Unit 13 · The Number System

A thermometer in Chicago in January might read −8°F. The minus sign means eight degrees below zero. Zero here is not ’nothing’; it is a mark on the scale. Numbers above the mark are positive, and numbers below it are negative. Every situation with a negative number has its own zero mark. For temperature, zero is a spot on the thermometer. For elevation, zero is sea level. For money, zero is having no money and owing none.

Try a few. A submarine 300 feet below the surface is at an elevation of −300 feet. A parking level two floors below the street is level −2. If you owe a friend $12 and have no cash, your money situation is −$12. A football team that loses 5 yards on a play has a gain of −5 yards. In each case, ask two questions: where is zero, and which direction counts as negative?

The whole numbers together with their negatives are called the integers: ..., −3, −2, −1, 0, 1, 2, 3, ... Zero is an integer, but it is neither positive nor negative. A common mistake is to read −300 feet as ’a big number’ and think it means high. Remember that the minus sign points below zero; −300 feet is deep under the water, not far above it.

Draw a line. Put a tick in the middle and label it 0. To the right, mark 1, 2, 3, 4 at equal spaces. To the left, mark −1, −2, −3, −4 at the same spacing. This is a number line that runs both ways. Moving right always means the numbers get larger. Moving left always means they get smaller. So −4 is smaller than −1 because it is farther to the left.

To find −3 on the line, start at 0 and take three steps left. To find 2, start at 0 and take two steps right. The distance between −3 and 2 is 5 units: count the jumps −3 to −2, −2 to −1, −1 to 0, 0 to 1, 1 to 2. That is five jumps. Students sometimes count the tick marks instead of the jumps and get 6. Count the spaces, not the marks.

A vertical number line works the same way, with positive numbers above zero and negative numbers below. A thermometer is a vertical number line. An elevator panel with basement levels is a vertical number line. A horizontal one is handier for writing on paper. Either way, the rule is the same: numbers grow in one direction and shrink in the other, and the negatives are on the shrinking side of zero.

Take 7 on the number line. Its opposite is −7: the same distance from zero, but on the other side. The opposite of −3 is 3. The opposite of 0 is 0 itself, since 0 is neither left nor right of zero. We write ’the opposite of 7’ as −7, and ’the opposite of −7’ as −(−7), which equals 7. Two minus signs in a row bring you back to where you started.

Opposites show up when two things cancel. If you climb 40 feet and then descend 40 feet, your change in elevation is 40 + (−40) = 0. A team gains 8 yards and then loses 8 yards; net change, 0. That is the key property: a number plus its opposite is always 0. Chicago’s Willis Tower is about 1,450 feet tall; going up to the top and back down is a change of +1,450 and −1,450, which sum to 0.

Do not confuse ’opposite’ with ’reciprocal.’ The opposite of 4 is −4 (flip the sign). The reciprocal of 4 is 1/4 (flip the fraction). Another slip: students see −(−5) and write −5, forgetting that the outer minus flips the sign of −5. Read it aloud: ’the opposite of negative five.’ The opposite of a negative number is positive, so −(−5) = 5.

How far is −6 from zero? Six steps. How far is 6 from zero? Also six steps. That distance is called the absolute value, and it is written with two bars: |−6| = 6 and |6| = 6. Distance is never negative, so absolute value is never negative. The absolute value of 0 is 0. A debt of −$35 has an absolute value of $35: that is the size of the debt, ignoring which direction it goes.

Absolute value answers ’how much’ when direction does not matter. Two boats are at elevations −20 feet and 15 feet. Which one is farther from sea level? Compare |−20| = 20 and |15| = 15; the boat at −20 feet is farther, even though −20 is the smaller number. A temperature of −25°F is farther from zero than 18°F because |−25| = 25 is bigger than |18| = 18.

Two mistakes to watch. First, |−8| is 8, not −8; the bars strip away the sign. Second, students sometimes think the bigger absolute value means the bigger number. Not so: −25 has a bigger absolute value than 18, but −25 is smaller than 18. Absolute value measures size, not order. Keep ’how far from zero’ and ’which is larger’ as two separate questions.

Which is greater, −2 or −9? Picture the number line. −2 is two steps left of zero; −9 is nine steps left. Since −2 is farther to the right, −2 > −9. The symbol > means ’is greater than’ and < means ’is less than.’ The open side of the symbol faces the larger number. So we can also write −9 < −2. Students who compare only the digits would say 9 beats 2, but on the negative side the bigger digit means farther left, which means smaller.

Three quick rules follow from the number line. Any positive number is greater than any negative number: 1 > −100. Zero is greater than every negative number and less than every positive one. And for two negatives, the one with the smaller absolute value is greater: −3 > −7 because |−3| = 3 is less than |−7| = 7.

Try a real one. On a cold night Rockford reads −11°F and Chicago reads −6°F. Which city is warmer? −6 > −11, so Chicago is warmer by 5 degrees. In a game, one player has −40 points and another −15. Since −15 > −40, the player at −15 is ahead. Whenever you compare negatives, ask which one sits closer to zero; that one is greater.

A rational number is any number that can be written as a fraction of two integers, like 3/4, −5/2, 0.6 or 7 (which is 7/1). To order a mix of them, put them in one form. Take −1.5, 3/4, −2, 0.25 and −1/2. Change the fractions to decimals: 3/4 = 0.75 and −1/2 = −0.5. Now the list is −1.5, 0.75, −2, 0.25, −0.5.

Place them on the line. Negatives go left: −2 is farthest left, then −1.5, then −0.5. Positives go right: 0.25, then 0.75. Read left to right: −2 < −1.5 < −0.5 < 0.25 < 0.75. Back in original form: −2 < −1.5 < −1/2 < 1/4 < 3/4. A common slip is to order the negatives as if they were positive, putting −0.5 before −2. Remember, on the left side of zero, the bigger absolute value is farther left.

Comparing two negative fractions works the same way. Which is greater, −2/3 or −3/4? As decimals, −2/3 ≈ −0.667 and −3/4 = −0.75. Since −0.667 is closer to zero, −2/3 > −3/4. Or use common denominators: −8/12 and −9/12; −8/12 is greater because it is only 8 twelfths below zero, not 9.

To write 3/8 as a decimal, divide 3 by 8. Since 8 does not go into 3, write 3.000 and divide: 8 into 30 is 3 with remainder 6; 8 into 60 is 7 with remainder 4; 8 into 40 is 5 with no remainder. So 3/8 = 0.375. The decimal terminates, meaning it stops. Fractions whose bottoms divide evenly into a power of ten, like 8, 4, 5 and 20, always terminate.

Now try 2/3. Divide 2 by 3: 3 into 20 is 6, remainder 2; 3 into 20 is 6 again, remainder 2 again. The same remainder keeps coming back, so the same digit repeats forever: 2/3 = 0.6666... We write a repeating decimal with a bar over the repeating part, or in plain text as 0.666... For 5/11 the remainders cycle 5, 6, 5, 6, giving 0.454545... The block ’45’ repeats.

Going the other way, a decimal that ends is easy to turn into a fraction: 0.35 = 35/100 = 7/20. A negative rational number works the same: −0.375 = −3/8. The important fact is that every fraction of integers gives a decimal that either stops or falls into a repeating pattern, because there are only so many possible remainders. A decimal that never stops and never repeats, like the digits of π, is not rational; you will meet those in Chapter 28.

Lay a horizontal number line on the page. Now stand a vertical number line on it so the two zeros touch. The horizontal line is the x-axis, the vertical line is the y-axis, and the point where they cross is the origin. Together they form the coordinate plane. The axes cut the plane into four regions called quadrants, numbered with Roman numerals I, II, III, IV, starting at the upper right and going counterclockwise.

Every point on the plane gets a pair of numbers (x, y). The first number tells how far left or right of the origin; the second tells how far up or down. The signs tell you the quadrant. Quadrant I: both positive, like (3, 2). Quadrant II: x negative, y positive, like (−3, 2). Quadrant III: both negative, like (−3, −2). Quadrant IV: x positive, y negative, like (3, −2).

Chicago’s street grid works like this. State Street and Madison Street cross at the city’s origin. Addresses are north, south, east or west of that corner. An address at 800 North and 1200 West is like the point (−1200, 800): west is negative x, north is positive y. That would be Quadrant II. A point on an axis, like (0, 4) or (−5, 0), is not in any quadrant; it sits on the border.

Plot (−4, 3). Start at the origin. The x-coordinate is −4, so move 4 units left along the x-axis. The y-coordinate is 3, so from there move 3 units up. Mark the point. Now plot (2, −5): from the origin move 2 right, then 5 down. The order matters. The ordered pair always lists x first, then y. Swapping them puts the point in a different spot: (3, 1) and (1, 3) are not the same.

Reading a point works in reverse. Look straight up or down from the point to the x-axis and read the number; that is x. Look left or right to the y-axis; that is y. A point directly below −6 on the x-axis and level with 2 on the y-axis is (−6, 2). A common error is reading a point on the left side as positive because the student forgets which way is negative. Left is negative, down is negative.

Coordinates make shapes. Plot A(−3, 1), B(3, 1), C(3, −2) and D(−3, −2) and connect them in order. You get a rectangle. Its width runs from x = −3 to x = 3, which is 6 units. Its height runs from y = −2 to y = 1, which is 3 units. Its area is 6 × 3 = 18 square units. Maps, video games and graphs all locate things this same way.

How far is (−3, 4) from (5, 4)? Both points have y = 4, so they sit on the same horizontal line. Only the x-coordinates differ. From −3 to 0 is 3 units, from 0 to 5 is 5 more, so the distance is 8. Using absolute value: |−3| + |5| = 3 + 5 = 8. When the two x-values have opposite signs, add their absolute values, because the path crosses the y-axis.

Now (2, −6) and (2, −1). Same x, so the line is vertical. Both y-values are negative, on the same side of zero. Distance is the difference of their absolute values: |−6| − |−1| = 6 − 1 = 5. Check by counting on the y-axis from −6 up to −1: five jumps. The general rule: same signs, subtract the absolute values; opposite signs, add them. Or, in one step, take the absolute value of the difference: |−6 − (−1)| = |−5| = 5.

A common mistake is subtracting the numbers without care and getting a negative distance. Distance is never negative, so if your answer has a minus sign, take its absolute value. Another slip is trying to find the distance between points that share neither coordinate this way; for those you need a right triangle and the Pythagorean theorem, which comes later. In this chapter, stick to points that line up horizontally or vertically.

Take the point (3, 5). Change the sign of x to get (−3, 5). On the plane, the new point is the mirror image of the old one across the y-axis: same height, same distance from the axis, opposite side. Now change the sign of y instead: (3, −5). That is the reflection across the x-axis, straight below the original. Change both signs, (−3, −5), and the point lands in the opposite quadrant, diagonally across the origin.

This is why opposites matter on the plane. (3, 5) and (−3, 5) are the same distance from the y-axis, 3 units each, so the distance between them is 6. (3, 5) and (3, −5) are each 5 units from the x-axis, so they are 10 apart. Symmetric designs, like a butterfly drawing or a logo, are often built by plotting one half and reflecting every point.

Try it. A triangle has corners at (1, 2), (4, 2) and (4, 6). Reflect it across the x-axis by changing every y to its opposite: (1, −2), (4, −2), (4, −6). Reflect the original across the y-axis instead: (−1, 2), (−4, 2), (−4, 6). A common mistake is changing the wrong coordinate. Remember that reflecting across the y-axis moves you left or right, so x changes; reflecting across the x-axis moves you up or down, so y changes.

Picture two kinds of chips: yellow for +1 and red for −1. One yellow and one red together are a zero pair; they cancel. To add 4 + 3, lay out 4 yellow and 3 more yellow: 7 yellow, so 4 + 3 = 7. To add (−4) + (−3), lay out 4 red and 3 red: 7 red, so (−4) + (−3) = −7. Same sign, same color: count them all and keep the sign.

Now add 5 + (−2). Lay out 5 yellow and 2 red. Pair up 2 yellows with the 2 reds; those pairs vanish. Three yellows are left, so 5 + (−2) = 3. Try (−6) + 4: 6 red, 4 yellow; four pairs cancel, 2 red remain, so (−6) + 4 = −2. With different signs, the answer takes the sign of the pile that had more chips, and its size is the difference of the two sizes.

Jaylen’s paycheck did this. His balance was −42 and he deposited 95: (−42) + 95. Forty-two of the 95 yellows cancel the 42 reds; 53 yellows remain, so the balance is 53. A common mistake is to add the sizes when the signs differ, getting 137. Ask first: are the chips the same color? If not, they cancel, so subtract.

The number line gives the same answers as chips and works better for big numbers. To add −3 + 8, put your finger on −3. Adding a positive means moving right, so move 8 units right: −3 to 0 is 3 steps, and 5 more steps land on 5. So −3 + 8 = 5. To add 2 + (−9), start at 2 and move 9 units left: 2 steps reach 0, then 7 more reach −7. So 2 + (−9) = −7.

Temperature is a natural number line. At 6 a.m. in Rockford it is −11°F. By noon the temperature has risen 17 degrees. Start at −11, move 17 right: 11 steps to zero, 6 more to 6. Noon temperature: 6°F. If instead it had dropped 5 degrees from −11, you would move 5 left: −11 + (−5) = −16°F.

Here is the rule in words. Same signs: add the absolute values, keep the sign. Different signs: subtract the smaller absolute value from the larger, and take the sign of the number with the larger absolute value. For −3 + 8: absolute values 3 and 8, difference 5, and 8 is bigger and positive, so 5. The common slip is a wrong direction: adding a negative moves left, never right.

Subtraction with negatives gets easier with one move: change the subtraction to adding the opposite. 9 − 4 is 9 + (−4) = 5; same answer as always. Now 3 − 7 becomes 3 + (−7). Start at 3, move 7 left: −4. And 3 − (−5) becomes 3 + 5 = 8. Subtracting a negative is adding a positive. On the number line, taking away a leftward move is the same as moving right.

Why does this work? Think of chips. To do 3 − (−5) with chips, you need to remove 5 reds from a pile of 3 yellows. There are no reds to remove. So add 5 zero pairs first: now you have 8 yellows and 5 reds. Remove the 5 reds. Eight yellows remain, so 3 − (−5) = 8. Adding the zero pairs and taking away reds has the same effect as adding yellows.

Jaylen’s overdraft fee was −8 − 34 = −8 + (−34) = −42. A weather example: yesterday’s low was −3°F, today’s is −12°F. How much did it change? Today minus yesterday: −12 − (−3) = −12 + 3 = −9. The temperature fell 9 degrees. The most common mistake is to see two minus signs and not flip: −12 − (−3) is not −15. Rewrite it as addition every time until it feels natural.

Two questions sound alike but differ. ’How much did it change?’ has a direction, so the answer can be negative. ’How far apart are they?’ is a distance, so the answer is never negative. For change, compute end minus start. A stock price goes from $14 to $9: change is 9 − 14 = −5, a drop of $5. A submarine rises from −250 feet to −90 feet: change is −90 − (−250) = −90 + 250 = 160, a rise of 160 feet.

For distance, take the absolute value of the difference. The distance between −250 and −90 is |−250 − (−90)| = |−160| = 160 feet. The distance between 7 and −7 is |7 − (−7)| = |14| = 14. Both orders give the same distance: |−7 − 7| = |−14| = 14. That is why distance is safe to compute either way, but change is not: 14 − 9 = 5 and 9 − 14 = −5 mean opposite things.

Chicago’s lakefront is about 580 feet above sea level, and the tallest floors of downtown towers stand more than 1,000 feet above the street. In a building with a garage at level −3 and an office on floor 21, an elevator ride from the garage to the office is a change of 21 − (−3) = 24 levels. From the office back to the garage the change is −3 − 21 = −24, and the distance both ways is 24.

Multiplication is repeated addition, and that explains the first case. 3 × (−4) means three groups of −4: (−4) + (−4) + (−4) = −12. A positive times a negative is negative. Jaylen paying a $9 fee four times is 4 × (−9) = −36. Since order does not matter in multiplication, (−4) × 3 is also −12. So a negative times a positive is negative too.

Now the puzzling one: (−3) × (−4). Look at a pattern. 3 × (−4) = −12, then 2 × (−4) = −8, then 1 × (−4) = −4, then 0 × (−4) = 0. Each time the first factor drops by 1, the product rises by 4. Keep going: (−1) × (−4) must be 4, (−2) × (−4) must be 8, and (−3) × (−4) must be 12. A negative times a negative is positive. Another picture: undoing a loss of $4 three times leaves you $12 better off.

So the rule: multiply the sizes, then decide the sign. Same signs (both positive or both negative), the product is positive. Different signs, the product is negative. (−6) × (−7) = 42, (−6) × 7 = −42, 6 × (−7) = −42, 6 × 7 = 42. With three or more factors, count the negatives: an even number of negative factors gives a positive answer, an odd number gives a negative one. (−2) × (−3) × (−1) has three negatives, so it is −6.

Division undoes multiplication, so the sign rules match. Since 4 × (−5) = −20, it must be that −20 ÷ 4 = −5 and −20 ÷ (−5) = 4. Check each by multiplying back: 4 × (−5) = −20 and (−5) × 4 = −20. So a negative divided by a positive is negative, and a negative divided by a negative is positive. And 20 ÷ (−4) = −5 because (−4) × (−5) = 20.

Real division problems with signs often involve sharing a loss or finding an average change. Four friends split a debt of $60 equally: −60 ÷ 4 = −15, so each owes $15. A pond’s water level drops 18 inches over 6 weeks at a steady rate: −18 ÷ 6 = −3, so it drops 3 inches per week. If the level changed by −18 inches and the rate was −3 inches per week, how many weeks? (−18) ÷ (−3) = 6 weeks.

A fraction bar is a division sign, so the sign of a fraction can sit in three places: −3/4, (−3)/4 and 3/(−4) are all the same number, −0.75. But (−3)/(−4) is positive 3/4. Two cautions. First, dividing by zero is not allowed; 5 ÷ 0 has no answer. Second, students sometimes give the quotient the wrong sign when both numbers are negative. Same signs means positive, for division just as for multiplication.

The commutative property says order does not change a sum or product: (−5) + 8 = 8 + (−5) = 3, and (−4) × 6 = 6 × (−4) = −24. The associative property says grouping does not matter: (−2 + 7) + (−7) equals −2 + (7 + (−7)). The second grouping is easier, because 7 + (−7) = 0, leaving −2. Look for opposites that cancel before you start computing.

The distributive property says a × (b + c) = a × b + a × c. It works with signs: −3 × (10 + (−2)) can be done as −3 × 8 = −24, or as (−3 × 10) + (−3 × (−2)) = −30 + 6 = −24. Both give −24. The distributive property also explains a subtraction shortcut: −3 × 8 = −3 × (10 − 2) = −30 + 6 = −24, which is easier for some people than 3 × 8 with a sign.

Two properties involve special numbers. Adding 0 changes nothing, and multiplying by 1 changes nothing. Multiplying by −1 flips the sign: (−1) × 9 = −9 and (−1) × (−9) = 9. That is why −x means ’the opposite of x’: it is (−1) × x. Subtraction is not commutative: 5 − 8 = −3 but 8 − 5 = 3. A common slip is swapping the numbers in a subtraction; keep them in order or rewrite as adding the opposite first.

Compute −3/4 + 1/2. First a common denominator: 1/2 = 2/4. Now −3/4 + 2/4. Different signs, so subtract the sizes: 3/4 − 2/4 = 1/4, and take the sign of the larger, −3/4. Answer: −1/4. Now −3/4 − 1/2: rewrite as −3/4 + (−1/2) = −3/4 + (−2/4). Same signs, add the sizes: −5/4, which is −1 1/4.

Decimals work the same. A balance of −$18.50 gets a deposit of $25.75: −18.50 + 25.75. Different signs, subtract sizes: 25.75 − 18.50 = 7.25, positive because 25.75 is larger. Balance: $7.25. A temperature of −2.5°C drops another 4.75 degrees: −2.5 + (−4.75) = −7.25°C. Line up the decimal points when you add the sizes: 2.50 + 4.75 = 7.25.

Mixed numbers with signs are easiest as improper fractions. 2 1/3 − 3 2/3: write 7/3 − 11/3 = 7/3 + (−11/3) = −4/3 = −1 1/3. Watch for the mistake of subtracting the whole parts and the fraction parts separately and losing the sign. Another slip: adding denominators, writing −3/4 + 1/2 as −2/6. Denominators name the piece size; only the numerators add.

Compute (−2/3) × (3/4). Multiply tops and bottoms: 2 × 3 = 6 and 3 × 4 = 12, so 6/12 = 1/2. Different signs, so the product is negative: −1/2. Now (−2/3) × (−3/4): same size, 1/2, but same signs, so positive 1/2. To divide, multiply by the reciprocal: (−5/6) ÷ (1/3) = (−5/6) × (3/1) = −15/6 = −5/2 = −2 1/2. Different signs, so negative.

Decimals: (−0.4) × 2.5. Ignore signs first: 0.4 × 2.5 = 1.0. Different signs, so −1. And (−3.6) ÷ (−0.9): 3.6 ÷ 0.9 = 4, same signs, so 4. A hiker descends 3/4 mile in elevation each hour for 2 1/2 hours: (−3/4) × (5/2) = −15/8 = −1 7/8 miles. The elevation change is −1 7/8 miles.

Sign rules for fractions come with a bonus: a negative fraction can carry its sign on top, on the bottom or out front. −2/3 = (−2)/3 = 2/(−3). Pick the placement that makes the arithmetic easy, usually out front. The common mistakes are the usual ones, applying the wrong sign rule or forgetting to flip the second fraction when dividing. Do the size first, then the sign, as two separate checks.

A football drive starts at the team’s own 30-yard line. Plays gain or lose yards: +7, −4, +12, −9, +3. Where is the ball? Add in order: 30 + 7 = 37, 37 + (−4) = 33, 33 + 12 = 45, 45 + (−9) = 36, 36 + 3 = 39. The ball is at the 39-yard line. A faster route: the gains total 7 + 12 + 3 = 22 and the losses total −4 + (−9) = −13, so the net change is 22 − 13 = 9, and 30 + 9 = 39.

Money problems mix operations. Jaylen has $53. He buys 3 lunches at $8.50 each and gets a refund of $15. Lunches: 3 × (−8.50) = −25.50. New balance: 53 + (−25.50) + 15 = 27.50 + 15 = 42.50. He has $42.50. Write the multiplication step before adding, and write each amount with its sign so nothing gets lost.

Average change is a division at the end. A pond’s level changes over four weeks by −3, +1, −5 and −1 inches. Total change: −3 + 1 + (−5) + (−1) = −8 inches. Average per week: −8 ÷ 4 = −2 inches. Two checks help. First, estimate the sign: mostly drops, so the average should be negative. Second, the size: the changes are all between 1 and 5, so an average of 2 is reasonable. If you got +2 or −8, something slipped.

Compute −3 + 4 × (−2). Multiplication comes first: 4 × (−2) = −8. Then −3 + (−8) = −11. If you added first you would get 1 × (−2) = −2, which is wrong. Now (−3 + 4) × (−2): the parentheses come first, −3 + 4 = 1, then 1 × (−2) = −2. The parentheses changed the answer from −11 to −2, so read them carefully.

Watch the difference between (−4)^2 and −4^2. With the parentheses, (−4)^2 = (−4) × (−4) = 16. Without them, −4^2 means the opposite of 4^2, which is −16. Try 12 ÷ (−3) − (−5): division first, 12 ÷ (−3) = −4; then −4 − (−5) = −4 + 5 = 1. And 2 − 3 × (−1 − 5): inside the parentheses, −1 − 5 = −6; then 3 × (−6) = −18; then 2 − (−18) = 2 + 18 = 20.

A real problem: a store’s weekly result is 5 days of profit at $120 each and 2 days of loss at $80 each. Weekly total: 5 × 120 + 2 × (−80) = 600 + (−160) = 440. Do both multiplications before adding. The most common errors are doing the operations left to right regardless of rank, and losing a sign inside parentheses. Write one line per step and carry every sign with its number.

In 2^5, the 2 is the base and the 5 is the exponent. The expression is read ’two to the fifth power’ and it means multiply five 2s: 2 × 2 × 2 × 2 × 2 = 32. It does not mean 2 × 5 = 10, which is the most common mistake. Try 3^4 = 3 × 3 × 3 × 3 = 81, and 10^3 = 10 × 10 × 10 = 1,000. Any base to the first power is itself: 7^1 = 7.

Powers of ten are easy to spot: 10^n is a 1 followed by n zeros. 10^6 is 1,000,000, a million. Powers of 2 come up in computers, where memory doubles: 2^10 = 1,024, which is why a ’kilobyte’ is a bit more than a thousand bytes. The chessboard squares follow 2^0, 2^1, 2^2, and so on, so the eighth square holds 2^7 = 128 grains.

Negative bases need parentheses. (−3)^2 means (−3) × (−3) = 9, but −3^2 means the opposite of 3^2, which is −9. And (−2)^3 = (−2) × (−2) × (−2) = −8, negative because there are three negative factors. Fractions can be bases too: (2/3)^2 = 2/3 × 2/3 = 4/9. Square both the top and the bottom.

Write out 2^3 × 2^4: (2 × 2 × 2) × (2 × 2 × 2 × 2). That is seven 2s multiplied together, which is 2^7. Counting factors shows the rule: to multiply powers with the same base, add the exponents. 5^2 × 5^6 = 5^8. The bases must match; 2^3 × 3^2 cannot be combined this way, so compute it as 8 × 9 = 72.

Division goes the other way. 2^7 ÷ 2^4 is seven 2s over four 2s. Four pairs cancel, leaving three 2s: 2^3 = 8. Check: 128 ÷ 16 = 8. So to divide powers with the same base, subtract the exponents. 10^9 ÷ 10^4 = 10^5. A billion divided by ten thousand is a hundred thousand, which is right.

The common error is multiplying the exponents: writing 2^3 × 2^4 = 2^12. Test it with small numbers: 8 × 16 = 128, and 2^12 = 4,096, so that cannot be right. Another error is adding bases: 2^3 × 2^4 is not 4^7. When in doubt, write the factors out and count.

What is (2^3)^2? It means 2^3 times itself: 2^3 × 2^3. By the product rule that is 2^(3+3) = 2^6 = 64. Check: 2^3 = 8 and 8^2 = 64. So a power of a power multiplies the exponents: (a^m)^n = a^(m × n). For (5^2)^4, multiply 2 × 4 to get 5^8. This is the one place where multiplying exponents is correct, and it is easy to mix up with the product rule, where you add.

A power of a product spreads to each factor. (2 × 5)^3 = 2^3 × 5^3 = 8 × 125 = 1,000, and indeed 10^3 = 1,000. With a variable: (3x)^2 = 3^2 × x^2 = 9x^2, not 3x^2. Students often forget to square the 3. Similarly, (x^2 y)^3 = x^6 y^3. A fraction works the same way: (2/3)^3 = 2^3 / 3^3 = 8/27.

Keep the three rules straight by testing with 2s. Product: 2^2 × 2^3 = 4 × 8 = 32 = 2^5, exponents add. Quotient: 2^5 ÷ 2^2 = 32 ÷ 4 = 8 = 2^3, exponents subtract. Power of a power: (2^2)^3 = 4^3 = 64 = 2^6, exponents multiply. Whenever you are unsure which rule applies, write out a tiny example and count.

Follow the pattern down. 2^3 = 8, 2^2 = 4, 2^1 = 2. Each step divides by 2. Keep going: 2^0 must be 2 ÷ 2 = 1. Then 2^−1 = 1 ÷ 2 = 1/2, 2^−2 = 1/4, 2^−3 = 1/8. So a zero exponent gives 1, and a negative exponent gives 1 over the positive power. In general a^0 = 1 for any nonzero a, and a^−n = 1/a^n.

The quotient rule says the same thing. 5^3 ÷ 5^3 is clearly 1, and by the rule it is 5^(3−3) = 5^0. So 5^0 = 1. And 5^2 ÷ 5^5 is 25/3,125 = 1/125 = 1/5^3, while the rule gives 5^(2−5) = 5^−3. So 5^−3 = 1/5^3. Negative exponents are not negative numbers: 5^−3 is a small positive fraction, 1/125.

Powers of ten with negative exponents name decimal places. 10^−1 = 0.1, 10^−2 = 0.01, 10^−3 = 0.001. A negative exponent in a denominator flips up: 1/2^−3 = 2^3 = 8. Two mistakes to avoid: thinking 4^0 = 0 (it is 1), and thinking 3^−2 = −9 (it is 1/9). Say ’negative exponent means reciprocal’ each time until it sticks.

A square garden has an area of 49 square feet. How long is each side? You need a number that times itself gives 49. That is 7, because 7 × 7 = 49. We write √49 = 7 and call 7 the square root of 49. Numbers like 49 that come from squaring a whole number are perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144. Memorize these; they make roots quick.

Every positive number has two square roots, one positive and one negative, because (−7) × (−7) = 49 too. The symbol √ means the positive one. When an equation says x^2 = 49, both x = 7 and x = −7 work. A negative number has no real square root: no number times itself gives −49, since the product of two same-sign numbers is positive.

Roots of fractions take the root of top and bottom: √(9/16) = 3/4, because (3/4)^2 = 9/16. Roots of decimals: √0.25 = 0.5, since 0.5 × 0.5 = 0.25. The common error is halving instead of rooting: √16 is 4, not 8. Ask ’what number squared gives this?’ rather than ’what is half of this?’

A cube-shaped box holds 64 cubic inches. How long is each edge? You need a number that, used three times as a factor, gives 64. Since 4 × 4 × 4 = 64, each edge is 4 inches. We write this as the cube root of 64, ³√64 = 4. Numbers made by cubing a whole number are perfect cubes: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1,000.

Unlike square roots, cube roots of negative numbers exist. (−4) × (−4) × (−4) = −64, because three negative factors give a negative product. So ³√(−64) = −4. Each number has exactly one real cube root, with the same sign as the number itself. ³√(−27) = −3 and ³√27 = 3.

Keep squares and cubes apart. √64 = 8 but ³√64 = 4. √27 is not a whole number, but ³√27 = 3. If a problem gives a volume, think cube root; if it gives an area, think square root. And a number can be both: 64 = 8^2 = 4^3, and 1 = 1^2 = 1^3. When you see ³√1,000, ask: what number times itself three times gives 1,000? It is 10.

Draw a square with sides of 1 unit. Its diagonal has length √2, because 1^2 + 1^2 = 2. What is √2 as a decimal? Since 1^2 = 1 and 2^2 = 4, it is between 1 and 2. Try 1.4^2 = 1.96 and 1.5^2 = 2.25, so it is between 1.4 and 1.5. Try 1.41^2 = 1.9881 and 1.42^2 = 2.0164, so between 1.41 and 1.42. The decimal is 1.41421356... and it never ends or repeats.

A number whose decimal never ends and never repeats cannot be written as a fraction of two integers. Such numbers are irrational. √2 is irrational; so is π, and so is the square root of any whole number that is not a perfect square, like √3, √5 and √10. Together, the rational numbers and the irrational numbers make up the real numbers, every point on the number line.

Even though √2 has no finite decimal, it has an exact position. Set a compass to the diagonal of a 1-by-1 square and swing it down to the number line; the mark lands at √2, a little past 1.4. To place √10 without a compass, note that 3^2 = 9 and 4^2 = 16, so √10 is between 3 and 4, and close to 3 because 10 is just past 9; 3.2^2 = 10.24, so √10 ≈ 3.16. The mistake to avoid is calling 0.333... irrational. It repeats, so it is 1/3, a rational number.

The distance from Earth to the Sun is about 93,000,000 miles. Writing all those zeros invites mistakes. In scientific notation the number becomes 9.3 × 10^7. The first part, 9.3, is at least 1 and less than 10. The second part, 10^7, tells how many places the decimal point moved. Start at 93,000,000. Move the point left until only one nonzero digit is in front: 9.3. It moved 7 places, so the exponent is 7.

Go the other way to expand. 2.05 × 10^5: move the decimal point 5 places right, filling with zeros: 205,000. Check by reasoning: 10^5 is 100,000, and 2.05 × 100,000 = 205,000. The rice on the last chessboard square, about 9,223,000,000,000,000,000, becomes about 9.2 × 10^18. The exponent counts places moved, which is the number of digits after the first one.

Two common slips. Writing 45 × 10^5 is not proper scientific notation because 45 is not less than 10; it should be 4.5 × 10^6. And counting zeros instead of places: 93,000,000 has six zeros but the point moves seven places, because the 3 counts too. Illinois has about 12.5 million people, which is 1.25 × 10^7 people.

A red blood cell is about 0.000008 meters across. To write it in scientific notation, move the decimal point right until one nonzero digit sits in front: 8. It moved 6 places to the right, so the exponent is −6: 8 × 10^−6 meters. Moving right means the original number was small, and small numbers get negative exponents. Check: 10^−6 = 0.000001, and 8 × 0.000001 = 0.000008.

Expand 2.7 × 10^−3 by moving the point 3 places to the left, filling with zeros: 0.0027. A quick check: 10^−3 is one thousandth, and 2.7 thousandths is 0.0027. The exponent counts how many places, and the sign says which way: negative means the number is less than 1.

Comparing small numbers: which is smaller, 4 × 10^−5 or 9 × 10^−6? Look at exponents first. 10^−6 is one tenth of 10^−5, so 9 × 10^−6 = 0.000009 is smaller than 4 × 10^−5 = 0.00004. Students often assume the bigger coefficient wins; the exponent matters more. And do not confuse a negative exponent with a negative number: 2.7 × 10^−3 is positive.

Light travels about 3 × 10^5 kilometers per second. How far does it go in 5 × 10^2 seconds? Multiply coefficients: 3 × 5 = 15. Add exponents: 10^5 × 10^2 = 10^7. So 15 × 10^7. But 15 is not between 1 and 10, so rewrite: 15 = 1.5 × 10, giving 1.5 × 10^8 kilometers. Always check the coefficient at the end.

Division: (8.4 × 10^9) ÷ (2 × 10^4). Divide coefficients: 8.4 ÷ 2 = 4.2. Subtract exponents: 9 − 4 = 5. Answer: 4.2 × 10^5. If the coefficient comes out below 1, adjust the other way: (2 × 10^6) ÷ (8 × 10^2) gives 0.25 × 10^4, which becomes 2.5 × 10^3. Moving the point right one place means the exponent goes down by one.

A population problem: the United States has about 3.3 × 10^8 people, and Illinois about 1.25 × 10^7. How many times larger is the U.S.? (3.3 × 10^8) ÷ (1.25 × 10^7) = 2.64 × 10^1 = 26.4, so about 26 times. Common slips: adding exponents when dividing, and forgetting to readjust a coefficient like 15 or 0.25.

Add 3.2 × 10^5 and 4.5 × 10^4. The exponents differ, so the powers of ten are different sizes, like adding dollars and dimes. Rewrite the smaller one to match: 4.5 × 10^4 = 0.45 × 10^5. Now add coefficients: 3.2 + 0.45 = 3.65, so the sum is 3.65 × 10^5. Check in standard form: 320,000 + 45,000 = 365,000. Yes.

Subtraction works the same: 7 × 10^6 − 2.5 × 10^5 = 7 × 10^6 − 0.25 × 10^6 = 6.75 × 10^6, which is 7,000,000 − 250,000 = 6,750,000. If the result’s coefficient falls outside 1 to 10, adjust it. For example, 9.5 × 10^3 + 8 × 10^2 = 9.5 × 10^3 + 0.8 × 10^3 = 10.3 × 10^3 = 1.03 × 10^4.

To compare, the exponent comes first. 2 × 10^9 is bigger than 9 × 10^8, because 10^9 is ten times 10^8, and 2 × 10 beats 9. Only when exponents are equal do you compare coefficients: 4.1 × 10^6 < 4.3 × 10^6. To find how many times bigger one number is than another, divide. Lake Michigan holds roughly 4.9 × 10^12 cubic meters of water; a large swimming pool holds about 2.5 × 10^3 cubic meters. The lake holds about (4.9 × 10^12) ÷ (2.5 × 10^3) = 1.96 × 10^9 pools, nearly two billion.