically represented as –
√(−1)
So can you identify the square root of -1? Yes, i can do that.
Imaginary Number Rules
Suppose, a pure quadratic equation is x*x= a, where the value of a is known. Hence, it will be presented as x = √a. Following rules are followed in imaginary number –
- i = √-1
- i2 = -1
- i3 = -i
- i4 = +1
- i4n = 1
- i4n-1= -i
Imaginary Number Chart
One of the most interesting properties of i is when you multiply. It shows four different values. For example, i x i = -1, then with, -1 x i = -i. -i x i = 1 and then 1 x i = i. Thus, you can easily trace the exponent.
Since this process is carried out through the exponents, profoundly known as the Imaginary Number Chart, this chart is a prerequisite for multiplication and division. It helps to simplify the imaginary equations.
Operations on Imaginary Number
In mathematics, arithmetic followed is – Addition, subtraction, division, and multiplication. Let us check their operations on imaginary numbers.
- Addition
Suppose two numbers a+bi and x+yi are added. Firstly, the real parts are simplified and added therein. Later, the imaginary parts are added and simplified. For instance–
(a+x) + i(b+y)
- Subtraction
If (x+yi) needs to be subtracted from (a+xi), we will first segregate the real and imaginary terms and apply simplification. For example –
(a+xi)-(b+yi) = (a-b) +i(x-y)
- Division
When numbers need to be divided, the following method is followed-
(a+bi) / (x+yi) = (a+bi) (x-yi)/ (x+yi) (x-yi)
(ax + by) (bx- ay)/x2 + y2
- Multiplication
When two numbers need to multiply, the following procedure is followed –
(a+bi) (x+yi) = (a+bi)x + (a+bi)yi
ax+bix+ayi+byi2
(ax -by) + i(bxi + ay)
Useful uses of Imaginary Number
There are a plethora of benefits to imaginary numbers. Some of them are as under –
- Spectrum Analyzer- Have you ever noticed those seamless displays when you play any music? It is done through complex numbers and by using Fourier transforms. With the help of a complex number, one can do signal processing, sound filtering, whispering sound, and much more.
- Electricity- As mentioned before, two AC currents are difficult to manage, and it a tough job to find a new current supply. But with the help of a complex number, it is possible.
- Mandelbrot Set– The Mandelbrot set is the arrangement of complex numbers {\displaystyle c}c for which the capacity {\displaystyle f_{c}(z)=z^{2}+c}{\displaystyle f_{c}(z)=z^{2}+c} doesn’t separate when iterated from {\displaystyle z=0}z=0, i.e., for which the succession {\displaystyle f_{c}(0)}{\displaystyle f_{c}(0)}, {\displaystyle f_{c}(f_{c}(0))}{\displaystyle f_{c}(f_{c}(0))}, and so on, stays limited in total worth. Its definition is credited to Adrien Douady who named it in accolade for the mathematician Benoit Mandelbrot, a pioneer of fractal calculation. It depends on complex numbers only.
- Quadratic Equation– When we talk about the quadratic equation, an imaginary number occurs when the radical portion’s quadratic formula’s value is negative. The roots that appear belong to the set of complex numbers, also known as imaginary roots. It can be referred to as a +bi.
Conclusion
From the above explanation, it is clearly understood the meaning and importance of i in algebra. But do you know the most exciting fact about i? Earlier, it was observed that the imaginary number is impossible and doesn’t exist. From that, it was named as an imaginary number out of the fun. But after extensive research, it happened to discover that it is useful and bridges the gap in mathematics.
Now that you have a fair idea of the key points to focus on, you will find yourself more confident and knowledgeable about imaginary numbers or i. So now you know what does i mean in algebra. Right? Well, Cuemath offers personalized learning material for every student and believes in fabricating a staunch learning foundation where they are math-loving rather than math fearing.
If you have any more suggestions or have any questions about the imaginary numbers, please comment below and let us know about it!
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