The Laws of Algebra

This week we’re exploring the laws of algebra with a guest post from my dad, Donald McGill. He majored in math back in the day and was more than happy to feed my own interest in math – and eventually physics – as I grew up. Since we’re diving into the kinematics equations next, I thought this would be a good time for an interlude about how algebra works. Enjoy!

The development of mathematics is one of the crowning achievements of the human mind, perhaps only second in importance to the development of language. I say “development” because it has been a slow and hard slog to get to where we are today. Along the way humans have developed lots of different number systems; for example, the Mayans developed a system of dots and bars representing ones, fives, and twenties, and the Romans used a number system of no base using letters from their alphabet to represent values, such as “X” for ten, “V” for five, and “I” for one and would represent a number by listing its component parts—for instance, 28 would be “XXVIII”.

Despite their various sophisticated number systems, it is interesting that none of the ancient peoples recognized zero as a number. The Babylonians came close, having invented a placeholder mark to denote an empty wire when recording calculations from an abacus onto clay tablets. However, that’s all “zero” was – a placeholder for the concept of “nothing.” In general, the older civilizations did just fine without conceptualizing zero as a number, so it wasn’t until the 7th Century in India that zero was finally recognized as a quantity that could be added, subtracted, or multiplied like any other number*.

It was a big deal when zero was used to anchor the counting numbers on the modern number line. Zero was the marker that allowed negative numbers to be invented. For that matter, math was so wrapped up in the idea of counting and measuring things in the real world that it was difficult for even the best mathematicians to accept the idea of negative numbers as being “real” even as late as the 18th and 19th centuries, referring to them as absurd or fictitious when forced to confront them.

Algebra is concerned with studying the properties of numbers and their manipulations according to rules of logic and basic laws which have been discerned during the passage of time as mathematical thought became increasingly sophisticated and separated from the physical world. But at its heart, math started with counting, and counting implies the idea of addition. Multiplication is really just repetitive addition but has its own quirk.

Before looking at the laws, let’s look at counting and the number line.

In some sense it can be argued that the very first known number line is what is called a tally stick, which is a stick or similar object which has notches carved in it to track the number of…something. One of the earliest was uncovered in 1937 in Moravia; it was a wolf bone dated back to about 30,000 BCE which has 55 notches in it. What was this prehistoric guy counting? The number of beasts he brought down? The number of days since it rained? History will never know. A tally stick isn’t quite a number line as it really just represents a one-to-one mapping of things to notches, and you don’t need to know specifically about numbers for that.

By convention a number line is a line which has equally-spaced marks on it, each assigned a sequential integer number in increasing order to the right. With only positive numbers considered to be “real” we can see that adding two positive numbers together is performed by starting at zero, counting forward the amount of the first number, then counting the amount of the second number which puts you at the sum of the two numbers. Subtraction is a separate operation from addition and doesn’t have the same properties as we shall see. However, the formalization and acceptance of negative numbers allowed addition to perform subtraction’s function.

A few more quick things. The addition sign “+” and multiplication sign “*” can be thought of as “verbs” in a math sentence, they indicate actions. On the other hand, the equal sign “=” is not an action word, it indicates a static condition: the math expression on the left side is equal to (or “is the same as”) the math expression on the right side. The two expressions have different forms, but they represent the same mathematical idea. This also means that, to preserve equality, whatever you do mathematically to one side must also be done to the (entire) other side. Also, parentheses are used to group numbers, variables, and math operators together to be treated as a single mathematical idea. For example, that means that (2 + 3 ) can be substituted in wherever the mathematical idea of 5 is used.

Invisible numbers

Mathematicians and scientists have adopted notational conventions to reduce clearly unnecessary symbols when there is no chance of misinterpretation. But you might see a “1” pop up where there wasn’t one before. The invisible numbers you don’t see are usually 0,1, and -1 (and you will shortly see why). Also, multiplication is usually indicated by simply juxtaposing two variables, or a constant and variable, directly next to each other without the multiplication symbol “*”. Also, constants, if present, precede any variables in each term of an expression.

And finally, a reminder about the order of operations used in algebra. Anything in parentheses gets evaluated first. Then exponents get applied and simplified. Then multiplications and divisions are executed in order left to right. Then additions and subtractions are performed in order left to right.

The Commutative Law of Addition

For all real numbers .

This law states that, when adding two numbers, the order of addition doesn’t matter; the numbers can “commute” back and forth across the addition sign. For example, . Note that subtraction is not commutative, that is, while . However, the invention of negative numbers took care of this issue. Hence, and .

The Associative Law of Addition

For all real numbers .

This law states that, in a list of numbers to be added or summed, the order of pairing together the intermediate sums doesn’t matter. Note that subtraction is not associative either. For example, while .

The Commutative Law of Multiplication

For all real numbers .

This law states that, when multiplying two numbers, the order doesn’t matter. Suppose you want to multiply the number “5” three times. You could line up 5 pebbles three times like this:

o o o o o
o o o o o
o o o o o

We see that it is the same number of 15 pebbles whether you look at it as 3 rows of 5 pebbles or 5 columns of 3 pebbles. We also note that division is not commutative. For example, 2 divided by 3 is not the same as 3 divided by 2.

The Associative Law of Multiplication

For all real numbers .

This law states that, in a list of numbers to be multiplied, the order of pairing together the intermediate products doesn’t matter.

The Additive Identity Law

For all real numbers .

This law states that there exists a number, the Additive Identity, which, when added to any real number , leaves the value of unchanged. The Additive Identity is, of course, the number Zero.

The Additive Inverse Law

For all real numbers .

This law states that every number has another number, its Additive Inverse, with which it can be combined under addition that sums to the Additive Identity. Another way of stating this is that any number can be transformed into zero by adding to it its additive inverse. Any number’s additive inverse is simply the negative of that number which is obtained by multiplying the number by -1. This law is one of the most useful and most used laws. Suppose you have an equation with two or more terms, one of which is an unknown that you need to isolate and identify, such as , that is, you want all by itself on one side and then whatever it is equal to will be on the other. So first apply the Additive Inverse Law, then the Associative Law, then the Identity Law:

The Multiplicative Identity Law

For all real numbers .

This law states that there exists a number, the Multiplicative Identity, which, when multiplied with any real number , leaves the value of unchanged. The Multiplicative Identity is, of course, the number One. This also is extremely useful, the key insight being that any number divided by itself is the number One, so this law can be used to change the way any number looks without changing its value, even within terms. For example, suppose you have 3/5 of a loaf and another 2/3 of a loaf, exactly how much bread do you have in total? You don’t want to change the value of each loaf portion but you want to change the way each fraction looks to make the computation easier. This entails multiplying each fraction by the number One, but One formed using the denominator of the other fraction. Remember, fraction multiplication is just multiplying across the numerators to get the product numerator, and multiplying across the denominators to get the product denominator. So:

The Multiplicative Inverse Law

For all real numbers , and not zero: .

Since division by zero is undefined, meaning that mathematicians don’t know what it means, zero is excluded from the list of valid values. This law is stating that you can transform any number to the number One by multiplying by its Multiplicative Inverse. This is handy when you need to isolate the unknown variable because it is being expressed as a multiple or fraction of other numbers or variables. For instance:

The Distributive Law (1)

For all real numbers . Note that here I indicate multiplication by juxtaposing variables instead of using “*”.

What this law is really saying is, if you have two numbers and , and can be represented as the sum of two parts and , then it doesn’t matter whether you multiply by number of times or you multiply each of the parts that make up by number of times and add them up, the answer is the same. For example, suppose and , is 3 and is 4, then:

The Distributive Law (2)

For all real numbers : .

This law really just states that if both numbers to be multiplied are each represented by a sum of their parts, then each part of the first number must multiply each part of the second number and these partial products then sum to the final answer. The Distributive Law could probably be renamed the Multiplication By Parts Law. It just extends the first distributive law for clarity using the same logic. And it goes without saying that this is not limited to an expression of just two terms, this applies in the same manner to an expression of an arbitrary number of terms; just make sure that each part of the first expression is multiplied with each part of the second.

The Law of Exponents

In this section I will discuss numbers written in what is called exponential form. As above, for brevity as well as clarity, I may indicate multiplication by juxtaposing variables instead of using “*”.

A number in exponential form has two parts, a base and an exponent, and represents a repetitive multiplying of (exponent number of) the base number. The base is said to be raised to the exponent number power.

Some things to mention about exponential form. By convention, exponents in a term should be shown as positive numbers. But an exponent in the denominator is there because it is actually a negative exponent. That is, . Also, remember the sign of the number is attached to the base, not the exponent. The exponent being positive or negative only determines whether it belongs in the numerator or denominator.

1. In simplest form an exponential number looks like this, a base and an exponent : . For example, .
2. Similarly, the base may be a product of two or more factors and the exponent is “distributed” to each of them: . For example, . Note that both the Commutative Law and Associative Law were used to manipulate the equation. This required a lot of intermediate steps not shown.
3. If two numbers being multiplied together have the same base, then the product is the base number raised to the sum of the two exponents; to wit, . For example, .
4. If a fraction is raised to a power, the exponent can be “distributed” to both the numerator and denominator. This is because of fraction multiplication, explained above. That is,
   .
5. If a base is in both the numerator and denominator, the fraction can be simplified by netting out that base’s exponents, that is, subtracting the denominator’s exponent from the numerator’s exponent. This is due to the fact that the fraction can be factored, with the (invisible!) number “1” appearing as needed to fill in fractional parts. So: .

Let’s say and . Then:

6. If a base has an exponent, and that exponent also has an exponent on it, then the simplified exponent for is the product of the two exponents. So: .

This makes sense because the exponent causes to be multiplied times, so the resulting exponent on is .

Conclusion

If you’ve made it this far you are undoubtedly saying, “This is really tedious! It would take forever to solve a problem.” And you’d be right. People who use math a lot rapidly become adept at combining steps and taking shortcuts, as I alluded to in some of the above examples. If a computer was programmed to solve algebra equations, all of these laws would be implemented and executed in excruciating detail. Which means, following these laws without making mistakes guarantees that you will get the correct answer.

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*For a quick read on the history and importance of zero as a number, check out the BBC Future article, “We couldn’t live without ‘zero’ – but we once had to” by Hannah Fry. For a longer read, check out Zero: The Biography of a Dangerous Idea, by Charles Siefe.