## Fractions and Division

One more post before I return to core algebra. We need to look a bit more at division and fractions.

Now, as you’ve seen, something like 8 ÷ 2 indicates division, but another way to show exactly the same thing is $\frac{8}{2}$. In other words, fractions are just another way to show division. Now before I expand on this, let’s review how fractions multiply together.

When two fractions are to be multiplied, the process is very simple. You just multiply the numerators (the numbers above the line) together and the denominators (the numbers below the line) together.

$\frac{1}{2}\hspace{0.33em}\times\hspace{0.33em}\frac{3}{4}\hspace{0.33em}{=}\hspace{0.33em}\frac{{1}\hspace{0.33em}\times\hspace{0.33em}{3}}{{2}\hspace{0.33em}\times\hspace{0.33em}{4}}\hspace{0.33em}{=}\hspace{0.33em}\frac{3}{8}$

Now this can be used to our advantage to simplify fractions. Each of the numbers in the above example are called “factors”. Factors are things that are multiplied together. So if we can show the factors of the parts of a fraction, we can effectively cancel factors that are common between the numerator and the denominator because we can split off the common factors as $\frac{\mathrm{number}}{\mathrm{number}}$ and any number divided by itself is 1 and anything multiplied by 1 is the same anything.

$\frac{8}{2}\hspace{0.33em}{=}\hspace{0.33em}\frac{{2}\hspace{0.33em}\times\hspace{0.33em}{4}}{{2}\hspace{0.33em}\times\hspace{0.33em}{1}}\hspace{0.33em}{=}\hspace{0.33em}\frac{2}{2}\hspace{0.33em}\times\hspace{0.33em}\frac{4}{1}\hspace{0.33em}{=}\hspace{0.33em}{1}\hspace{0.33em}\times\hspace{0.33em}{4}\hspace{0.33em}{=}\hspace{0.33em}{4}$

A shortcut version of this is

$\frac{8}{2}\hspace{0.33em}{=}\hspace{0.33em}\frac{\rlap{/}{2}\hspace{0.33em}\times\hspace{0.33em}{4}}{\rlap{/}{2}\hspace{0.33em}\times\hspace{0.33em}{1}}\hspace{0.33em}{=}\hspace{0.33em}\frac{4}{1}\hspace{0.33em}{=}\hspace{0.33em}{4}$

So note that when you cross out the only factor in the numerator or denominator, a “1” is left, and this “1” can be left out of the result since it does not change the value of the remaining numbers. Also note that this works for known and unknown numbers such as  x which we will see in my next post.

A couple more examples:

$\begin{array}{c} {\frac{16}{4}\hspace{0.33em}{=}\hspace{0.33em}\frac{\rlap{/}{4}\hspace{0.33em}\times\hspace{0.33em}{4}}{\rlap{/}{4}\hspace{0.33em}\times\hspace{0.33em}{1}}\hspace{0.33em}{=}\hspace{0.33em}{4}}\\ {\frac{6}{9}\hspace{0.33em}{=}\hspace{0.33em}\frac{\rlap{/}{3}\hspace{0.33em}\times\hspace{0.33em}{2}}{\rlap{/}{3}\hspace{0.33em}\times\hspace{0.33em}{3}}\hspace{0.33em}{=}\hspace{0.33em}\frac{2}{3}} \end{array}$

## Signs and Multiplication and Division

Before I start this topic, let me demonstrate a shortcut in maths. I can show a multiplication by using the “×” symbol. For example, 8 × 2 = 16. But another way, which is especially useful when dealing with negative numbers is to use brackets with no “×” symbols: (8)(2) = 16, that is (8)(2) means 8 × 2.

So, if two positive numbers are multiplied or divided by one another, you already know that the result is positive.

(8)(2) = 16 (Remember the “+” symbol is assumed to be there for positive numbers)

8 ÷ 2 = 4

Now if the signs of the numbers are different, the result of the arithmetic is negative:

(-8)(2) = (8)(-2) = -16

(8) ÷ (-2) = (-8) ÷ (2) = -4

The reason for this is because multiplication is really just successive adding. For example, (-8)(2) is a shortcut for adding (-8) twice, That is (-8)(2) means (-8) + (-8) which is -16.

Now if both numbers are negative, the result is positive. It’s that double negative effect again. So

(-8)(-2) = 16

(-8) ÷ (-2) = 4

These rules are easy to remember: Like signs are positive, unlike signs are negative.

Tomorrow, let’s talk about fractions and more maths shortcuts.

## 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.

## Algebra: Welcome to the Unknown

So now we know how to create new true equations from other true equations involving known numbers. But as you know, algebra is all about “solving for x” or “finding x”. “x” is the most common letter used in algebra to represent a number that is unknown. Now other letters are also used, and the following discussion works with other letters as well.

Now suppose you had a brother Joe and a sister Tammy who gave you some money. You needed $10 and your siblings came to the rescue. It is now time to pay them back. You remember that Joe gave you$7, but you cannot remember what Tammy gave you. What Tammy gave you is, at the moment, an unknown number. Let’s call that “x”. You know that what Tammy gave you plus what Joe gave you is \$10. The equation (sentence) in algebra that says this same thing is:

x + 7 = 10

Now I know you’re all screaming the answer as this one can be done in your head without formal algebra – but this blog is about algebra, and the techniques here will help you solve equations that cannot be done in your head.

Now this equation is a true one and we can “solve for x” using the principle stated in the last post: you can create a new true equation from a true equation by doing the same arithmetic on both sides of the equation. It doesn’t matter if you have an unknown number in the equation. The principle still applies.

So the objective is to “solve for x”. This means use algebra to create an equation that looks like:

x = known number

So now look at x + 7 = 10. What can we do to get x by itself on the left side of the equation? What about subtracting 7 on both side?

x + 7 – 7 = 10 – 7 and then doing the arithmetic gives x + 0 = 3. But 0 added to anything is the same anything so it can be removed from the equation without changing the value of the left side. So we have

x = 3. We have solved for x. You can now stop screaming.

So the point of this post is that you can manipulate equations, even those with unknown numbers, as long as you do the same thing to both sides of the equation.

Now before I get into more complex equation, I would like to review the number line (the different types of numbers) and some basics around fractions. I will cover the number line tomorrow.

## Algebra: The Beginnings

As mentioned in a previous post, our math abilities are a by-product of our language skills. Indeed, mathematics can be thought of as another language, limited in its subject matter but powerful in developing its own sentences (equations). Maths has its own set of words (symbols), its own dialects (scientists use a different set of symbols and equations than do engineers), and its own syntax and grammar. Algebra is the mathematical version of syntax and grammar, which is why it is such and important subject. It underpins all that is mathematics.

You are already familiar with many of the symbols (words) used in maths:

+ means “plus” or “positive” depending on context

− means “subtract” or “negative” depending on context

× means “multiplied by” or “times”

÷ means “divided by”

= means “equals”

So you already can read math sentences (equations) like 2 + 3 = 5 and you already know that there are rules (syntax) to be followed when writing an equation. For example, 2 3 + 5 = uses the same symbols, but it doesn’t make sense.

Other equations are 3 = 3, 4 = 4, or 2 = 1. Just as in language, maths sentences can be true or false. But let’s start with an obviously true sentence: 3 = 3. Now if I add 1 to both sides of the “=” sign, I get:

3 + 1 = 3 + 1 or 4 = 4, another true equation. Now let’s divide both sides by 2:

4 ÷2 = 4 ÷2 or 2 = 2, Another true equation. I can do this all day, but let’s just do one more:

2 – 2 = 2 – 2 or 0 = 0. In each step, notice that I had to do the same thing on both sides. If there is only one algebra rule you can remember, it’s this one: you can create a new true equation from a true equation by doing the same arithmetic on both sides of the equation. We will see some caveats (cautions) later regarding this, but this is the most important rule in algebra.

Tomorrow, I will introduce the idea of equations with unknown numbers.