Integrated Math 1 · Unit 0 · Review of the Essentials

Decimals

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

The longest section in the unit — it ends by naming every kind of number the course uses.

Decimals · bookSHelf Integrated Math 1§0.4

Outline — by the end of this section you will be able to

Objectives

Read, write, compare and round decimals using place value Def 0.4.1
Round money amounts to the nearest cent Ex 0.4.2
Add, subtract, multiply and divide decimals, and say why each rule works §0.4.2
Convert among decimals, fractions and percents Def 0.4.3–0.4.5
Simplify square roots and recognize when one is not real Def 0.4.6–0.4.8
Classify a number: counting, whole, integer, rational, irrational, real Def 0.4.9–0.4.11

The six section SLOs verbatim.

Decimals · bookSHelf Integrated Math 1§0.4

§0.4.1 — Decimal notation and rounding

Decimal point

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.

Moving right divides by ten each time.

Decimals · bookSHelf Integrated Math 1§0.4

§0.4.1 — Two letters, four orders of magnitude

The places on both sides

Place	Position	As a fraction	As a decimal
Tens	2 left	$10$	$10$
Ones	1 left	$1$	$1$
Tenths	1 right	$\dfrac{1}{10}$	$0.1$
Hundredths	2 right	$\dfrac{1}{100}$	$0.01$
Thousandths	3 right	$\dfrac{1}{1{,}000}$	$0.001$
Ten-thousandths	4 right	$\dfrac{1}{10{,}000}$	$0.0001$

Table 0.4.1: There is no "oneths" place — ones is the pivot the pattern reflects around.

The "-th" ending does real work. Ten thousand is large; one ten-thousandth is tiny. Two letters are the entire difference between $10{,}000$ and $0.0001$ .

36.407 = 3(10) + 6(1) + 4\!\left(\tfrac{1}{10}\right) + 7\!\left(\tfrac{1}{1{,}000}\right)

The hundredths zero is a placeholder. Delete it and $36.407$ collapses to $36.47$ — a different number.

Same placeholder job the zero in 4,072 did in §0.1.

Decimals · bookSHelf Integrated Math 1§0.4

§0.4.1 — Reading decimals aloud

A four-step routine

Procedure — naming a decimal

1. Name the whole-number part.

2. Say "and" for the decimal point.

3. Name the digits right of the point as if they were a whole number.

4. Name the place of the last digit.

Decimal	Read as
$0.062$	sixty-two thousandths
$7.4$	seven and four tenths
$7.04$	seven and four hundredths
$12.385$	twelve and three hundred eighty-five thousandths
$0.0009$	nine ten-thousandths

Table 0.4.2: $7.4$ and $7.04$ differ by one zero — and that zero costs the 4 a factor of ten.

36.407

is "thirty-six and four hundred seven thousandths."

Decimals · bookSHelf Integrated Math 1§0.4

Context Pause — why "and" is off limits inside a whole number

The word is reserved

§0.1 said 204 is "two hundred four," never "two hundred and four." Now you can see why.

"And" is reserved for the decimal point, so "two hundred and four" would be indistinguishable from $200.4$ when spoken aloud.

A §0.1 convention that only makes sense once decimals arrive.

Decimals · bookSHelf Integrated Math 1§0.4

§0.4.1 — Worked example

From words to digits and back

Example 0.4.1 — write "nine and forty-six thousandths" in standard form; then read $0.207$ aloud

The phrase names thousandths, so the last digit goes three places right of the point. The digits are 46 — the 6 takes thousandths, the 4 takes hundredths, and tenths needs a placeholder zero.

9.046

$0.207$ has no whole part; the last digit sits in thousandths — "two hundred seven thousandths."

† "Five and eighty-one thousandths" is $5.081$ , not $5.81$ . Writing the second multiplies the fractional part by ten.

The named place tells you where the last digit lands; fill gaps with zeros.

Decimals · bookSHelf Integrated Math 1§0.4

Insight — dollars before dimes

Compare the big places first

Think of tenths as dimes and hundredths as pennies. Someone with 7 dimes has more than someone with 6 dimes and 8 pennies, no matter how many coins each is holding.

You compare the big places first and only look at the small ones to break a tie. So $0.7 > 0.68$ , even though $0.68$ has more digits.

"More digits means bigger" is true for whole numbers and false here — the most stubborn misconception in the section.

Decimals · bookSHelf Integrated Math 1§0.4

§0.4.1 — Comparing decimals

Pad, then compare place by place

Comparison	Padded to match	First place they differ	Result
$0.7$ and $0.68$	$0.70$ and $0.68$	tenths: 7 beats 6	$0.7 > 0.68$
$0.325$ and $0.33$	$0.325$ and $0.330$	hundredths: 2 loses to 3	$0.325 < 0.33$
$4.9$ and $4.90$	$4.90$ and $4.90$	they never differ	$4.9 = 4.90$
$0.08$ and $0.1$	$0.08$ and $0.10$	tenths: 0 loses to 1	$0.08 < 0.1$
$-0.4$ and $-0.35$	$-0.40$ and $-0.35$	negatives flip the order	$-0.4 < -0.35$

Table 0.4.3: Padding is safe — $0.7 = 0.70 = 0.700$ , since $\tfrac{7}{10} = \tfrac{70}{100}$ .

The §0.2 rule still runs everything: a number is greater than whatever sits to its left, no matter how many decimal places it carries.

Stop at the first place where the digits differ.

Decimals · bookSHelf Integrated Math 1§0.4

§0.4.1 — Rounding a decimal

Same four steps, one change at the end

Procedure — rounding a decimal

1. Find the digit in the place you are rounding to.

2. Look at the digit immediately to its right.

3. If it is 5 or more, add 1. Otherwise leave it alone.

4. Drop every digit to the right — do not replace them with zeros.

Round $8.4763$ to	Digit right of it	Result
Tenths	7 — round up	$8.5$
Hundredths	6 — round up	$8.48$
Thousandths	3 — leave it	$8.476$
Whole number	4 — leave it	$8$

Table 0.4.4: Four places, four different answers — every one from the original.

† Step 4 is the change. Rounding 6,473 to hundreds needed the zeros in 6,500, because those places carry value. Right of the point a trailing zero carries none.

Start over from the original number every time — the §0.1 chained-rounding warning again.

Decimals · bookSHelf Integrated Math 1§0.4

Context Pause — rollover carries just like addition

When the rounding digit is 9

It becomes 10 and carries into the next place. Rounding $2.397$ to hundredths turns the 9 into 10, which carries into the tenths, giving $2.40$ .

That trailing zero stays, because here it reports how precise the answer is — the one case where a trailing zero earns its place.

Same rollover as 3,962 → 4,000 in §0.1.

Decimals · bookSHelf Integrated Math 1§0.4

§0.4.1 — Rounding money

Round to the nearest cent, once, at the end

Exact amount	To the nearest cent	Why
$\$47.3826$	$\$47.38$	thousandths digit is 2, so leave the 8 alone
$\$47.3852$	$\$47.39$	thousandths digit is 5, so the 8 becomes 9
$\$9.995$	$\$10.00$	the nines roll over and carry into the dollars
$\$0.4499$	$\$0.45$	thousandths digit is 9, so the 4 becomes 5

Table 0.4.5: Tax, per-unit prices, interest and gasoline all compute to more than two places before anyone rounds.

Carry the exact value all the way through and round only at the very end. Rounding partway is the same chained-rounding mistake — and on a long receipt those errors add into a total that visibly disagrees with the register.

The nearest cent is the hundredths place — a fixed convention, not a choice.

Decimals · bookSHelf Integrated Math 1§0.4

§0.4.1 — Worked example

Rounding a tax calculation

Example 0.4.2 — a jacket costs $\$18.95$ ; tax at $7.25\%$ is exactly $\$1.373875$

In $\$1.373875$ the hundredths digit is 7 and the digit to its right is 3. Since 3 is less than 5, the 7 stays and the rest is dropped.

\$1.373875 \rightarrow \$1.37

Line up the decimal points and add.

\$18.95 + \$1.37 = \$20.32

The tax was rounded once, at the end of the tax computation — which is what a register does.

$1.37 in tax, $20.32 total.

Decimals · bookSHelf Integrated Math 1§0.4

§0.4.1 — Your turn

Try It Now 0.4.1

Try It Now 0.4.1 — write "six and ninety-four hundredths"; decide which is larger, $0.409$ or $0.41$ ; round $0.409$ to the nearest hundredth

94 fills tenths and hundredths exactly — no placeholder needed.

6.94 \qquad 0.409 \;\text{vs}\; 0.410 \;\Rightarrow\; 0.409 < 0.41 \qquad 0.409 \rightarrow 0.41

Tenths match at 4; hundredths are 0 against 1. Then rounding: the hundredths digit is 0 and 9 follows it, so the 0 becomes 1.

Note the comparison and the rounding give the same two digits for different reasons.

Decimals · bookSHelf Integrated Math 1§0.4

§0.4.2 — Operations with decimals

Trailing zero

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.

\begin{array}{r} 12.40 \\ 3.75 \\ +\;0.60 \\ \hline 16.75 \end{array}

Definition 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.

Right-aligning

12.4

and

3.75

stacks 4 over 5 — tenths onto hundredths, and nonsense.

40 - 6.28

works because 40 is 40.00.

Decimals · bookSHelf Integrated Math 1§0.4

§0.4.2 — Multiplying: count the decimal places

A statement about exponents in disguise

Product	Ignoring the points	Places	Answer
$2.4 \times 3.5$	$24 \times 35 = 840$	$1 + 1$	$8.4$
$0.7 \times 0.03$	$7 \times 3 = 21$	$1 + 2$	$0.021$
$1.6 \times 0.45$	$16 \times 45 = 720$	$1 + 2$	$0.72$
$0.5 \times 0.5$	$5 \times 5 = 25$	$1 + 1$	$0.25$
$12 \times 0.08$	$12 \times 8 = 96$	$0 + 2$	$0.96$

Table 0.4.6: Places in the product = places in the first factor + places in the second.

0.7 \times 0.03 = \frac{7}{10} \times \frac{3}{100} = \frac{21}{1{,}000}

One place plus two places gives three because $10^1 \cdot 10^2 = 10^3$ . The zeros in the denominator are the decimal places.

† Row 4: $0.5 \times 0.5 = 0.25$ is smaller than either factor. Multiplying by a number between 0 and 1 shrinks things — an instinct built on whole numbers has to go before Chapter 2.

It is §0.3 fraction multiplication wearing a disguise.

Decimals · bookSHelf Integrated Math 1§0.4

§0.4.2 — Worked example

Placing the point in a product

Example 0.4.3 — compute $0.24 \times 3.5$

Drop the points: $24 \times 35 = 840$ . Two places in $0.24$ and one in $3.5$ means $2 + 1 = 3$ places, so count three left from the right end of 840.

0.840 = 0.84

Sanity check: $0.24$ is about a quarter, and a quarter of 3.5 is a little less than 1 — so 0.84 is believable.

The estimate is the habit worth teaching, not the digits.

Decimals · bookSHelf Integrated Math 1§0.4

§0.4.2 — Dividing by a decimal

Move both points

Problem	Shift both by	Rewritten	Answer
$8.61 \div 0.3$	1 place	$86.1 \div 3$	$28.7$
$0.144 \div 0.06$	2 places	$14.4 \div 6$	$2.4$
$7 \div 0.25$	2 places	$700 \div 25$	$28$
$5.5 \div 1.1$	1 place	$55 \div 11$	$5$

Table 0.4.7: Moving only the divisor's point changes the answer by a factor of ten.

\frac{8.61}{0.3} = \frac{8.61 \times 10}{0.3 \times 10} = \frac{86.1}{3}

Multiplying both parts by the same thing leaves the value unchanged — the equivalent-fractions idea from §0.3.

Row 3 is the other instinct to retrain: dividing by a number between 0 and 1 makes things bigger.

"How many quarter-units fit inside 7?" has a large answer. Dividing by a whole number just needs the point brought straight up.

Decimals · bookSHelf Integrated Math 1§0.4

§0.4.2 — Multiplying and dividing by powers of ten

Count the zeros, slide the point

Operation	Point moves	Example
$\times 10$	1 place right	$4.62 \times 10 = 46.2$
$\times 100$	2 places right	$4.62 \times 100 = 462$
$\times 1{,}000$	3 places right	$4.62 \times 1{,}000 = 4{,}620$
$\div 10$	1 place left	$4.62 \div 10 = 0.462$
$\div 100$	2 places left	$4.62 \div 100 = 0.0462$
$\div 1{,}000$	3 places left	$4.62 \div 1{,}000 = 0.00462$

Table 0.4.8: Append placeholder zeros whenever the digits run out — row 3 needs one in the ones place.

This shortcut is behind every metric-unit conversion you will ever do, and §1.3 comes back to it as a statement about exponents.

Multiplying by 10 bumps every digit up one place — same thing as sliding the point.

Decimals · bookSHelf Integrated Math 1§0.4

§0.4.2 — Worked example

Order of operations with decimals

Example 0.4.4 — simplify $\;0.6(4.5 - 1.25) + 0.8$

Grouping first — pad $4.5$ to $4.50$ so the places line up. Then multiply: $6 \times 325 = 1950$ , needing $1 + 2 = 3$ places. Then add, padding $0.8$ to $0.80$ .

3.25 \;\rightarrow\; 1.95 \;\rightarrow\; 1.95 + 0.80 = 2.75

† Nothing about the §0.1 order changes. Almost everyone gets the digits right; what separates a correct answer from a wrong one is whether the point in the product landed in the right spot. Count the places before you write the product down.

Answer 2.75. The point is where the errors hide.

Decimals · bookSHelf Integrated Math 1§0.4

§0.4.2 — Your turn

Try It Now 0.4.2

Try It Now 0.4.2 — compute $0.9 \times 0.04$ ; then $6.3 \div 0.7$ ; then simplify $\;2.5 + 0.4(6 - 1.5)$

9 \times 4 = 36,\; 1 + 2 = 3 \text{ places} \;\Rightarrow\; 0.036

\frac{6.3}{0.7} = \frac{63}{7} = 9 \qquad 2.5 + 0.4(4.5) = 2.5 + 1.8 = 4.3

Part 1 needs a placeholder zero in the tenths spot.

Answers 0.036, 9, 4.3.

Decimals · bookSHelf Integrated Math 1§0.4

§0.4.3 — Decimals, fractions, and percents

Three notations for the same quantities

Each is convenient somewhere

Fractions are exact and are what algebra prefers. Decimals are easiest to compare and are what a calculator hands you. Percents are the clearest way to report a proportion to another person.

Decimal → fraction: read it aloud

The name is the fraction. $0.36$ is "thirty-six hundredths," so $\tfrac{36}{100} = \tfrac{9}{25}$ . The denominator is always the place value of the last digit.

Being fluent means converting without stopping to think, because a problem almost never arrives in the notation you want to work in.

Fraction → decimal is just division:

\tfrac{3}{8} = 3 \div 8 = 0.375

.

Decimals · bookSHelf Integrated Math 1§0.4

§0.4.3 — Definition

Terminating decimal

Definition 0.4.3 — Terminating decimal

A terminating decimal is a decimal whose digits stop. The division reaches a remainder of zero.

Definition 0.4.3: it stops because the division reaches a remainder of zero.

Every fraction converts, but the division can end in one of two ways.

Decimals · bookSHelf Integrated Math 1§0.4

§0.4.3 — Definition

Repeating decimal

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 $0.8333\ldots$ only hints at.

Definition 0.4.4: a remainder that comes back around is what makes the digits cycle forever.

Watch where each bar starts. In $0.8\overline{3}$ only the 3 repeats, so it is $0.8333\ldots$ . In $0.\overline{36}$ the pair repeats as a block: $0.363636\ldots$ .

\tfrac{5}{6} = 0.8\overline{3}

,

\tfrac{4}{11} = 0.\overline{36}

,

\tfrac{2}{3} = 0.\overline{6}

.

Decimals · bookSHelf Integrated Math 1§0.4

§0.4.3 — Predicting which fractions terminate

Only 2s and 5s terminate

Fraction	Denominator factored	Only 2s and 5s?	Prediction	Actual
$\dfrac{3}{8}$	$2 \cdot 2 \cdot 2$	yes	terminates	$0.375$
$\dfrac{7}{20}$	$2 \cdot 2 \cdot 5$	yes	terminates	$0.35$
$\dfrac{5}{6}$	$2 \cdot 3$	no, there is a 3	repeats	$0.8\overline{3}$
$\dfrac{4}{11}$	$11$	no, there is an 11	repeats	$0.\overline{36}$

Table 0.4.9: 2 and 5 are the two primes that divide 10 — any other prime factor forces a repeat.

Reduce first — not optional. $\tfrac{6}{15}$ has a 3 in the denominator and predicts a repeat, but in lowest terms it is $\tfrac{2}{5} = 0.4$ , which terminates. Test the fraction in lowest terms, not whatever form you were handed.

You can tell which ending you get before you divide.

Decimals · bookSHelf Integrated Math 1§0.4

§0.4.3 — Definition

Percent

Definition 0.4.5 — Percent

A percent is a fraction whose denominator is 100, written with $\%$ in place of the denominator.

So $42\% = \dfrac{42}{100} = 0.42$ . 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.

Both conversions are mechanical because dividing by 100 moves the point two places.

Decimals · bookSHelf Integrated Math 1§0.4

Insight — a percent is a score out of 100

Rewriting so they line up

If you got 17 out of 20 on a quiz, nobody compares that to 43 out of 50 in their head.

Rewriting both as scores out of 100 makes them line up instantly — and that rewriting is all a percent is.

A common denominator, chosen once, for everybody.

Decimals · bookSHelf Integrated Math 1§0.4

§0.4.3 — A percent need not land between 0 and 100

Read the symbol as "per hundred"

Percent	Decimal	Fraction in lowest terms
$75\%$	$0.75$	$\dfrac{3}{4}$
$8\%$	$0.08$	$\dfrac{2}{25}$
$125\%$	$1.25$	$\dfrac{5}{4}$
$0.8\%$	$0.008$	$\dfrac{1}{125}$
$250\%$	$2.5$	$\dfrac{5}{2}$

Table 0.4.10: Rows 3 and 4 contradict a widespread belief.

$125\%$ is a perfectly good number — it is $1.25$ , more than the whole — and it turns up any time something grows past its original size.

$0.8\%$ is less than one percent. Writing it as $0.8$ is an error by a factor of one hundred: $\dfrac{0.8}{100} = 0.008$ .

Read "per hundred" every time and both rows lose their drama.

Decimals · bookSHelf Integrated Math 1§0.4

§0.4.3 — Worth recognizing on sight

Common equivalents

Fraction	Decimal	Percent
$\dfrac{1}{2}$	$0.5$	$50\%$
$\dfrac{1}{3}$	$0.\overline{3}$	$33\tfrac{1}{3}\%$
$\dfrac{2}{3}$	$0.\overline{6}$	$66\tfrac{2}{3}\%$
$\dfrac{1}{4}$	$0.25$	$25\%$
$\dfrac{3}{4}$	$0.75$	$75\%$
$\dfrac{1}{5}$	$0.2$	$20\%$

Fraction	Decimal	Percent
$\dfrac{2}{5}$	$0.4$	$40\%$
$\dfrac{1}{8}$	$0.125$	$12.5\%$
$\dfrac{3}{8}$	$0.375$	$37.5\%$
$\dfrac{1}{10}$	$0.1$	$10\%$
$\dfrac{1}{100}$	$0.01$	$1\%$
$1$	$1.0$	$100\%$

Table 0.4.11: The $\tfrac{1}{3}$ row is why you see $33\tfrac{1}{3}\%$ rather than $33.3\%$ — the decimal repeats forever, so a decimal percent is rounded while the fraction is exact.

Split into two columns so nothing shrinks below readable size. The same handful of equivalents reappears throughout Chapter 4's data work; when exactness matters, keep the fraction.

Decimals · bookSHelf Integrated Math 1§0.4

§0.4.3 — Worked example

All the way around the triangle

Example 0.4.5 — convert $\dfrac{3}{8}$ to a decimal then a percent; convert $24\%$ to a decimal then a fraction

\frac{3}{8} = 3 \div 8 = 0.375 \;\rightarrow\; 37.5\%

24\% \;\rightarrow\; 0.24 \;\rightarrow\; \frac{24}{100} = \frac{6}{25}

The bar means division; moving the point two places converts either way; the denominator is the place value of the last digit.

Both 24 and 100 are divisible by 4, reducing in one step.

Decimals · bookSHelf Integrated Math 1§0.4

§0.4.3 — Your turn

Try It Now 0.4.3

Try It Now 0.4.3 — convert $0.45$ to a fraction and a percent; then decide which is largest: $\dfrac{5}{8}$ , $0.63$ , or $61\%$

0.45 = \frac{45}{100} = \frac{9}{20} = 45\%

Put all three in one notation — decimals rank by place value with no common denominator to hunt for.

\frac{5}{8} = 0.625 \qquad 0.63 = 0.630 \qquad 61\% = 0.610

All three have 6 tenths; hundredths are 2, 3 and 1, so $0.63$ is largest.

Converting to one notation before comparing is the transferable move.

Decimals · bookSHelf Integrated Math 1§0.4

§0.4.4 — Square roots and the real numbers

Square root and the radical sign

Definition 0.4.6 — Square root and the radical sign

A square root of $m$ is a number whose square is $m$ , written $\sqrt{m}$ ; the symbol $\sqrt{\;\;}$ is the radical sign.

\sqrt{81} = 9 \quad \text{because} \quad 9^2 = 81

Definition 0.4.6: squaring asks what 9 rows of 9 come to; the square root asks the reverse.

A subtlety hides in that line: both 9 and $-9$ square to 81. If $\sqrt{81}$ meant "either one" it would not name a single number, and every expression containing it would be ambiguous.

Like signs multiply to a positive — §0.2 — which is what creates the ambiguity.

Decimals · bookSHelf Integrated Math 1§0.4

§0.4.4 — Definition

Principal square root

Definition 0.4.7 — Principal square root

The principal square root of a nonnegative number is its nonnegative square root. $\sqrt{m}$ always means the principal root, so $\sqrt{81} = 9$ and never $-9$ .

To ask for the negative root, put the sign outside: $-\sqrt{81} = -9$ . The radical groups — finish everything underneath, then apply the outside sign.

Definition 0.4.7: two numbers square to 81, so the radical returns only the nonnegative one.

Same "negate afterward" structure as the absolute-value expressions in §0.2.

Decimals · bookSHelf Integrated Math 1§0.4

§0.4.4 — Definition

Perfect square

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.

Squares of 1 through 15.

Decimals · bookSHelf Integrated Math 1§0.4

§0.4.4 — When a square root is not a real number

Inside the radical or outside it

Expression	Value	Reason
$\sqrt{49}$	$7$	the principal, nonnegative root
$-\sqrt{49}$	$-7$	take the root first, then negate
$\sqrt{-49}$	not real	nothing real squares to a negative
$\sqrt{0}$	$0$	$0^2 = 0$
$(-7)^2$	$49$	like signs give a positive product

Table 0.4.12: Rows 2 and 3 differ only in whether the sign is inside the radical.

A positive squared is positive, a negative squared is positive, and zero squared is zero — so no real number squares to a negative.

This is a real gap in the number system, not an oversight, and it is why a later course introduces imaginary numbers. For now treat it like division by zero: an expression with no value among the numbers we have.

Do not confuse

\sqrt{-49}

with

-\sqrt{49} = -7

, which is perfectly fine.

Decimals · bookSHelf Integrated Math 1§0.4

Context Pause — a radical does not split across a plus sign

Simplify underneath first

$\sqrt{25 - 9} = \sqrt{16} = 4$ , while $\sqrt{25} - \sqrt{9} = 5 - 3 = 2$ .

Different answers, so the grouping is not optional. The radical groups everything under it, exactly as the fraction bar and the absolute-value bars do.

Third grouping symbol in three sections — the pattern is the point.

Decimals · bookSHelf Integrated Math 1§0.4

§0.4.4 — Worked example

Simplifying, and estimating a root

Example 0.4.6 — simplify $\;2\sqrt{36} - \sqrt{25 - 9}$

Grouping first, then both roots, then multiplication before subtraction.

2\sqrt{36} - \sqrt{16} = 2(6) - 4 = 8

Most numbers are not perfect squares. Pin the root between the perfect squares on either side:

7 = \sqrt{49} < \sqrt{55} < \sqrt{64} = 8

Squaring candidates closes in: $7.41^2 = 54.9081$ , $7.42^2 = 55.0564$ , so $\sqrt{55} \approx 7.42$ .

† Estimating is not a lesser substitute for a calculator; it is how you catch a mistyped key. If a calculator reports $\sqrt{55} \approx 24.1$ , the estimate tells you instantly that something went in wrong.

Answer 8.

Decimals · bookSHelf Integrated Math 1§0.4

§0.4.4 — Definition

Rational number

Definition 0.4.9 — Rational number

A rational number can be written as a ratio $\dfrac{p}{q}$ of two integers, $q \neq 0$ . The name comes from ratio, not from reasonable.

It swallows nearly everything so far: $12 = \tfrac{12}{1}$ , $-7 = \tfrac{-7}{1}$ , $2.6 = \tfrac{13}{5}$ , $0.\overline{6} = \tfrac{2}{3}$ .

Definition 0.4.9: five familiar numbers, each rewritten as a ratio of two integers.

A test you can run by eye: a number is rational exactly when its decimal terminates or repeats. Both directions hold — the remainders in a division can only take finitely many values, so eventually one repeats and the digits cycle.

That leaves room for a third kind of decimal — one that runs forever without settling.

Decimals · bookSHelf Integrated Math 1§0.4

§0.4.4 — Definition

Irrational number

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 $\sqrt{55} = 7.4161984\ldots$ , and $\pi = 3.14159265\ldots$ .

Definition 0.4.10: one decimal stops, one repeats, and one does neither.

The radical sign is not what makes a number irrational. $\sqrt{36} = 6$ is rational; $\sqrt{35}$ is not. Whether the number underneath is a perfect square is what decides. And $3.14$ and $\tfrac{22}{7}$ are rational approximations — which is exactly why neither equals $\pi$ .

The square root of any whole number that is not a perfect square is irrational.

Decimals · bookSHelf Integrated Math 1§0.4

§0.4.4 — Definition

Real number

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.

\text{counting} \subset \text{whole} \subset \text{integer} \subset \text{rational} \subset \text{real}

Definition 0.4.11: the sets nest outward, the irrationals sit beside them, and together they fill the line.

The irrationals are the one set that does not nest — they sit beside the rationals rather than inside them. Together the two make up the reals.

A number in an inner set automatically belongs to every set outside it.

Decimals · bookSHelf Integrated Math 1§0.4

§0.4.4 — Three rows carry the whole idea

Which sets each number belongs to

Number	Counting	Whole	Integer	Rational	Irrational	Real
$12$	yes	yes	yes	yes		yes
$0$		yes	yes	yes		yes
$-7$			yes	yes		yes
$-2.6$				yes		yes
$0.\overline{45}$				yes		yes
$\sqrt{9}$	yes	yes	yes	yes		yes
$\sqrt{35}$					yes	yes
$\pi$					yes	yes
$\sqrt{-16}$						not real

Table 0.4.13: $\sqrt{9}$ looks exotic and is simply 3 — always simplify before you classify.

\sqrt{35}

sits one line away and belongs to none of the inner sets;

\sqrt{-16}

belongs to nothing on the chart.

Decimals · bookSHelf Integrated Math 1§0.4

§0.4.4 — Worked example

Classifying numbers

Example 0.4.7 — list every set each belongs to: $\sqrt{64}$ , $-\dfrac{5}{2}$ , $\sqrt{20}$

$\sqrt{64} = 8$ — simplify first. A counting number, so also whole, integer, rational $\left(\tfrac{8}{1}\right)$ , and real.

$-\dfrac{5}{2}$ — already a ratio of integers, so rational and real. Negative and not whole, so none of counting, whole, integer. As a decimal it is $-2.5$ , which terminates.

$\boldsymbol{\sqrt{20}}$ — 20 sits between 16 and 25, so $4 < \sqrt{20} < 5$ and it is not an integer. Irrational and real.

Simplify, then classify — in that order, every time.

Decimals · bookSHelf Integrated Math 1§0.4

§0.4.4 — No gaps anywhere

The reals fill the line completely

§0.2's number line held the integers as evenly spaced ticks. Fractions and decimals fill the space between them — $2.6$ sits six tenths of the way from 2 to 3, and $-\tfrac{3}{4}$ three quarters of the way from 0 toward $-1$ .

The irrationals take the positions that are left over: $\sqrt{55}$ has a definite spot just past 7.4, even though no finite decimal names it exactly.

That completeness is what lets Chapter 5 talk about the graph of a line as one solid unbroken object instead of a dotted trail of points — and why §0.5's properties can be stated for all real numbers at once.

The payoff slide — it is why the whole classification exercise mattered.

Decimals · bookSHelf Integrated Math 1§0.4

§0.4.4 — Your turn

Try It Now 0.4.4

Try It Now 0.4.4 — simplify $\;\sqrt{100} - 3\sqrt{4}$ ; is $\sqrt{-9}$ real; name every set $\sqrt{49}$ belongs to

\sqrt{100} - 3\sqrt{4} = 10 - 3(2) = 4

$\sqrt{-9}$ is not real — it would need a number whose square is $-9$ , and positives, negatives and zero all square to non-negatives.

$\sqrt{49} = 7$ — counting, whole, integer, rational, real.

Answers 4; not real; all five sets.

Decimals · bookSHelf Integrated Math 1§0.4

Key Terminology — the words this section defines

Key terms

decimal point — separates the whole-number places from the fractional ones.

trailing zero — a zero at the right end of the fractional part; never changes the value.

terminating decimal — its digits stop; the division reaches remainder zero.

repeating decimal — a block repeats forever, written with a bar over it.

percent — a fraction with denominator 100, written with $\%$ .

square root — a number whose square is $m$ , written $\sqrt{m}$ .

principal square root — the nonnegative one; what $\sqrt{\;\;}$ always means.

perfect square — the square of an integer.

rational number — a ratio of two integers; terminates or repeats.

irrational number — real but not rational; neither terminates nor repeats.

real number — rational or irrational; anything with a spot on the line.

Second column reveals on click.

Decimals · bookSHelf Integrated Math 1§0.4

The headline result

A number is rational exactly when its decimal terminates or repeats

Both directions hold, and the reason is finite: a division has only finitely many possible remainders, so one must eventually recur — and from that point the quotient digits cycle.

That single fact links the notation half of this section to the classification half, and it is why you can sort a number by looking at its decimal.

† It also tells you what irrationality has to look like: a decimal with no end and no pattern. $\pi$ and $\sqrt{55}$ qualify; $3.14$ and $\tfrac{22}{7}$ do not, which is precisely why they are approximations rather than equalities.

One headline result — the rest folds into the conclusions cards.

0.4

Decimals · bookSHelf Integrated Math 1§0.4

§0.4 — Conclusions

What to carry forward

The one idea

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.

Where it goes wrong

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.

Every listed failure has a size or place-value check that catches it.