Integers and the Number Line: Chapter Review

Vocabulary and notation

Positive and negative numbers

A negative number is less than zero and sits left of zero on the number line; a positive number sits right. Read $-3$ as "negative three"; a bare $3$ is understood to be positive.

Number line

A line with $0$ marked and the integers on equally spaced ticks, negatives running left and positives right, with no last tick either way. Position on it decides order.

Integer

One of $\{\ldots, -2, -1, 0, 1, 2, \ldots\}$: a whole number or the opposite of one. A value landing between two ticks, such as $\tfrac{1}{2}$ or $2.7$, is not an integer.

Zero

Neither positive nor negative: the dividing point the two sides are measured from. It is its own opposite, and $-0$ is just $0$.

Opposite of $x$, written $-x$

The number that folding the line at zero lands $x$ on. The symbol $-x$ names the opposite of $x$, not "a negative number": when $x$ is negative, $-x$ is positive.

Absolute-value bars $\lvert \; \rvert$

A grouping symbol, like parentheses: simplify everything inside to one number first, then measure its distance from zero.

Signed quantity

A measurement whose sign records which side of a chosen reference it falls on: $-8$ degrees is eight below zero, $-40$ dollars is forty owed, $-30$ meters is thirty below sea level.

Formulas and theorems

• Order by position

  $$\begin{gathered} a < b \quad \text{exactly when} \\ a \text{ lies to the left of } b \end{gathered}$$

  Text description

  A number line with negative five marked further left than negative two, and arrows showing that leftward means smaller.

  Use when Every integer. Every negative number is therefore less than $0$ and less than every positive number.

  e.g. $-5 < -2$ while $5 > 2$; and $-1000 < 4$ without any arithmetic.

• Opposites

  $$-(-a) = a$$

  Use when Every integer has exactly one opposite, the same distance from zero. $0$ alone is its own opposite; every other pair sits on opposite sides.

  e.g. The opposite of $-15$ is $15$; the opposite of the opposite of $-8$ is $-8$.

• Zero pair

  $$a + (-a) = 0$$

  Use when Every integer, equivalently $a - a = 0$. A number and its opposite, such as $7$ and $-7$, form a zero pair, so matched units cancel.

• Adding two numbers with the same sign

  $$(-m) + (-n) = -(m + n)$$

  Use when $m$ and $n$ are the two distances from zero, both positive. Two negatives add their distances and keep the minus sign; matching signs never cancel.

  e.g. $(-9) + (-4) = -(9 + 4) = -13$.

• Adding two numbers with different signs

  $$\begin{gathered} m + (-n) = m - n \quad \text{if } m > n \\ m + (-n) = -(n - m) \quad \text{if } n > m \end{gathered}$$

  Use when $m$ and $n$ are the two distances from zero, both positive. Always subtract the smaller from the larger. The answer takes the sign of the number farther from zero; equal distances give $0$.

  e.g. $(-12) + 5 = -(12 - 5) = -7$.

• Subtraction is adding the opposite

  $$a - b = a + (-b)$$

  Use when Every pair of integers. Subtracting a NEGATIVE is where this bites: $a - (-n) = a + n$ for $n > 0$, a step to the right. Rewrite first, then use the addition rules. Order still matters: $a - b$ and $b - a$ are opposites.

  e.g. $3 - (-8) = 3 + 8 = 11$.

• Sign rule for multiplication

  $$\begin{gathered} \text{in } a \times b: \\ \text{matching signs give a positive result,} \\ \text{differing signs a negative one} \end{gathered}$$

  Use when Both factors nonzero: multiply the two distances from zero, then attach $+$ when the signs match and $-$ when they differ. A factor of $0$ has no sign, and any product with it is $0$.

  e.g. $(-7) \times (-8) = +(7 \times 8) = 56$.

• Counting negative factors

  $$\begin{gathered} \text{count the negative factors:} \\ \text{an even count gives a positive product,} \\ \text{an odd count a negative one} \end{gathered}$$

  Use when A product of any number of nonzero factors. If even one factor is $0$ the product is $0$ and the count decides nothing.

  e.g. $(-1) \times (-2) \times (-3) = -6$: three negatives, an odd count.

• Sign rule for division

  $$\begin{gathered} \text{in } a \div b: \\ \text{matching signs give a positive result,} \\ \text{differing signs a negative one} \end{gathered}$$

  Use when The divisor $b$ is not zero. Divide the two distances from zero, then attach the sign by the same match-or-differ rule as multiplication. A dividend of $0$ has no sign, and $0 \div b = 0$.

  e.g. $(-30) \div (-6) = 5$, checked by $5 \times (-6) = -30$.

• Division by zero

  $$a \div 0 \ \text{is undefined for every } a$$

  Use when Undefined means the expression names no number at all. It covers $0 \div 0$, where every number fits and none can be singled out. Only division BY zero fails.

• Absolute value

  $$\begin{gathered} \lvert x \rvert = x \ \text{ when } x \ge 0 \\ \lvert x \rvert = -x \ \text{ when } x < 0 \end{gathered}$$

  Use when $\lvert x \rvert$ is the distance from $x$ to $0$, so $\lvert x \rvert \ge 0$ always, while $\lvert x \rvert > 0$ fails at $x = 0$. On the second line $-x$ is the opposite of a negative number, so it is positive. A number and its opposite share one value: $\lvert x \rvert = \lvert -x \rvert$.

  e.g. $\lvert -12 \rvert = -(-12) = 12$, and $\lvert 0 \rvert = 0$.

• Numbers with a given absolute value

  $$\lvert x \rvert = c \ \text{ gives } \ x = c \ \text{ or } \ x = -c$$

  Use when $c > 0$ gives exactly two solutions, $c = 0$ exactly one ($x = 0$), and $c < 0$ none at all, since a distance is never negative.

  e.g. $\lvert n \rvert = 6$ has the two integer solutions $6$ and $-6$; $\lvert n \rvert = -6$ has none.

• Distance between two numbers

  $$\text{distance from } a \text{ to } b = \lvert a - b \rvert$$

  Use when Any two integers, subtracted in either order, since $a - b$ and $b - a$ are opposites. The distance is $0$ exactly when $a = b$.

  e.g. The distance between $-6$ and $2$ is $\lvert -6 - 2 \rvert = \lvert -8 \rvert = 8$.

Problem types, step by step

Exam traps

• Trap Ordering negatives by their digits, so $-5 > -2$ and $-100 > -3$.

  Fix Farther left is smaller, so among negatives a bigger numeral means a smaller number: $-5 < -2$ and $-100 < -3$.

• Trap Carrying "two negatives make a positive" into addition, so $(-3) + (-4) = 7$.

  Fix Two negative factors make a positive; two negative addends pile up: $(-3) + (-4) = -7$.

• Trap Adding the distances when the signs differ, or taking the sign from the wrong number, so $(-8) + 5$ comes out $-13$ or $3$.

  Fix Differing signs cancel: subtract $8 - 5 = 3$, then take the sign of $-8$, the number farther from zero, for $-3$.

• Trap Dropping the rewrite when a negative is subtracted, so $-9 - (-4)$ is worked as $-9 - 4 = -13$.

  Fix Subtracting a negative adds its opposite and moves right: $-9 - (-4) = -9 + 4 = -5$.

• Trap Answering $(-6) \times (-2) = -12$, on the feeling that negative factors keep an answer negative.

  Fix Each negative factor flips the sign once, so two of them flip it back: $(-6) \times (-2) = 12$.

• Trap Counting only the numerator's negative factors under a fraction bar.

  Fix Count above and below together. In $\dfrac{(-3) \times (-4) \times (-2)}{-6}$ there are four negatives, an even count, so the answer is $+4$, not $-4$.

• Trap Reading $\lvert x \rvert = -x$ as a negative output, so $\lvert -5 \rvert = -5$.

  Fix There $-x$ is the opposite of a negative number, so it is positive: $\lvert -5 \rvert = -(-5) = 5$. An absolute value is never negative.

• Trap Splitting the bars across an operation, so $\lvert (-3) + 5 \rvert$ is worked as $\lvert -3 \rvert + \lvert 5 \rvert = 8$.

  Fix The bars group what is inside, so finish the sum first: $\lvert (-3) + 5 \rvert = \lvert 2 \rvert = 2$.

• Trap Letting the bars swallow a minus sign that sits outside them, so $-\lvert -4 \rvert = 4$.

  Fix Only what is inside the bars loses its sign: the bars give $4$, and the outside minus makes it $-4$.

• Trap Answering "which is greatest" with the greatest absolute value, or ranking $-9$ above $6$ as a number.

  Fix Greatest asks which is farther right: $6 > -9$. Greatest absolute value asks which is farther from zero: $\lvert -9 \rvert = 9 > 6$. A larger debt is a smaller number.

• Trap Treating $\dfrac{-7}{0}$ and $\dfrac{0}{-7}$ as the same kind of expression.

  Fix $\dfrac{0}{-7} = 0$, an ordinary quotient. $\dfrac{-7}{0}$ is undefined: no number times $0$ gives $-7$.