Introduction to Integers

At over $29{,}000$ feet, Mount Everest stands as the tallest peak on land. Describing the drastic change in elevation that climbers experience, and the change in temperature as they climb, requires numbers that extend both above and below zero. In this chapter, we describe these kinds of numbers and the operations using them.

Locate positive and negative numbers on the number line

Have you ever experienced a temperature below zero? A negative number is a number that is less than $0$. Very cold temperatures are measured in degrees below zero. For example, $-1^\circ\text{F}$ (read “negative one degree Fahrenheit”) is $1$ degree below $0$. A minus sign is shown before a number to indicate that it is negative.

Temperatures are not the only negative numbers. A bank overdraft is another example: if a person writes a check for more than is in the account, the balance is negative. Elevations, too, can be negative — the elevation at sea level is $0$ feet, elevations above sea level are positive, and elevations below sea level are negative. The elevation of the Dead Sea, which borders Israel and Jordan, is about $1{,}302$ feet below sea level, so its elevation is $-1{,}302$ feet. Depths below the ocean surface are described the same way: a submarine at a depth of $500$ feet is at position $-500$ feet.

Both positive and negative numbers can be shown on a number line. The counting numbers ($1, 2, 3, \dots$) are all positive; we could write a plus sign before a positive number, such as $+2$, but it’s customary to omit it — if a number has no sign, it is assumed positive. To include negative numbers, we extend the number line to the left of zero, marking off intervals the same width as those on the positive side, and label them $-1, -2, -3,$ and so on:

The arrows at either end of the line show that the number line extends forever in each direction: there is no greatest positive number, and no smallest negative number.

Example. Plot the numbers $3$, $-3$, and $-2$ on a number line.

To plot $3$, start at $0$ and count three units to the right. To plot $-3$, start at $0$ and count three units to the left. To plot $-2$, start at $0$ and count two units to the left.

Order positive and negative numbers

We can use the number line to compare and order positive and negative numbers. Going from left to right, numbers increase in value; going from right to left, numbers decrease. Just as with positive numbers, we use $a < b$ ("$a$ is less than $b$") when $a$ is to the left of $b$ on the number line, and $a > b$ ("$a$ is greater than $b$") when $a$ is to the right of $b$.

For example, $-2$ is to the left of $1$ on the number line, so $-2 < 1$ — and equivalently, $1 > -2$. Likewise, $-1$ is to the right of $-3$, so $-1 > -3$, and $-3 < -1$.

Example. Order each pair using $<$ or $>$: (a) $14 \_\_ 6$, (b) $-1 \_\_ 9$, (c) $-1 \_\_ -4$, (d) $2 \_\_ -20$.

Plot both numbers of each pair on a number line, then compare their positions: (a) $14$ is to the right of $6$, so $14 > 6$; (b) $-1$ is to the left of $9$, so $-1 < 9$; (c) $-1$ is to the right of $-4$, so $-1 > -4$; (d) $2$ is to the right of $-20$, so $2 > -20$.

Find opposites

On the number line, the negative numbers are a mirror image of the positive numbers with zero in the middle. Because $2$ and $-2$ are the same distance from zero, they are called opposites. The opposite of $2$ is $-2$, and the opposite of $-2$ is $2$.

The symbol “$-$” is used in three different ways, and its meaning depends on how it’s used:

Example. Simplify $-(-6)$. The opposite of $-6$ is $6$, so $-(-6) = 6$.

The set of counting numbers, their opposites, and $0$ is the set of integers.

We must be careful with signs when evaluating the opposite of a variable.

Example. Evaluate $-x$: (a) when $x = 8$; (b) when $x = -8$.

(a) Substitute $8$ for $x$: $-x = -(8) = -8$. (b) Substitute $-8$ for $x$: $-x = -(-8) = 8$.

Simplify expressions with absolute value

Numbers such as $5$ and $-5$ are opposites because they are the same distance from $0$ on the number line — they are both five units from $0$. The distance between $0$ and a number on the number line is called the absolute value of that number. Because distance is never negative, the absolute value of any number is never negative.

The symbol for absolute value is two vertical bars around a number: the absolute value of $5$ is written $|5|$, and the absolute value of $-5$ is written $|-5|$. Since both are five units from $0$, $|5| = 5$ and $|-5| = 5$.

Example. Simplify: (a) $|3|$, (b) $|-44|$, (c) $|0|$.

(a) $3$ is $3$ units from zero, so $|3| = 3$. (b) $-44$ is $44$ units from zero, so $|-44| = 44$. (c) $0$ is already at zero, so $|0| = 0$.

Absolute value bars act like grouping symbols. First simplify inside the bars as much as possible; then take the absolute value; then continue with any operations outside the bars.

Example. Simplify $|9-3|$ and $4|-2|$.

For $|9-3|$: simplify inside the bars first, $|9-3| = |6|$, then take the absolute value: $6$.

For $4|-2|$: take the absolute value first, $|-2| = 2$, then multiply: $4 \cdot 2 = 8$.

Example. Simplify $24 - |19 - 3(6-2)|$.

Follow the order of operations, treating the absolute value bars like grouping symbols. Simplify the innermost parentheses first: $24 - |19 - 3(4)|$. Multiply: $24 - |19-12|$. Subtract inside the bars: $24 - |7|$. Take the absolute value: $24 - 7$. Subtract: $17$.

Translate word phrases into expressions with integers

Now we can translate word phrases into expressions with integers. Look for words that indicate a negative sign — negative and opposite both signal one.

Example. Translate each phrase into an expression with integers: (a) the opposite of positive fourteen; (b) the opposite of $-11$; (c) negative sixteen; (d) two minus negative seven.

(a) $-14$; (b) $-(-11) = 11$; (c) $-16$; (d) $2 - (-7)$.

Negative numbers describe many real-world situations. Look for key phrases, then look for words that indicate a negative sign, and don’t forget to include units of measurement.

Example. Translate into an expression with integers: (a) the temperature is $12$ degrees Fahrenheit below zero; (b) the football team had a gain of $3$ yards; (c) the elevation of the Dead Sea is $1{,}302$ feet below sea level; (d) a checking account is overdrawn by $\text{\textdollar}40$.

(a) Below zero signals a negative number: $-12^\circ\text{F}$. (b) Gain signals a positive number: $3$ yards. (c) Below sea level signals a negative number: $-1{,}302$ feet. (d) Overdrawn signals a negative number: $-\text{\textdollar}40$.

Key terms

negative number — a number that is less than $0$. opposite — the number that is the same distance from zero on the number line, but on the opposite side of zero. integers — counting numbers, their opposites, and zero. absolute value — the distance of a number from $0$ on the number line, written $|n|$.

Practice

Locate positive and negative numbers on the number line

Order positive and negative numbers

Find opposites

Simplify expressions with absolute value

Translate word phrases to expressions with integers

This section is adapted from Prealgebra 2e, Section 3.1: Introduction to Integers by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, © OpenStax, licensed under CC BY-NC-SA 4.0. Access the original for free at openstax.org. Changes: recreated the number-line figures as accessible inline graphics; condensed prose; omitted the Be Prepared quiz, Manipulative Mathematics callout, and media links; converted the practice problems (“Try Its”) into interactive exercises with instant feedback; and adapted selected end-of-section exercises into the interactive Practice block, with each multipart exercise expanded into one question per part.