Foundations of Arithmetic: Chapter Review

Vocabulary and notation

Base 10 (the decimal system)

Naming every number with the ten digits $0$ through $9$, letting a digit's column decide its size. Base $10$ means bundling in tens: ten ones make one ten, ten tens one hundred.

Expanded form

A number written as the sum of its digit values: $4{,}572 = 4{,}000 + 500 + 70 + 2$. A place holding $0$ contributes nothing and is left out.

Placeholder zero

A $0$ digit whose job is to hold a column open so the digits around it stay put. Deleting it slides everything right: $3{,}205$ becomes $325$.

Period

A group of three digits marked off by commas from the right: ones, thousands, millions. The commas ARE the period boundaries, which is why they fall every three digits.

Property (of an operation)

A statement holding for EVERY choice of numbers, not one computed fact. Proving one needs a general argument; disproving one needs a single failing case.

Commutative versus associative

Commute means move, so commutativity changes the ORDER of two numbers. Associate means group, so associativity changes which pair is combined first. Commutativity needs two numbers, associativity three.

Counterexample

One choice of numbers at which a claimed property fails. A single one disproves it; no number of successful examples can prove it.

Grouping symbols

Parentheses $(\;)$, brackets $[\;]$, and braces $\{\;\}$, all meaning the same thing: treat what is inside as one quantity. The differing shapes only make matching pairs easy to spot.

Greatest common factor

The largest number dividing every term of a sum, where the terms are above zero. It is what a fully factored answer pulls out front.

Formulas and theorems

• Value of a digit

  $$\text{digit value} = \text{digit} \times \text{place value}$$

  Use when Any digit in any place. The digit alone is only a symbol; multiply it by the worth of the column it sits in. A $0$ digit has value $0$ and still holds its column.

  e.g. In $6{,}308$ the $3$ sits in the hundreds column, so its value is $3 \times 100 = 300$.

• The ten-times ladder of places

  $$\ldots, 100{,}000, 10{,}000, 1000, 100, 10, 1$$

  Text description

  A ladder of the whole-number places from ten thousands down to ones, joined by times-ten arrows, with the ones place at the bottom.

  Use when Every place is exactly $10$ times the place to its right, so a digit one column left is worth $10$ times as much, and one column right a tenth as much. The ones place is the right-hand end of the ladder for the whole numbers this chapter works in.

  e.g. The $5$ in $508$ is worth $10$ times the $5$ in $58$, since $500 \div 50 = 10$.

• How many of a place a number contains

  $$\begin{gathered} \text{tens in } n = \frac{n}{10} \\ \text{hundreds in } n = \frac{n}{100} \end{gathered}$$

  Use when Divide the WHOLE number by the place value, which is not the digit sitting in that column. Exact only when the division comes out whole.

  e.g. $3{,}400$ contains $340$ tens, even though its tens digit is $0$.

• Commutative property

  $$a + b = b + a, \qquad a \times b = b \times a$$

  Use when Addition and multiplication only. Subtraction and division are not commutative: swapping the two inputs is NOT guaranteed to preserve the value, so you get no licence to reorder them.

  e.g. $5 - 3 = 2$, while $3 - 5$ takes $5$ away from only $3$ and cannot give $2$.

• Associative property

  $$\begin{gathered} (a + b) + c = a + (b + c) \\ (a \times b) \times c = a \times (b \times c) \end{gathered}$$

  Use when Addition and multiplication only, which is why $a + b + c$ and $abc$ need no parentheses. Chains of $-$ or $\div$ are pinned by the left-to-right convention instead.

  e.g. $(10 - 4) - 2 = 4$ but $10 - (4 - 2) = 8$, and $(16 \div 4) \div 2 = 2$ but $16 \div (4 \div 2) = 8$.

• Identity elements and the zero property

  $$\begin{gathered} a + 0 = a, \qquad a \times 1 = a \\ a \times 0 = 0 \end{gathered}$$

  Use when Every number. $0$ is the additive identity, $1$ the multiplicative identity. The third is a separate fact, not an identity: one zero factor collapses the whole product.

  e.g. $(37 + 58) \times (6 - 6) = 0$, with the first factor never needed.

• A chain removes the total, or divides by the product

  $$a - b - c = a - (b + c), \qquad a \div b \div c = a \div (b \times c)$$

  Use when Removing $b$ then $c$ removes $b + c$ in total; dividing by $b$ then $c$ cuts into $b \times c$ parts. Both follow from the left-to-right reading, not associativity. So $b$ and $c$ may trade places; $a$ stays put. Keep $b + c \le a$, and each division exact.

  e.g. $30 - 12 - 8 = 30 - 20 = 10$, and $720 \div 12 \div 3 = 720 \div 36 = 20$.

• The order of operations

  $$\begin{gathered} \text{Grouping} \;\to\; \text{Exponents} \;\to \\ \times,\ \div \;\to\; +,\ - \end{gathered}$$

  Use when Multiplication and division share ONE tier, addition and subtraction share ONE tier, and inside a tier the work runs strictly left to right. Nested groups settle innermost first, with the tiers applying inside each group.

  e.g. $2 \times (3 + 1)^2 - 24 \div 6 = 2 \times 16 - 4 = 28$: group, then power, then the $\times, \div$ tier, then the subtraction.

• The fraction bar as a grouping symbol

  $$\frac{a + b}{c + d} = (a + b) \div (c + d)$$

  Use when The bar wraps invisible parentheses round the WHOLE numerator and the WHOLE denominator, so each is settled before dividing. Needs $c + d \neq 0$. Anything outside the fraction stays outside.

  e.g. $\dfrac{8 + 4}{2 + 2} = \dfrac{12}{4} = 3$, not the flattened $8 + 4 \div 2 + 2 = 12$.

• Distributive property

  $$\begin{gathered} a(b + c) = ab + ac \\ a(b - c) = ab - ac \end{gathered}$$

  Text description

  A rectangle of height a, cut into parts of width b and c labelled ab and ac, whose full width is b plus c.

  Use when The outside factor reaches EVERY term inside, for any number of terms: $a(b + c + d) = ab + ac + ad$. Over addition and subtraction only, never over a product. With only whole numbers, the difference form needs $b \ge c$.

  e.g. $7(100 - 2) = 700 - 14 = 686$, while $4(3 \times 5)$ is just $4 \times 15 = 60$.

• Factoring, the same rule backward

  $$ab + ac = a(b + c)$$

  Use when $a$ must divide every term. Fully factored means $a$ is the greatest common factor, so what is left inside shares nothing above $1$. Distributing back returns the original, which is the check.

  e.g. $45 + 27 = 9(5 + 3)$, while $3(15 + 9)$ is a true equality that is not finished.

Problem types, step by step

Exam traps

• Trap Reading the tiers as a strict ranking, so multiplication always beats division and addition always beats subtraction.

  Fix Each pair shares one tier and runs left to right. $16 \div 4 \times 2 = 8$, not $2$, and $10 - 4 + 1 = 7$, not $5$.

• Trap Finishing a parenthesis and then sweeping left to right through whatever is left.

  Fix A settled group is only a number, and the tiers resume around it: $30 - 2 \times (3 + 4) = 30 - 14 = 16$, not $28 \times 7 = 196$.

• Trap Flattening a stacked fraction onto one line without writing the parentheses the bar was supplying.

  Fix The bar groups the whole top over the whole bottom: $\dfrac{3^2 + 6}{1 + 2} \times 2 = \dfrac{15}{3} \times 2 = 10$, while $3^2 + 6 \div 1 + 2 \times 2 = 19$ divides one term instead.

• Trap Calling a regrouping "commutative" because the expression merely looks rearranged.

  Fix $(5 \times 8) \times 2 = 5 \times (8 \times 2)$ leaves the order $5$, $8$, $2$ untouched and only moves the parentheses, so it is associative. Commutative would need two numbers to trade places.

• Trap Letting the outside factor reach only the first term inside the parentheses.

  Fix Every term gets multiplied: $9 \times 53 = 9(50 + 3) = 450 + 27 = 477$, never $450 + 3 = 453$. With three terms inside, none may be skipped either.

• Trap Answering "hundreds" when asked for the VALUE of the $3$ in $6{,}308$, or answering $300$ when asked for its PLACE.

  Fix The place is the column's name and the value is the digit times that column's worth, so the place is the hundreds and the value is $300$.

• Trap Reassembling $40{,}000 + 2{,}000 + 90 + 5$ as $42{,}905$.

  Fix A column no part mentions gets a $0$. There is no hundreds term, so the hundreds digit is $0$ and the number is $42{,}095$.

• Trap Answering "how many tens are in $9{,}060$" with the tens digit, $6$.

  Fix The question counts groups of ten in the whole number, so divide: $9{,}060 \div 10 = 906$ tens.

• Trap Reading "a subtraction chain may be reordered" as licence to move ANY of its numbers.

  Fix Only the numbers being REMOVED may trade places: $80 - 25 - 15$ and $80 - 15 - 25$ are both $40$. The first number never moves, and $80 - (25 - 15) = 70$ shows regrouping stays barred.