Real Numbers, Exponents, and Radicals

This is Unit One of College Algebra: real numbers, exponents, and radicals. The unit runs long, so it comes in two parts, and this is the first. The unit tells one story, and it is worth having in advance. Every new kind of number and every new piece of notation gets introduced for the same reason: so that rules already working keep working in a wider setting. That is why a zero exponent equals one, and why a negative exponent means a reciprocal. Those look like arbitrary decisions until you see what they are protecting — and once you do, they stop needing to be memorised. Part One covers the number sets, the order of operations, the arithmetic properties, and the exponent rules. Part Two takes the same story into roots and rational exponents. A word about the errors. Six entries here are named mistakes rather than rules, which is unusual for a mathematics course, and they are the entries most worth your attention — each one is a rule applied where it does not reach. By the end you will be able to do five things. One: classify a real number by testing the nested sets in order. Two: evaluate an expression by the fixed sequence, including one with a minus sign in front. Three: say which exponent rule adds and which multiplies. Four: say what a zero exponent and a negative exponent mean. Five: simplify an expression using every rule together.

Five sets, and the arrangement is what matters: each one contains the one before it, so classifying a number is a question of how far down the list it reaches. Watch what gets added at each step, because each addition solves a problem the previous set could not.

The whole line.

Alg 001 The Real Numbers

Everything on the number line, with nothing missing between any two points. The other four sets in this topic are all parts of it.

Where counting starts.

Alg 002 Natural Numbers and Whole Numbers

One, two, three and onwards is the counting set; add zero and you have the next one out. Zero belongs to the second and not the first.

Where the negatives arrive.

Alg 003 The Integers

Take that set and add the opposite of every counting number, and the line runs both ways from zero.

Anything written as a fraction.

Alg 004 Rational Numbers

Any number expressible as one integer over another, with the bottom not zero — and its decimal either stops or falls into a repeating pattern. Every integer qualifies; most of these are not integers.

And everything left over.

Alg 005 Irrational Numbers

Real numbers that no fraction of two integers equals, whose decimals run on without ever settling into a pattern.

The procedure runs down the list in order, and there is one instruction that has to come before any of it — because numbers are routinely written in a form that hides what they are.

Watch it run.

Alg 007 Classifying a Real Number

Work down the list and name every set the number belongs to. But simplify first: the square root of twenty-five is five, and four over two is two, and both look like something they are not until you compute them.

Compact notation for repeated multiplication, the sequence for evaluating anything, and then two entries on a single error — one that calculators themselves disagree about, which tells you the confusion is real.

The notation.

Alg 009 Exponential Notation

The lower number is what gets multiplied and the raised one counts how many copies. It counts factors — nothing here is being added.

The sequence.

Alg 010 The Order of Operations

Brackets first, then powers, then multiplying and dividing, then adding and subtracting. Everything after this in the course assumes it, and the next two entries are about one place it surprises people.

Where the minus sign waits.

Alg 011 Exponents Before Negation

A raised number attaches to whatever sits directly beneath it, so a minus sign in front is applied afterwards. Negative three squared, written without brackets, means the opposite of three squared — which is negative nine.

And the error.

Alg 012 The Invisible-Parentheses Error

Reading it as though the minus sign were inside brackets gives positive nine. The brackets are not written, so the value is negative nine — and different calculators will hand you different answers for the same keystrokes.

Five rules that permit rearranging. They look too obvious to be worth naming, and the value is in the limits — each one has a place where it stops, and every one of those stopping points is a standard error.

Order.

Alg 014 The Commutative Properties

Adding and multiplying do not care which comes first. Subtracting and dividing care enormously — eight take away three is not three take away eight.

Grouping.

Alg 015 The Associative Properties

With all additions or all multiplications, it makes no difference which pair you handle first. The brackets can move without changing anything.

Spreading a factor.

Alg 016 The Distributive Property

A multiplier outside a bracketed sum reaches both terms inside. And it runs one way only — multiplication spreads over addition, and addition never spreads over multiplication.

What changes nothing.

Alg 017 The Identity Properties

Adding zero leaves a number as it was, and multiplying by one does the same. Two numbers that do nothing, each to its own operation.

And what undoes.

Alg 018 The Inverse Properties

Adding a number's opposite returns zero; multiplying by its reciprocal returns one. Note the exception: zero has no reciprocal, which is the same fact as division by zero being undefined.

The first two exponent rules. Both share a condition that gets forgotten constantly — they apply only when the thing being raised is the same in both places.

Multiplying.

Alg 022 The Product Rule of Exponents

Two powers of the same base multiplied together: add the exponents. If the bases differ there is nothing to do — two cubed times five squared combines into nothing at all.

Dividing.

Alg 023 The Quotient Rule of Exponents

Same base above and below: subtract the exponents, and the order matters — the one on top minus the one underneath, never the reverse.

One more rule, and then an entry devoted entirely to confusing it with the previous one — because these two rules look almost identical on the page and do opposite arithmetic.

A power raised to a power.

Alg 025 The Power Rule of Exponents

Something already carrying an exponent, raised again: multiply the two together. The previous rule added; this one multiplies.

And the confusion.

Alg 026 The Power-Product Confusion

Adding where this rule multiplies, or multiplying where the earlier rule adds. The brackets are what tell them apart, so read those before touching any exponent.

Two definitions that look invented, and the opening of this lecture said why they are not. Follow the dividing rule down past an exponent of one and it forces both of these — they are what keeps the rule true rather than decisions someone made.

The zero power.

Alg 028 The Zero Exponent Rule of Exponents

Any base other than zero, raised to the zero power, equals one — which is exactly what the dividing rule produces when the two exponents match.

The error.

Alg 029 The Zero-Power Error

Answering with zero instead of one. The value is one for every base except zero itself, where the course leaves it undefined.

The negative power.

Alg 031 The Negative Rule of Exponents

A negative exponent means one over that power. The sign moves the expression across the fraction bar, and it does not touch whether the value is positive or negative.

And the error there.

Alg 032 The Negative-Exponent Sign Error

Reading the result as a negative number. A positive base raised to any real power stays positive — reciprocating and negating are different operations.

Two rules for applying an exponent to something already assembled. Both work by reaching every piece — and the warning is the same one from the distributive property: this reaches products and never sums.

A product raised to a power.

Alg 034 The Power of a Product Rule of Exponents

The exponent applies to each factor separately. Products only — a sum inside brackets does not come apart this way, which is one of the most expensive errors in the subject.

A fraction raised to a power.

Alg 035 The Power of a Quotient Rule of Exponents

The exponent reaches the numerator and the denominator both. Applying it only to the top is a standard slip, and it inflates the answer.

One expression needing every rule so far, and the instruction to carry is about coefficients — the outer exponent reaches everything inside the brackets, including the plain number sitting in front.

Watch them all run.

Alg 037 Simplifying with the Exponent Rules

Apply the rules for products, quotients, powers of powers, zero, and negatives in turn, until every base appears exactly once with a positive exponent. The outer power reaches every factor inside — the coefficient is a factor.

One notation to close Part One, and it is where the exponent rules earn their keep — arithmetic on numbers too large or too small to write out becomes arithmetic on the exponents.

The notation.

Alg 039 Scientific Notation

A number written as something between one and ten, multiplied by a whole-number power of ten. Multiplying two of these is the product rule doing the work, which is what the notation is for.

Five things you can now do — and Part Two takes the same story into roots. You can classify a real number by working down the nested sets, and you know to compute first, because a fraction or a root may be an integer in disguise. You can evaluate by the fixed sequence, and you know where a leading minus sign goes — the power attaches to what is under it, and the sign is applied after. You can say which rule adds and which multiplies: same base multiplied means add the exponents, and a power raised to a power means multiply them. The brackets are what tell them apart. You can say what the zero and negative exponents mean, and why — both are what the dividing rule forces once you follow it past an exponent of one. And you can simplify with every rule at once, remembering that an outer exponent reaches every factor inside the brackets, the coefficient included.