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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 see why these different groups exist. We will also pay special attention to real and imaginary numbers because these two concepts cause so much difficulty for students of algebra.
To better understand the different groups, it helps to pretend you don’t know anything about numbers and then think back to when you or your children first learned numbers. The first thing that children learn in relation to mathematics is to know how to count: 1, 2, 3, 4, 5,… This group is also the first group to which we give a name. We use this notation, , which reads “the set of”. The set of counted numbers is written 1, 2, 3, 4, 5,… and reads as “the set of 1, 2, 3, 4, 5, etc.” This set is also called the set of natural numbers. The sentence I tell myself to remember this name is “it is natural to start counting with one”. This set lasts forever, but it doesn’t include zeros, fractions, decimals, percents, radicals, or negative numbers – just what’s needed for counting.
When children learn to write their numbers, it quickly becomes necessary to introduce them to zero so that they can write 10, 20, 30, etc. When zero is added to the natural numbers, this new set has a new name: the whole numbers. The set of integers is written 0, 1, 2, 3, 4, 5,…. The only difference between natural numbers and whole numbers is the inclusion of zero. Remember: it is natural to start counting with one. This set also lasts indefinitely, but it does not include fractions, decimals, percentages, radicals, or negatives.
The reason there are so many groups of different numbers is that the previous group(s) does not meet our needs, nor does the need to include zero. With the set of integers, we can do any addition, which means that if we take two integers and add them together, the answer exists in the set (is an integer). The word we use for this concept is closure. We say that the set of integers is closed on addition (or under addition). Unfortunately, this set of numbers is NOT closed on (or under) subtraction. We can definitely do subtraction problems like 10 – 7 since the answer 3 is in the set of numbers. But, if we reverse this problem, 7 – 10, no answer exists in this set of numbers. In elementary school, children would say “you can’t do that” or “you can’t get a lot out of a small number”.
Every time we encounter a “you can’t do that” situation, a new symbol is created to solve the problem. We need an answer to problem 7 – 10, so negatives have been created to provide that answer. 7 – 10 = -3. Adding the negatives to the numbers that exist at this point gives us this set: …, -3, -2, -1, 0, 1, 2, 3,… which is called the integers. Whole numbers do not include fractions, decimals, percentages, or radicals. Integers are closed on addition, subtraction and multiplication. You can choose two integers and multiply them together and the answer exists in the set of integers. The difficulty comes from division. Again, some division problems are correct: 10 divided by 2 is 5. But, turn that into 2 divided by 10 and we have a “you can’t do that” again. For their level of knowledge, no answer exists.
So math is getting harder now because fractions are introduced as solutions to these division problems, but fractions can also be expressed in decimals and 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 warning here: the set of rationals does not include all decimals. It only understands repeating and trailing (terminal) decimals. These decimal numbers CAN BE written as the ratio of two whole numbers. For example, 0.333… = 1/3.
So far, each of these sets of numbers has been created by adding a new type of number to what already exists. If we had started by representing the numbers to be counted as a small circle, each new group would simply be a larger circle surrounding the smaller one.
But, now we have to deal with non-repeating, non-terminating decimals. For example: 0.01001000100001000001… These decimals cannot be written as the ratio of two integers (a fraction). This is also where we find numbers like pi, or the square root of two. As decimals, these numbers never repeat but also never end. These strange numbers are called irrational and do not belong to the other group. They exist in their own circle.
If we draw a great circle to include the rationals and irrationals, we call it the set of real numbers. So, are all numbers real? It certainly looks like it. 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 sometimes come across things like the square root of -1. At this point, high school students 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. Yet again, a new symbol is created to solve the problem. The new symbol is i and 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 is a poor choice of word. Numbers with an i are just as legitimate as fractions. We just forgot when fractions seemed “funny” to us.
These imaginary numbers exist in a circle by themselves, but if we make a giant circle that includes the real numbers and the imaginary ones, we call it the complex number system. And that’s all ! This makes up the entire number system that your child will work with in high school.
As to why there are so many groups of numbers, we answered in terms of new symbols needed, but we also use the group that suits our situation. If we’re designing a new plane and want to know how many seats it will have, we don’t need negatives or fractions. You won’t need -72.8 seats. We just need natural numbers.
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