Lecture: Real Numbers, Exponents & Algebraic Expressions
Lecture script · about 15–20 minutes spoken · Week 1
Opening
Picture four friends at the end of a dinner. The check comes to 84 dollars, and the group wants to add a tip and split it four ways. One friend works on the whole check. Another works on one share of 21 dollars and multiplies by four. A third punches the numbers into a phone in a different order. All three land on the same total. That isn't luck. It's a sign that numbers follow rules, and the rules never change.
Those same few rules show up far from the dinner table. A nursing student uses them to check a dosage calculation before a dose is given. A small-business owner uses them to split costs and add a fee. A computer-science student uses them to count memory, which comes in powers of two. Learn the rules once, and you can check every answer yourself.
This is the first week of the course, so we're starting at the foundation. Nothing earlier is assumed beyond the arithmetic you already use. In this lecture we'll take five ideas in order. First, the families of real numbers. Second, the order of operations. Third, the properties that let you swap and regroup numbers safely. Fourth, the exponent rules, which we'll build on the powers of two. And fifth, simplifying and evaluating an expression. Next week, in Week 2, Linear Equations & Inequalities, these rules become the tools for solving an equation.
I want you to leave with one habit above all: when you get an answer, check it. Every rule I show you can be checked, so let's check each one as we go.
1. The real numbers: every number has a family
Think about the numbers on a grocery receipt. You see counts, like 3 cans of beans. You see a coupon worth minus 2 dollars. You see five eighths of a pound of cheese and a price of 0.45 dollars. Those numbers feel different, and mathematicians agree. They sort numbers into families, and the families fit inside one another like boxes.
The smallest box holds the counting numbers. A natural number is one of the counting numbers 1, 2, 3, and so on. Add zero and you get the whole numbers. Add the negatives of the counting numbers and you get the integers, numbers like minus 3, 0, and 7. Next comes a much bigger box. A rational number is a number that can be written as a quotient of two integers with a nonzero denominator. Its decimal either stops or repeats. Every integer is rational, because 7 is the same as 7 over 1.
Some numbers will not fit in that box at all. An irrational number is a real number that cannot be written as a quotient of two integers. Its decimal goes on forever without repeating. The square root of 2 is one, at about 1.41. Pi is another. Put the rational and irrational numbers together and you have the real numbers, every number with a place on the number line.
Here's a trick that catches a lot of people. The square root of 16 looks unfinished, so it feels irrational. But work it out first. 4 squared is 16, so the square root of 16 is 4, a plain counting number. Evaluate first, then name the family.
Pause and try this. Of these four numbers, which two are irrational: 0.45, the square root of 16, the square root of 2, and pi?
Say your two picks out loud before I go on.
Here's the answer. The square root of 2 and pi are the irrational ones. The decimal 0.45 is 45 hundredths, which simplifies to 9 over 20, so it's rational, and the square root of 16 is 4, a natural number.
So of the eight numbers in the chapter, six are rational and only two are irrational. The root sign alone tells you nothing until you evaluate it.
2. The order of operations: one agreed order
Here's a small puzzle. Type 2 plus 3 times 4 into a simple calculator one key at a time, and it may add first and show 20. A scientific calculator multiplies first and shows 14. They can't both be right, so mathematicians made an agreement. The order of operations is the agreed sequence for evaluating an expression: grouping symbols first, then exponents, then multiplication and division from left to right, then addition and subtraction from left to right.
Notice the two partner pairs. Multiplication and division are partners, so you do whichever comes first as you read. Addition and subtraction are partners in the same way. That's why the scientific calculator is right: 3 times 4 is 12, and 2 plus 12 is 14.
Now let's add an exponent. In a power, the base is the number being multiplied by itself, and the exponent is the raised number that says how many factors of the base to use. Take 2 plus 3 times 4 squared. There are no parentheses, so the exponent comes first. 4 squared is 16. Then multiply: 3 times 16 is 48. Then add: 2 plus 48 is 50. If you multiplied too early, you'd square 12 and get 146 instead, a completely different number from the same digits.
The order also settles a famous question about minus signs. Compare negative 4 squared, written with no parentheses, against negative 4 in parentheses, squared. Without parentheses, the exponent grabs only the 4. You square 4 to get 16, then apply the minus sign, and the result is minus 16. We call that a negated power: the exponent applies only to the number, and the minus sign comes afterward. With parentheses, the base is negative 4, and negative 4 times negative 4 is positive 16.
Pause and try this. Negative 2 to the fourth power, with no parentheses: is it 16 or minus 16?
Ask yourself where the minus sign lives.
Here's the answer. It's minus 16. Without parentheses, the exponent applies to 2 alone, so 2 to the fourth is 16, and the minus sign then makes it minus 16. Put the negative 2 inside parentheses, and the fourth power gives positive 16.
So the question to ask every time is short: is the minus sign inside the parentheses with the number?
3. The properties of real numbers: why the shortcuts are safe
Back to dinner for a moment. Three friends order meals of 14, 9, and 11 dollars. One adds 14 and 9 first, then 11. Another adds 9 and 11 first, then 14. Both get 34 dollars, and nobody argues. That calm is a property at work.
The commutative property is the rule that changing the order of two numbers in a sum or in a product does not change the result. The associative property is the rule that changing how three numbers are grouped in a sum or a product does not change the result. The dinner check used grouping, so that was the associative property.
The most useful one is the distributive property, the rule that a factor multiplies every term of a sum. Think of doubling a recipe that uses 3 cups of flour and 2 cups of milk. You can add first and double: 5 cups doubled is 10. Or you can double each ingredient: 6 cups plus 4 cups is also 10. Same 10 cups either way.
Two more round out the set. The identity property says adding 0 or multiplying by 1 leaves a number unchanged. The inverse property says a number plus its opposite is 0, and a nonzero number times its reciprocal is 1. A reciprocal of a nonzero number is the number that multiplies it to give 1, so the reciprocal of 4 is one fourth.
Read the distributive property backward and it combines like terms, terms with the same variable raised to the same exponent. Three x plus five x is eight x, the same way 3 apples plus 5 apples is 8 apples. A variable is a letter, like x, that stands for a number.
Pause and try this. A meal costs 40 dollars and you leave a 20 percent tip. Which is the same total: 40 times 1.2, or 40 plus 1.2?
Picture the tip as part of every dollar.
Here's the answer. It's 40 times 1.2, which is 48 dollars. Factoring out the 40 is the distributive property read backward: 40 plus 8 dollars of tip is 48, the same as 40 times 1.2.
One warning before we move on. Subtraction and division are not commutative. 10 minus 4 is 6, but 4 minus 10 is minus 6. Swap only what the properties allow.
4. The exponent rules: built on the powers of two
Picture a tournament bracket where each round cuts the teams in half. A bracket for 8 teams takes three rounds, because 2 times 2 times 2 is 8. The team counts are powers of two, and they make exponents easy to check. Here's the list we'll use: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, and 1024. Each one is double the one before. The first is 2 to the zero power, and the last is 2 to the tenth.
Start with multiplying powers. The product rule says that when two powers with the same base are multiplied, you keep the base and add the exponents. So 2 cubed times 2 to the fourth is 2 to the seventh. Check it with the list: 8 times 16 is 128, and 2 to the seventh is 128.
Dividing goes the other way. The quotient rule says that when one power of a nonzero base is divided by another power of the same base, you keep the base and subtract the exponents. So 2 to the tenth divided by 2 to the seventh is 2 cubed. Check it: 1024 divided by 128 is 8.
The power rule handles a power of a power: keep the base and multiply the exponents. So 2 squared, all cubed, is 2 to the sixth, which is 64. Check it: 2 squared is 4, and 4 cubed is 64. The power of a product says every factor gets the power, so 2x, all cubed, is 8 times x cubed.
Now the two rules that surprise people. Divide 2 cubed by itself. The quotient rule gives 2 to the zero, and plain arithmetic says 8 divided by 8 is 1. So the zero exponent rule says any nonzero base to the zero power is 1. Next, divide 2 cubed by 2 to the fifth. The rule says 2 to the minus 2, and arithmetic says 8 divided by 32 is one fourth. So the negative exponent rule says a negative exponent calls for a reciprocal. It never makes the value negative.
Pause and try this. Someone says 2 squared times 2 cubed is 2 to the sixth. What's the correct power, and how can the list prove it?
Think about whether to add or multiply the exponents.
Here's the answer. It's 2 to the fifth, because multiplying powers adds the exponents, and 2 plus 3 is 5. The list proves it: 4 times 8 is 32, which is 2 to the fifth, while 2 to the sixth is 64.
Keep the pair straight like this. Multiply powers, add exponents. Raise a power to a power, multiply them.
5. Simplify and evaluate: distribute, combine, substitute
Picture a campus print shop that charges 12 dollars a poster plus a 4-dollar fee per order. The cost for t posters is 12 t plus 4. An algebraic expression like that is a mix of numbers, variables, and operation signs with no equals sign. To evaluate it is to substitute a number for the variable and then follow the order of operations. For 3 posters, 12 times 3 is 36, and 36 plus 4 is 40 dollars.
Simplifying is the tidying step. Take 5x minus 2 times the quantity 3x minus 4. The minus 2 has to reach both terms inside. Minus 2 times 3x is minus 6x. Minus 2 times minus 4 is plus 8. That gives 5x minus 6x plus 8, and combining like terms leaves negative x plus 8.
Now check it, because an answer without a check is unfinished. Put in x equals 2. The original gives 10 minus 2 times 2, which is 10 minus 4, or 6. The simplified form gives minus 2 plus 8, which is also 6. They match.
Evaluating at a negative input is where signs go wrong. Take 3x squared minus 2x plus 5, and let x be minus 2. Put the minus 2 in parentheses everywhere. The exponent comes first: negative 2 squared, in parentheses, is 4. Then 3 times 4 is 12, and minus 2 times minus 2 is plus 4. Add 12, 4, and 5, and you get 21.
Pause and try this. A classmate evaluates 3x squared at x equals minus 2 and gets 36. What went wrong?
Ask which number the exponent belongs to.
Here's the answer. They squared the 3 along with the x, treating it as 3x all squared. The exponent belongs to x alone, so negative 2 squared is 4, and 3 times 4 is 12. The 3 is the coefficient, the number that multiplies the variable, and it comes after the square.
That one step, putting a negative input in parentheses, saves more points than any other habit in algebra.
Closing
Let me pull it together. Every number has a family, and the families nest: natural numbers inside integers, integers inside rational numbers, and all of them inside the real numbers, with the irrational numbers filling the gaps. The order of operations is one agreement: grouping, exponents, then two left-to-right passes, and a minus sign outside the parentheses waits until after the power. The properties of real numbers tell you which swaps and regroupings are safe, and the distributive property reaches every term. The exponent rules are shortcuts for counting factors, and the powers of two check every one. And simplifying ends with a check by substitution.
Later in this module, the slide deck, the Readings & Resources page, the lecture tutorial, the practice set, and the quiz give you a chance to use these rules yourself. Next week, in Week 2, Linear Equations & Inequalities, you'll use the same properties to solve an equation by doing the same thing to both sides.
Built with The Course Maker · samples.thecoursemaker.com