What are irrational and complex numbers?

Question

what are rationl/irrational number. and what are complex numbers and what are imaginary numbers?
i am so cunfused.

Answer 1

A rational number is any number that can be written as a ratio of two integers (ratio is even part of the word rational).

In other words any number that can be written as a fraction comparing integers.

Examples:
A. 2, 0, 1, are all whole numbers and can be written as fractions over 1 such as -2/1, 0/1, and 1/1.

B. Any fraction is by definition a ratio and therefore a rational number. For example 2/5, 5/2, -11/13 are all rational.
It does not matter if the ratio is positive or negative or whether or not the number is positive or negative.

C. All repeating numbers are rational. For example .333333.... is often shown as .3 with a bar over it, where the bar marks that the digit repeats continuously.
This may not seem "rational" at first but it is really the fraction 1/3 which is a ratio.

As a matter of fact all repeating decimals can be written as a ratio as follows.
Numerator:The number that is repeating.
Denominator: The same number of nines as digits repeating.
For Example - .234234234234234.... would be written as .234 with a bar over it.
The numerator would be 234.
3 numbers repeat so the denominator is 999.
So the repeating decimal .234 (bar)= 234/999

Irrational numbers cannot be written as ratios of integers. Classic examples are pi, the number e, and a variety of never ending sequences. Many square roots are also irrational as a ratio cannot be formed.

An easy way to remember irrational numbers is that they are numbers whose decimal places do not repeat or end.

Examples:

A. pi is a decimal that goes on forever but never repeats. It was found by comparing area and circumference of a circle. it starts 3.1415926535897932384....and the numbers keep going but do not repeat.

B. The number e is found with the equation(1+1/n)^n. As you plug in larger numbers the answers approach the same numbers but the decimals at the end keep going.
Though the calculator rounds,
Plugging in 1000 gives 2.716923932
Plugging in 10,000 gives 2.7181145927
Plugging in 100,000 gives 2.718268237
Plugging in 1,000,000 gives 2.718280469...
(See how 2.718 is not changing but the latter decimal places still are)

As you plug in even higher numbers we see the number approach 2.718281828459...
but the decimals at the end of the number dont repeat and keep growing ...so e represents the number this equation will approach as you plug in even higher numbers. Since the number does not repeat or end it is said to be irrational.

C. the Square Root of 2 is irrational as you cannot find one number with a decimal that ends to multiply by itself to equal two. The calculator rounds to 1.414213562....but the decimal is still going and not repeating.

Now On to Complex Numbers: Real verse Imaginary.

All numbers that we have discussed as rational or irrational are real numbers as they can be arrived at using an equation or ratio.

The idea of the imaginary number was formed for use in finding the solution to quadratic equations.

The solution to a quadratic equation is the point or points where the parabola(graph) hits the x axis, in other words where y =0. These equations can be solved using the quadratic formula. Common answers might be 2 +or- 6, where the answers are 2+6 or 2-6 which equals 8 or -4. Both of these answers exist and are real.

Certain Quadratic functions exist however that never cross the x axis as they are entirely above or below it. While they have no "real solutions" If you plug these equations into the quadratic formula you get an answer such as 2 plus or minus the square root of -4. While 2 is a real number the sqare root of -4 is not. There is no number you can multiply by itself to get -4.(note:-2 times -2 equals 4 not -4)

While answers to these negative square roots do not exist, it is helpful in graphing to use them as a measure (I will not get into why at this point). In order to get around the fact these measures are not real "the imaginary unit" or simply "i" was created. This measure represents the square root of -1 (even though the square root of -1 is not a real number).Any time the square root of -1 comes up it is referred to as i.

So in the answer above of 2 plus or minus the sqare root of -4, the number 2 is reffered to as are 2 real parts and the square root of -4 can be written (the square root of 4 times the square root of -1). The square root of 4 is 2 and the square root of negative 1 is known as i. so the answer to the square root of -4 is 2i or 2 imaginary parts.

So in the equation above when it said 2 plus or minus the square root of -4 it would be two real parts (the number 2) plus or minus the 2 imaginary parts (as found from the square root of -4 as shown above).

Complex numbers are numbers that incorporate both the real part and the imaginary part found in solving a equation such as the quadratic equation.

Complex numbers are written in the form a+bi, where a is the real part and b is the imaginary part.

So the answer above of 2 plus or minus 2i is really 2 complex numbers. 2 + 2i and 2 - 2i.

Any real number has no imaginary part so the number 5 would be written 5 + 0i as a complex number since it has 5 real parts and no imaginary part.

You can assume the imaginary part of a number is 0i unless a square root(or other even root) of a negative is part of your answer.

Answer 2

Here's the condensed version:

Complex numbers (where a, b are any real numbers including 0)
--> a + bi

Imaginary numbers (where a, b are any real numbers, except b ≠ 0)
--> a + bi

Irrational numbers (anything when reduced that includes):
--> √ π e Φ etc. (often sin, cos, log, ln)

Rational numbers (everything else)
--> fractions, decimals, repeating decimals

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Here's my original, verbose version:

Rational numbers are any numbers that can be represented as a *ratio* of two integers. For example, 1/3 is rational, 2 is rational, 15/7 is rational. In addition, the decimal versions of these numbers are also rational. For example 0.333..., 2.0 and 2.142857... are all rational. Notice that even though they may have repeating decimals, they repeat *in a pattern* so they can be turned into a fraction.

Irrational numbers are things like square roots (√3), pi, e, etc. that cannot be represented as a ratio of integers. When these are represented as decimals, they continue forever *without repeating in a pattern*.

Next there are real and imaginary numbers. Real numbers are all the numbers you are used to counting with or representing values.

Imaginary numbers are any numbers that have i within them where i = sqrt(-1). Examples are 5i, √2i, -7 + i, etc. The formal definition is if you square an imaginary number it is negative. (Real numbers will always have non-negative squares).

Complex numbers are numbers with a real and imaginary part (a + bi). For example 5 + 2i is a complex, imaginary number. Another example is √3 + (3/5)i.
And technically, real numbers are a subset of the complex numbers (in the case where b is 0). Thus √7 + 0i is a real number because it is just √7, a real number.

So at the highest level you have complex numbers:
a + bi

If they have i within them they are irrational. If they have no i within them they are real.

Next within the set of real numbers you have rational and irrational numbers. Irrational are numbers that have √ within them, things related to e, logs, natural logs, pi, etc. In decimal form these numbers would go forever without repeating in a pattern.

Finally, there are the rational numbers (anything than can be expressed as a fraction of integers).

Take your pick on which one helps you more...

Answer 3

Rational numbers are any numbers that may be written as fractions. Whole numbers n are rational, because they can be written as n/1. Decimal numbers are also rational, because, for example, 41.25 can be written as 4125/100.

Irrational numbers are all the rest of the numbers, that can't be written as fractions. Most numbers are irrational.

A complex number is a number with a real and an imaginary part. It's always written in the form a + bi, where a and b are real numbers. So a real number is a kind of complex number, as it can be written a + 0i. Imaginary numbers are also a kind of complex number, as it can be written 0 + bi. Complex numbers were invented to give solutions to equations such as x^2 + 1 = 0. No real number satisfies this equation, but if we invent a number, i, with the strange property that i^2 = -1, then i (and its negative -i) will satisfy the equation.

Answer 4

a million) There aren't any irrational numbers in this record. D is complicated. relax are genuine, rational numbers. as an social gathering E = 2-sqrt(-9)i = 2 - (3i)i = 2 + 3 = 5 2) a) sqrt(2)^2 = 2 b) (a million+i)(a million-i) = 2 clarification: rational variety might want to be written as a/b irrational variety won't be able to be written as a/b (as an social gathering sqrt(2), pi, e, ...) complicated variety might want to be written as a + bi the position b isn't 0 remember i*i = -a million