Integrated Math 1 · Unit 0 · Review of the Essentials
The place-value pattern from §0.1 run the other way — then fractions, percents, square roots, and the number line filled in completely.
bookSHelf · Integrated Math 1 · §0.4 · a self-paced section
Investigate — Problem
Here is the number 4,072 from Section 0.1 — each place is worth ten times the place to its right. Moving one place left multiplies the value by 10.
Investigate
What should moving one place right do instead?
Worked example — Step
Moving right divides by ten each time: tens, ones, tenths, hundredths, and so on.
Investigate
Now look at 36.407. Where does the whole-number part end and the fractional part begin?
Critique this claim
Jordan thinks there should be a "oneths" place next to "tens," the same way tens sits next to ones. Is Jordan correct? Why or why not?
Worked example — Answer
Right there — mark it with a _____________, the symbol that separates the whole-number places from the fractional places. Every place to its right is worth one tenth of the place immediately before it.
§0.4.1 — Decimal notation and rounding
Definition 0.4.1 — Decimal point
The decimal point separates the whole-number places from the fractional places. Every place to its right is worth one tenth of the place immediately before it.
§0.1's places ran ones, tens, hundreds — each ten times the one to its right. Nothing stops the pattern running the other way.
Definition 0.4.1: the places mirror around the ones place, and the point marks where the whole-number part ends.
Investigate — Problem
A bar is shaded to 0.7 of its length — 7 of 10 equal parts.
Investigate
Divide the same bar into 100 equal parts instead of 10. How many of the 100 parts are shaded? Did the shaded amount change?
Worked example — Step
70 of 100 parts are shaded: 0.7 = 0.70. The amount didn’t move — only how finely it’s sliced.
Investigate
Divide it into 1,000 parts. How many are shaded now? Could you keep going forever?
Critique this claim
Priya thinks 0.70 is bigger than 0.7, since 0.70 has one more digit. Is Priya correct? Why or why not?
Worked example — Answer
700 of 1,000: 0.7 = 0.70 = 0.700. Every zero you add to the right end is a _____________ — a zero at the right end of the fractional part that never changes the value.
§0.4.2 — Operations with decimals
Definition 0.4.2 — Trailing zero
A trailing zero sits at the right end of the fractional part. It never changes the value, so you may add or remove them freely to make places line up.
12.403.75+0.6016.75Definition 0.4.2: appending zeros re-slices the same amount into finer pieces without moving its edge.
You can only add quantities of the same kind. Tenths add to tenths — never straight to hundredths, any more than 3 hours add to 8 minutes. Line up the points, not the edges.
Investigate — Problem
Divide 3 by 8, one digit at a time.
Investigate
30 ÷ 8 gives a quotient digit and a remainder. What are they? Now 60 ÷ 8. Then 40 ÷ 8. What is the remainder each time?
Worked example — Step
30 ÷ 8 → 3, remainder 6. 60 ÷ 8 → 7, remainder 4. 40 ÷ 8 → 5, remainder 0.
Investigate
What happened the moment the remainder hit 0?
Critique this claim
Diego thinks if you divide long enough, every decimal’s digits eventually stop. Is Diego correct? Why or why not?
Worked example — Answer
The digits stopped: 3/8 = 0.375. A decimal whose digits stop like this is a ___________________ — the division reaches a remainder of zero.
Investigate — Problem
Now divide 5 by 6 the same way, one digit at a time.
Investigate
50 ÷ 6 gives a quotient digit and a remainder. What are they? Now 20 ÷ 6. What is the remainder this time — and does it look familiar?
Worked example — Step
50 ÷ 6 → 8, remainder 2. 20 ÷ 6 → 3, remainder 2. That remainder, 2, already showed up once.
Investigate
If the SAME remainder comes back around, what happens to the quotient digits from here on?
Critique this claim
Maya thinks 0.8333… and 0.83 are basically the same number, since they look almost identical written down. Is Maya correct? Why or why not?
Worked example — Answer
They cycle forever: 5/6 = 0.8333… A decimal where a block of digits repeats forever, because a remainder comes back around, is a _________________.
Definition 0.4.3 — Terminating decimal
A terminating decimal is a decimal whose digits stop. The division reaches a remainder of zero.
Definition 0.4.4 — Repeating decimal
A repeating decimal is one in which a block of digits repeats forever. The division never reaches remainder zero; a remainder comes back around, and the quotient digits cycle.
The bar is not decoration — it states that the block goes on without end, which 0.8333… only hints at.
Investigate — Problem
Sam scored 17 out of 20 on one quiz and 43 out of 50 on another.
Investigate
Which quiz did Sam do better on? How do you know?
Worked example — Step
Rewrite each score out of 100: 17/20 = 85/100. 43/50 = 86/100.
Investigate
Now which quiz was actually better? What made 85 and 86 easy to compare, that 17-out-of-20 and 43-out-of-50 weren’t?
Critique this claim
Marcus thinks 43/50 must be the better score, since 43 is bigger than 17. Is Marcus correct? Why or why not?
Worked example — Answer
Once both scores are out of the same 100, they line up instantly. A _______ is exactly that: a fraction whose denominator is 100, written with the symbol % in place of the denominator.
§0.4.3 — Definition
Definition 0.4.5 — Percent
A percent is a fraction whose denominator is 100, written with % in place of the denominator.
So 42%=10042=0.42. Percent → decimal: drop the %, move the point two places left. Decimal → percent: move it two right, attach the %.
Definition 0.4.5: the percent sign stands in for a denominator of 100.
Investigate — Problem
A square has an area of 81 square units.
Investigate
What is its side length? How do you know?
Worked example — Step
9 × 9 = 81, so the side is 9. Going from area back to side asks: what number, squared, gives 81?
Investigate
Does any OTHER number, squared, also give 81?
Critique this claim
Elena thinks √81 should equal both 9 and −9, since both square to 81. Is Elena correct? Why or why not?
Worked example — Answer
Yes — (−9)² = 81 too, since a negative times a negative is positive. A number whose square is 81 is a ___________ of 81, written √81.
Investigate — Problem
Both 9 and −9 square to 81.
Investigate
If √81 is going to name ONE single number, not two, which of the two should it point to — and why might that be the more useful choice?
Worked example — Step
The radical sign is defined to hand back only the nonnegative branch: √81 = 9, never −9.
Investigate
How would you write the OTHER root — the negative one — using this same symbol?
Critique this claim
Wesley thinks −√49 is not a real number, since there’s a negative sign right next to a radical. Is Wesley correct? Why or why not?
Worked example — Answer
Put the sign outside the radical: −√81 = −9. This nonnegative-only root is the _____________________ — the symbol √m always means this one.
Definition 0.4.6 — Square root and the radical sign
A square root of m is a number whose square is m, written m; the symbol is the radical sign.
81=9because92=81Definition 0.4.7 — Principal square root
The principal square root of a nonnegative number is its nonnegative square root. m always means the principal root, so 81=9 and never −9.
To ask for the negative root, put the sign outside: −81=−9. The radical groups — finish everything underneath, then apply the outside sign.
Investigate — Problem
Try to arrange 16 dots into a complete square shape — no gaps, no leftover dots.
Investigate
What size square works? Now try the same thing with 20 dots. Can you make a complete square?
Worked example — Step
16 dots close a 4×4 square exactly. 20 dots don’t — 16 fill a 4×4 block, but 4 more spill over with nowhere to go.
Investigate
Which two counts — 1, 4, 9, 16, 25, 36, and so on — does 20 fall between?
Critique this claim
Tyler thinks 20 must be a perfect square too, since it’s sandwiched right between two of them. Is Tyler correct? Why or why not?
Worked example — Answer
20 sits between 16 and 25. A count of dots that DOES close a complete square, like 16 or 25, is a ______________ — the square of an integer.
§0.4.4 — Definition
Definition 0.4.8 — Perfect square
A perfect square is a number that is the square of an integer. Its square root is an integer.
Knowing the first fifteen on sight is worth the small effort — it makes simplifying radicals fast, and it is the recognition step behind factoring quadratics in Chapter 7.
Definition 0.4.8: a perfect square fills a complete square of dots with nothing left over.
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225.
Investigate — Problem
Here are five numbers: 3/4, 12, −7, 2.6, and 0.6666… (repeating).
Investigate
3/4 is already one whole number over another. Can you write 12 that same way? What about −7?
Worked example — Step
12 = 12/1. −7 = −7/1. Any whole number can be written as itself over 1.
Investigate
2.6 and 0.6666… are decimals, not fractions on the page. Can each still be written as one integer over another?
Critique this claim
Isabella thinks −7 can’t be written as a fraction, since it’s already a whole number. Is Isabella correct? Why or why not?
Worked example — Answer
2.6 = 13/5. 0.6666… = 2/3. All five numbers turn out to be some integer over another integer. A _______________ is a number that can be written as a ratio p/q of two integers, where q ≠ 0.
Investigate — Problem
Every fraction you divide out eventually stops or starts cycling — you just watched that happen with 3/8 and 5/6. Now try √55 on a calculator: 7.4161984…
Investigate
Keep watching the digits. Do they ever stop? Do they ever lock into a repeating block?
Worked example — Step
No — they keep going, with no block that repeats. √55 never terminates and never repeats.
Investigate
Every ratio of two integers has to terminate or repeat when you divide it out. Can √55 be written as p/q for any integers p and q?
Critique this claim
Owen thinks √36 must belong in this same category, since it has a radical sign too, just like √55. Is Owen correct? Why or why not?
Worked example — Answer
No integer ratio behaves like that, so √55 can’t be one. A decimal that runs forever without ever settling into a repeating block is an _________________.
Investigate — Problem
You now have two kinds of numbers: ones that terminate or repeat, like 3/4 and 2.6, and ones that never settle into a pattern, like √55.
Investigate
Does every number you could mark on a number line fall into one of those two groups?
Worked example — Step
Yes — together they fill the entire line with no gaps.
Investigate
Now try √−16. Is there any number whose square is negative?
Critique this claim
Grace thinks √−16 has to belong to SOME category, since every number she’s seen before fits somewhere. Is Grace correct? Why or why not?
Worked example — Answer
No — positive² is positive, negative² is positive, 0² is 0. Nothing squares to a negative, so √−16 fits in neither group. Any number that is either rational or irrational is a ___________ — equivalently, any number with a location on the number line.
Definition 0.4.9 — Rational number
A rational number can be written as a ratio qp of two integers, q=0. The name comes from ratio, not from reasonable.
It swallows nearly everything so far: 12=112, −7=1−7, 2.6=513, 0.6=32.
Definition 0.4.10 — Irrational number
An irrational number is a real number that is not rational. Its decimal neither terminates nor repeats.
Two families supply most examples: roots of non-perfect-squares, like 55=7.4161984…, and π=3.14159265….
Definition 0.4.11 — Real number
A real number is any number that is either rational or irrational — equivalently, any number with a location on the number line.
counting⊂whole⊂integer⊂rational⊂real§0.4 — Conclusions
Decimals, fractions and percents are one number in three notations, and place value governs all of them. Beyond that, the reals divide cleanly into rationals and irrationals — and together they fill the number line with no gaps.
Judging size by digit count, right-aligning instead of aligning the points, miscounting decimal places in a product, moving only the divisor's point, writing 0.8% as 0.8, splitting a radical across a plus sign, and classifying before simplifying.
Next: §0.5 Properties of Real Numbers — the rules that let you rearrange all of this on purpose instead of by guessing. Back to start.