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A Parent’s Guide to Algebra’s Basic Concepts – From Counting Numbers To Imaginary Numbers
In this article, we’ll look at the different groups of numbers used in algebra: their names and what is and isn’t included in each group. We will review why these different groups exist. We will also pay a lot of attention to real and imaginary numbers because these two concepts cause a lot of difficulties for Algebra students.
To better understand the different groups, it’s helpful to pretend you don’t know anything about numbers and then think about when you or your children learned numbers. The first thing children learn related to mathematics is to count: 1, 2, 3, 4, 5,… This group is also the first group we name. We use this notation, , which reads “the set of”. The set of counting numbers is written as 1, 2, 3, 4, 5,… and read as “the set of 1, 2, 3, 4, 5, etc. This set also it’s called the set of natural numbers. The phrase I say to myself to remember this name is “it’s natural to start counting with one.” This set goes on forever, but it doesn’t include zero, fractions, decimals, percents, radicals, or negative numbers, just what it takes to count.
When children learn to write their numbers, it quickly becomes necessary to enter zeros so that they can write 10, 20, 30, etc. When we add zero to the natural numbers, this new set has a new name: the whole numbers. . The set of integers is written as 0, 1, 2, 3, 4, 5,…. The only difference between natural numbers and integers is the inclusion of zero. Remember: it’s natural to start counting on one. This set also lasts forever, but does not include fractions, decimals, percents, radicals, or negatives.
The reason there are so many different groups of numbers is that the previous groups do not meet our needs, such as having to include zero. With the set of whole numbers we can do any addition, that is, if we take two whole numbers and add them, the answer exists in the set (it is a whole number). The word we use for this concept is closure. We say that the set of integers is summationally (or subsummally) closed. Unfortunately, this set of numbers does NOT close above (or below) the rest. We can certainly do some subtraction problems like 10 – 7 since the answer 3 is in the set of numbers. But if we turn this problem around, 7 – 10, there is no answer in this set of numbers. In elementary school, kids would answer “you can’t do that” or “you can’t get a big number out of a small number.”
Every time we encounter a “you can’t do this” situation, a new symbol is created to solve the problem. We need an answer to problem 7 – 10, so negatives were created to provide that answer. 7 – 10 = -3. By adding the negatives to the numbers that exist at this point we get this set: …, -3, -2, -1, 0, 1, 2, 3,… which is called whole numbers. Whole numbers do not include fractions, decimals, percentages, or radicals. Integers are closed over addition, subtraction, and multiplication. You can pick any two integers and multiply them together and the answer exists in the set of integers. The difficulty comes with division. Again, some division problems are fine: 10 divided by 2 is 5. But turn it back to 2 divided by 10 and we’re back to “you can’t do that.” For your level of knowledge, there is no answer.
So the math gets harder now because fractions are introduced as solutions for these division problems, but fractions can also be expressed as decimals and as percentages. Two divided by ten can be expressed as a fifth, 0.2 or 20%. When we add fractions and their decimal and percentage equivalents to whole numbers, we now have a set of numbers called rational numbers, but we don’t have a set notation for it. A word of caution here: the set of rationals does not include all decimals. Only includes repeating and trailing (terminating) decimals. These decimals CAN be written as the ratio of two whole numbers. For example, 0.333… = 1/3.
Until now, each of these sets of numbers was created by adding a new type of number to one that already existed. If we had started by representing the counting numbers as a small circle, each new group would be a larger circle surrounding the smaller one.
But, now, we have to deal with non-repeating decimals, which are not terminating. For example: 0.01001000100001000001… These decimals cannot be written as a ratio of two whole numbers (a fraction). Here we also find numbers like pi, or the square root of two. As decimals, these numbers never repeat, but they also never end. These odd numbers are called irrational and do not belong to the other group. They exist in their own circle.
If we draw a large circle to include the rationals and irrationals, we call it the set of real numbers. So are all the numbers real? It certainly seems so. But surprisingly, we’re going to run into another one of those “you can’t do that” situations.
When we get to solving quadratic equations, we occasionally run into things like the square root of -1. At this point, high schoolers will say “you can’t do that”. We know that the square root of 4 is 2 because 2 times 2 is four. So the square root of -1 seems impossible since it is not an existing number that can be multiplied by itself to produce -1. However, again, a new symbol is created to solve the problem. The new symbol is ii means the square root of -1. The square root of -1 is i. The square root of -4 is 2i. The square root of -9 is 3i, and so on. These numbers are called imaginary numbers, although that was a bad choice of word. Numbers with an i are as legitimate as fractions. We’ve just forgotten when fractions seemed “fun”.
These imaginary numbers exist in a circle by themselves, but if we make a giant circle that includes the real and imaginary numbers, we call it a complex number system. And that’s all folks! This makes up the entire number system your child will be working with in high school.
As for why there are so many groups of numbers, we have answered that in terms of new symbols needed, but we also use the group that makes sense for our situation. If we’re designing a new plane and want to know how many seats it will hold, we don’t need negatives or fractions. You won’t need -72.8 seats. We only need natural numbers.
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