The imaginary numbers are needed to fill in the last gap in exponentiation. The real numbers do not allow some negative numbers to be raised to fractional powers. For example they do not allow (-1)(1/2) = √(-1).
To get around this problem we can just make up an answer. We'll call this answer i. It can't be a real number, so it is outside the real numbers.
Using i allows us to find roots for all negative numbers. For example
√(-2) = √(2 × -1) = √2 × √(-1)= (√2)i = 1.414i
Both negative and positive numbers have two square roots. Just as both 42 and (-4)2 are 16, both (4i)2 and (-4i)2 are -16.
When we look at cube roots, thing get more complicated. Now every number has three cube roots, not just one, like it did with the real numbers. So not only does 33 equal 27, but so does (-3/2 + (3√3/2)i)3 and (-3/2 + (3√3/2)i)3
For more on this see the complex numbers.
Saturday, April 18, 2009
Transcendental Numbers
The transcendental numbers are a part of the irrational numbers. Transcendental numbers are numbers that do not solve any polynomial equation that has rational coefficients. π and e are transcendental numbers so they don't solve any equations like:
4x - 7 =0
5x2 - 3x + 4 = 0
(4/5)x7 - 1.24x3 - 8 = 0
Strangely, it is not known whether π + e or π × e is transcendental, though at least one of them must be.
4x - 7 =0
5x2 - 3x + 4 = 0
(4/5)x7 - 1.24x3 - 8 = 0
Strangely, it is not known whether π + e or π × e is transcendental, though at least one of them must be.
Real Numbers
If we combine the rational numbers with the irrational numbers we get the real numbers. In the real numbers we find positive numbers, negative numbers, integers, fractions, square roots, cube roots, fractional roots, π, e, zero...
Like the rational numbers, we can add, subtract, multiply and divide any two numbers (except dividing by zero). But with the real numbers we can also raise any positive number to the power of any other number. We can now do fractional powers and get a real answer because the irrationals are part of the reals. However, we cannot raise negative numbers to every fraction. (-1)(1/2) = √(-1) is not allowed with the real numbers, although (-1)(1/3) is allowed because it equals -1.
The real numbers are given the symbol: ℝ
Like the rational numbers, we can add, subtract, multiply and divide any two numbers (except dividing by zero). But with the real numbers we can also raise any positive number to the power of any other number. We can now do fractional powers and get a real answer because the irrationals are part of the reals. However, we cannot raise negative numbers to every fraction. (-1)(1/2) = √(-1) is not allowed with the real numbers, although (-1)(1/3) is allowed because it equals -1.
The real numbers are given the symbol: ℝ
Friday, April 17, 2009
Irrational Numbers
Just as rational numbers were made up by ratios, the irrational numbers cannot be expressed as a ratio. For example the square root of 2 does not equal one integer divided by another. You can get closer and closer with fractions, but you will never exactly equal √2.
This also means that the decimal expansion of an irrational number goes on forever.
Square roots, cube roots and other roots can all be irrational numbers. Transcendental numbers like pi and e are also irrational.
Raising an integer to a fraction will give an irrational number (unless it comes out exactly and gives an integer.) For example:
21/2 = √2 = 1.41421356...
53/2 = (√5)3 = 11.180339877...
This also means that the decimal expansion of an irrational number goes on forever.
Square roots, cube roots and other roots can all be irrational numbers. Transcendental numbers like pi and e are also irrational.
Raising an integer to a fraction will give an irrational number (unless it comes out exactly and gives an integer.) For example:
21/2 = √2 = 1.41421356...
53/2 = (√5)3 = 11.180339877...
Rational Numbers
We grouped the negative numbers with the natural numbers to get the integers. Now we increase our set of numbers further to get the rational numbers.
The rational numbers include all of the fractions made by dividing one integer by another, except that you can't divide by zero. So positive and negative fractions, fractions smaller than one, fractions larger than one, and all the integers (you can have 1 on the bottom your fraction) make up the rational numbers.
You can remember that the RATIOnals are made up from RATIOs.
Rational numbers do not have to be written as a fraction, but can be in decimal form - they are still rational.
Some rational numbers:
1/2, 6, -4/5, 10/3, 54 3/4, 1.75, -0.33333
With the inclusion of fractions in the rational numbers we are better off for division that we were with the integers. We can now divide any two numbers and get another rational number, with only one exception. We cannot divide by zero.
Like the integers, we can add, subtract and multiply without restriction. But we still have to be careful about exponentiation.
In most cases we can only raise rationals to the power of an integer. 32 is OK, and, unlike the integers, we can now do 3-2, but we cannot do 31/2 as that is outside of the rational numbers. We can only raise numbers to a fraction when the answer lies in the rational numbers. For example 25(1/2) = 5 or (8/27)(1/3) = 2/3.
The rational numbers are given the symbol: ℚ
The rational numbers include all of the fractions made by dividing one integer by another, except that you can't divide by zero. So positive and negative fractions, fractions smaller than one, fractions larger than one, and all the integers (you can have 1 on the bottom your fraction) make up the rational numbers.
You can remember that the RATIOnals are made up from RATIOs.
Rational numbers do not have to be written as a fraction, but can be in decimal form - they are still rational.
Some rational numbers:
1/2, 6, -4/5, 10/3, 54 3/4, 1.75, -0.33333
With the inclusion of fractions in the rational numbers we are better off for division that we were with the integers. We can now divide any two numbers and get another rational number, with only one exception. We cannot divide by zero.
Like the integers, we can add, subtract and multiply without restriction. But we still have to be careful about exponentiation.
In most cases we can only raise rationals to the power of an integer. 32 is OK, and, unlike the integers, we can now do 3-2, but we cannot do 31/2 as that is outside of the rational numbers. We can only raise numbers to a fraction when the answer lies in the rational numbers. For example 25(1/2) = 5 or (8/27)(1/3) = 2/3.
The rational numbers are given the symbol: ℚ
Thursday, April 16, 2009
Integers
"Dad, can you buy me those butterfly wings?"
"They are seven dollars. How about you use some of your own money?"
"But I don't have my money with me."
"If I buy the wings, you can owe me the money and give it to me when we get back home."
Shortly afterwards my daughter has the butterfly wings and -$7 in her pocket and is thus introduced to negative numbers, and spending on credit.
The first extension we can make to the natural numbers is to go backwards as well as forwards to get the integers. The natural numbers start and zero and count 1, 2, 3 and so on. The integers allow us to count backwards as well. So -1, -2, -3... are integers, but are not natural numbers.
Now we can subtract any two integers and get another integer. We are not restricted like we were for the natural numbers. We can take a larger number from a smaller one and get an answer: a negative number.
And like with the natural numbers we can add and multiply any integer and get another integer. We are still restricted with division, though. 12 divided by 3 is fine, but 13 divided by 3 is a problem for the integers, just as it was for the natural numbers.
Strangely, exponentiation (raising a number to the power of something), which was OK for all natural numbers, is a problem for the integers. If we raise an integer to the power of a negative number we do not get an integer back. What is 3-1? or 2-2? Not an integer, that's for sure. So we can only use zero and positive powers in the land of the integers.
The Integers are given the symbol: ℤ
"They are seven dollars. How about you use some of your own money?"
"But I don't have my money with me."
"If I buy the wings, you can owe me the money and give it to me when we get back home."
Shortly afterwards my daughter has the butterfly wings and -$7 in her pocket and is thus introduced to negative numbers, and spending on credit.
The first extension we can make to the natural numbers is to go backwards as well as forwards to get the integers. The natural numbers start and zero and count 1, 2, 3 and so on. The integers allow us to count backwards as well. So -1, -2, -3... are integers, but are not natural numbers.
Now we can subtract any two integers and get another integer. We are not restricted like we were for the natural numbers. We can take a larger number from a smaller one and get an answer: a negative number.
And like with the natural numbers we can add and multiply any integer and get another integer. We are still restricted with division, though. 12 divided by 3 is fine, but 13 divided by 3 is a problem for the integers, just as it was for the natural numbers.
Strangely, exponentiation (raising a number to the power of something), which was OK for all natural numbers, is a problem for the integers. If we raise an integer to the power of a negative number we do not get an integer back. What is 3-1? or 2-2? Not an integer, that's for sure. So we can only use zero and positive powers in the land of the integers.
The Integers are given the symbol: ℤ
Wednesday, April 15, 2009
Natural Numbers
"What do you get when you take away 7 from 3?", I ask my daughter. "You can't do that. That's silly", she replies. "What about if I divide 5 in half?" "You get 2 in one group and 3 in the other."
Such in the world of a child, and such is the world of the natural numbers. In this world you start at zero and count 1, 2, 3 and so on. That's all the numbers you've got.
There's no problem with addition or multiplication or raising numbers to the power of something. The problems come with subtraction and division. When you try and subtract a larger number from a smaller one the answer is negative, which is outside the natural numbers. If you try to divide up a number that won't go evenly the answer is a fraction, once again outside the natural numbers.
What's 13 divided by 3? "4, and 1 remainder" in the world of the natural numbers.
The natural numbers are given the symbol: ℕ
Such in the world of a child, and such is the world of the natural numbers. In this world you start at zero and count 1, 2, 3 and so on. That's all the numbers you've got.
There's no problem with addition or multiplication or raising numbers to the power of something. The problems come with subtraction and division. When you try and subtract a larger number from a smaller one the answer is negative, which is outside the natural numbers. If you try to divide up a number that won't go evenly the answer is a fraction, once again outside the natural numbers.
What's 13 divided by 3? "4, and 1 remainder" in the world of the natural numbers.
The natural numbers are given the symbol: ℕ
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