An Irrational Post

In my posts to date, you have seen different types of numbers: Positive numbers, Negative numbers, Integers, Fractions, and Prime numbers. Some numbers can fit into more than one category. Today I want to introduce two more types: Rational and Irrational numbers.

It is easier to define what a rational number is then anything that is not rational, is an irrational number.

A rational number is any number that can be put into the form

\[\frac{a}{b}\]

where a and b are integers. So all fractions are rational numbers, but so are all  the integers like 0, -1, 3, -9,999 and so on because they can all be represented as the integer over “1”:

\[\frac{0}{1}{,}\hspace{0.33em}\frac{{-}{1}}{1}{,}\hspace{0.33em}\frac{3}{1}{,}\hspace{0.33em}\frac{{-}{9}{,}{999}}{1}\]

Now decimal numbers can be rational as well. The way to tell is if the decimal part repeats a pattern no matter how short or long the pattern is. That is because all repeating pattern decimals can be put into the a/b form. So the following are all rational numbers. Note that a line over a section of the decimal means that that section is repeated over and over: 

\[{3}{.}{0}\overline{0}{,}\hspace{0.33em}{0}{.}{111}\overline{1}{,}\hspace{0.33em}{6}{.}{25}\overline{25}{,}\hspace{0.33em}{-}{2}{.}{1275}\overline{1275}\]

That is in fact another way to test for whether a number is rational: if the decimal part (even if it is “0”) repeats eventually, it is a rational number. Any number that cannot be expressed as a repeating decimal is irrational. You have been exposed to one famous irrational number: 𝜋. This number has a non-repeating decimal that goes on forever without repeating. This has been proven to be true many times. Other numbers that are irrational are many (but not all) square roots and other roots like cube roots. For example, the following are also irrational numbers:

\[\sqrt{2}{,}\hspace{0.33em}\sqrt{3}{,}\hspace{0.33em}\sqrt[3]{7}{,}\hspace{0.33em}\sqrt[4]{2}\]

In fact, using the square root symbols or other symbols like 𝜋 are the only way we can express the number exactly. Even modern calculators with many digits of accuracy can only represent an irrational number to a limited number of decimal places. So when using an irrational number in a calculation, you use the number of decimal places required for the accuracy required in the final answer.

The set of rational numbers and irrational numbers comprise all the numbers on the number line. This complete set of numbers on the number line are called real numbers.

There are many other types of numbers. For example transcendental numbers and things called imaginary numbers. So we have real numbers and imaginary numbers. These can be combined to form another type of number, complex numbers. These are used in many branches of engineering and have physical meaning, even though imaginary numbers are used. Imaginary numbers are not on the real number line, they are plotted using another different number line. Maybe I will cover these later when I have a real, rational moment.

Negative Numbers and Signs

There are several topics we need to cover before we can continue our discussion on algebra so that we can solve more interesting equations. The first topic is Negative Numbers and introducing another interpretation of the “+” and “-” symbols. First, negative numbers.

All real numbers can be located on the number line below. By “real” I mean numbers you are familiar with as opposed to numbers called “complex” which we will cover some time in the far future.

The numbers to the right of 0 are called positive numbers. They can technically have the “+” symbol in front of them, but when a number is positive, the “+” sign can be removed and the number is assumed to be positive if there is no “+” in front. The numbers to the left of 0 are called negative numbers. The “+” and “-” symbols are called signs and this introduces another way to view these: one way that you are familiar with is as the arithmetic operation (+ means add, – means subtract), but the other interpretation is as the sign of a number. As you will see, these interpretations can be interchanged quite freely as you work with equations.

Numbers at the tick marks are the integers. Numbers between the tick marks are fractions or other kind of number called irrational. But this post is mainly about negative numbers.

Negative numbers can represent many things you are familiar with: money you owe instead of have, distance to the west of a city instead of distance to the right, deceleration rate instead of acceleration, distance down instead of up, etc. So how do you work with negative numbers?

Now I will use brackets (parenthesis if you’re from the States) to separate numbers with their sign from the arithmetic symbols. Let’s look at the following examples:

(+7) + (+3) = 7 + 3 = 10 because the + sign in front of the numbers can be assumed and removed leaving only the + (plus) arithmetic symbol. So on the number line, you can represent this by starting at 7, then moving to the right 3 tick marks where you arrive at 10.

(+7) – (+3) = 7 – 3 = 4 for the same reason but this time the -(subtract) arithmetic operation remains.So on the number line, you can represent this by starting at 7, then moving to the left 3 tick marks where you arrive at 4. Now this is where it gets interesting.

(+7) + (-3) = 7 – 3 = 4. Adding a negative number is the same as subtracting a positive number. This is where I said the sign of a number and the arithmetic operation it represents can be freely interchanged and swapped. So you see from this and the last example, if you see “+” followed by “-” or vice versa, you can replace both with just the “-” symbol. What about

(+7) – (-3) = 7 + 3 = 10. Think of subtracting a negative number as having the same effect as a double negative in English (I will not spend no money technically is saying that you will spend money). So from this example and the first one, if you see “+” followed by “+” or “-” followed by “-“, you can replace them both by a “+” symbol.

(-7) + (+3) = -7 + 3 = -4. Start at -7 on the number line and move 3 ticks to the right, landing on -4.

(-7) – (-3) = -7 + 3 = -4. Same as above.

(-7) – (+3) = -7 – 3 = -10. Start at -7 on the number line and move to the left 3 ticks to land on -10.

Tomorrow, I will cover multiplying numbers of varying signs.