In mathematics, a special number is called an algebraic integer. These numbers can be represented by a finite combination of integers and the operations of addition, subtraction, multiplication, and division.
Algebraic integers can be represented by the symbols ± (plus or minus), 1, 2, 3, 4, 5, 6 (1 through 6). You can add 1 to any of these numbers or subtract 1 from any of these numbers and you will get another algebraic integer.
For example: ±1 + 2 = 3; -1 + 2 = -1 + 3; 1 – 2 = -1; 1 – 2 – 3 = -1; 4 ÷ 2 = 2; 4 ÷ 3 = 1; 5 ÷ 6 = -1.
There are infinite algebraic integers, but there are some very special ones that have specific names. These special ones are called elementary algebraic integers and they are named: i (the number whose successor is 0), i² (the square of i), j (the number whose successor is 0), j² (the square of j), k (the number whose successor is 0), k² (the square of k), ldots|ldots|ldots|ldots|ldots|ldots||lldots|lldotsof order n where n is any natural number greater than or equal to one.(Dr. James A Kanzelmann).\r
The sum of any combination (n)of these special numbers nis always another one of these special numbers: i^n + j^n + k^n + ldots|lldotsof order n.
- i^0 + j^0 + k^0 + ldotsall order 0= ijk all order 0= i(0+0)= i
- i^(-1)+(j^(-1))+(k^(-1))+ ldotsof odd orders= (-i)j(-k)+(-i) all odd orders= (-i)(all odd orders)= all even orders
This article will discuss more about these very special algebraic integers.
What are the fractions?
A fraction is a ratio of two numbers, where the smaller number is called the denominator and the larger number is called the numerator.
There are several types of fractions, including reduced fractions, mixed fractions, and improper fractions. A reduced fraction is one where the denominator is equal to 1.
For example, one-third (1/3) is a reduced fraction because 1/3 = 1/1 = 1/3. A mixed fraction is one that has a denominator that isn’t equal to 1. For example, two-thirds (2/3) isn’t equal to 2/1 = 2, so it’s a mixed fraction. An improper fraction is one where the numerator isn’t equal to the denominator. For example, one-fourth (1/4) isn’t equal to 4/4 = 1, so it’s an improper fraction.
The fraction 1/6
The first fraction most people learn is 1/6. Most people learn this in elementary school as part of the math curriculum. This fraction represents one sixth, or one part out of six parts.
One way to think about 1/6 is five divided by six, or five parts out of six parts. One part is left over, which is how 1/6 is conceptualized.
1/6 can be visualized through objects as well. Imagine a box with six compartments inside it, each containing a different color of paint. One of the compartments contains one sixth of the total amount of paint in the box, or one color.
Another way to think about 1/6 is to imagine a length of wire that is five times as long as another length of wire. There are only two possible fractions that could represent this: 5/1 or 1/5. The first number in both cases is the same number of times as long as the second number, but the second number is half the size.
The fraction 2/3
A fraction, or ratio, is when you compare two numbers with each other and represent them as a single number. For example, 2/3 is the ratio of two parts to three parts, so 2 parts are compared to 3 parts and represented as a single number.
Fractions can be ranked from lowest to highest based on their denominators (bottom number) – lowest denominator means lowest ranking. In the case of 1/2, it has a denominator of 2, which is the highest ranking.
The fraction 1/6 has a ranking of lowest since it has a denominator of 6, which is lower than 1/2’s denominator of 2. The higher the number in the denominator, the lower the fraction’s rank.
1/6 + 1/4 = 1/3
The fractions can be combined into a new fraction using what is called an inverse operation. In this case, division is used to combine the fractions.
This process can be done in two ways: reconverting the fractions into whole numbers and then adding them or performing the addition operation first and then converting back to fractions.
The sum of these fractions is 1/3 + 2/6 + 4/12 = 6/18 = 1/2.
This demonstration shows that even though these three different-looking fractions appear to be different ratios between one-sixth and one-fourth and one-half, they are actually equal!
Watch this video by <a href=”https://www .youtube .
The fraction 1/4
The fraction 1/4 refers to the amount that is one quarter of a whole. One way to think about 1/4 is how many parts of a pound of something there are.
For example, there are four parts of a pound in an apple, so an apple is 1/4 of a pound. Like with apples, anything can be divided into four parts and still be the same thing.
Another use for the fraction 1/4 is in music. When musicians are writing music, they will write down notes on paper in terms of fractions of a beat. One beat is equal to one quarter of a full song rhythm.
Like with apples, any object can be divided into four parts and still be the same thing. This applies to anything that has dimensions- it can be divided into four identical pieces that fit the original whole.
The fraction 1/4 can also refer to twenty-fourths, or half of sixty-fourths.
Putting it all together
Now that you can solve the 1/6, 2/3, and 1/4 equations individually, you can put them all together to solve for any of them in relation to each other.
If you have the solution to one of these ratios, you can use this method to find the solutions for the others. For example, if you know the total number of drinks and how many of each type of drink there are, you can find out how many of each type of drink are in a serving.
This is especially useful when serving drinks at a party. You can keep track of how many total drinks there are and how many drinks of each type there are to make sure everyone gets what they want. Or, you can keep track of how many drinks are left so that you can refresh the supplies.
As always, be careful when drinking alcoholic beverages! Make sure to know how many drinks you’ve had so that you do not over-consume.