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The Integers

Define and identify numbers that are integers.

The Integers

After completing this section, you should be able to:

Define and identify numbers that are integers.
Graph integers on a number line.
Compare integers.
Compute the absolute value of an integer.
Add and subtract integers.
Multiply and divide integers.

Defining and Identifying Integers

Extending the counting numbers to include negative numbers and zero forms the integers. Any other number that cannot be written as $\{\ldots -3,-2,-1,0,1,2,3,\ldots \}$ is not an integer.

Identifying Integers

Try it.

Which of the following are integers and which are not?

−3	Is an integer, as it is the negative of a counting number
$\sqrt{24}$	This is not written as an integer. Entering the square root of 24 in a calculator, such as desmos, the result is 4.899 (rounded off). Since this is not an integer, then $\sqrt{24}$ is not an integer.
36/4	Since 36 divided by 4 is 9, and 9 is an integer, then 36/4 is an integer.
45	Is an integer, as it is a counting number
63.9	Is not an integer, because it is not a counting number and not the negative of a counting number.
2/7	Dividing 2 by 7 results in a number less than 1, but greater than 0, so is between two consecutive integers. So, 2/7 is not an integer.
−16.0	Is an integer, since the decimal part is 0

Graphing Integers on a Number Line

Integers are often imagined as steps along a path. You start at 0, and going to the left is going backward, or in the negative direction, while going to the right is going forward, or in the positive direction. A number line () helps envision the integers. This also means that an integer gives magnitude (size) and direction (positive is to the right, negative is to the left). Graphing an integer on the number line means placing a solid dot at the integer on the number line.

Graphing Numbers on the Number Line

Try it.

Graph the following on the number line:

1
−4
3

Solution

Comparing Integers

When determining if one quantity or size is larger than another, we know it means there is more of whatever is being discussed. In terms of positive integers, we can envision that larger integers are further to the right on the number line. This idea applies to negative integers also. This means that $a$ is greater than $b$ when $a$ is to the right of $b$ on the number line. We write $a>b$. When $a$ is greater than $b$, we can also say that $b$ is less than $a$. On the number line, $b$ would be to the left of $a$. We write \(b

We need to recognize that $a>b$ means the same thing as $bb$. But this means $b$ is to the left of $a$, so \(b

Comparing Integers Using a Number Line

Try it.

Determine which of −6 and 4 is larger using a number line, and express that using both the greater than and the less than notations.

Solution

To illustrate this, we use a number line ().

Since −6 is to the left of 4, then −6 is less than 4. We can write this as −6 < 4. Another way of expressing this is that 4 is greater than −6. So we can also write $4>-6$.

Comparing Negative Integers

Try it.

Determine which of −6 and −2 is larger, and express that using both the greater than and the less than notations.

Solution

To illustrate this, we use a number line ().

Since −6 is to the left of −2, then −6 is less than −2. We can write this as $-6<-2$.

Another way of expressing this is that −2 is greater than −6. So we can also write $-2>-6$.

Comparing Integers by Quantity

Try it.

Determine which of 27 and 410 is larger, and express that using both the greater than and the less than notations.

Solution

When thinking about quantity, 410 is more than 27. So, 410 is greater than 27 and 27 is less than 410. We can write this as $410>27$ or as $27<410$.

The Absolute Value of an Integer

When talking about graphing integers on the number line, one interpretation suggests it is like walking along a path. Negative is going to the left of 0, and positive going to the right. If you take 30 steps to the right, you are 30 steps away from 0. On the other hand, when you take 30 steps to the left, you are still 30 steps away from 0. So, in a way, even though one is negative and the other positive, these two numbers, 30 and −30, are equal since both are 30 steps away from 0. The absolute value of an integer $n$ is the distance from $n$ to 0, regardless of the direction. The notation for absolute value of the integer $n$ is $|n|$.

If we think of an integer as both direction and magnitude (size), absolute value is the magnitude part.

Calculating the absolute value of an integer is very straightforward. If the integer is positive, then the absolute value of the integer is just the integer itself. If the integer is negative, then to compute the absolute value of the integer, simply remove the negative sign. Keeping in mind the number line as a path, when you’ve gone 10 steps to the left of 0, you have still taken 10 steps, and the direction does not matter.

Calculating the Absolute Value of a Positive Integer

Try it.

Calculate |19|.

Solution

Since the number inside the absolute value symbol is positive, the absolute value is just the number itself. So |19| = 19.

Calculating the Absolute Value of a Negative Integer

Try it.

Calculate |−435|.

Solution

Since the number inside the absolute value is negative, the absolute value removes the negative sign. So |−435| = 435.

Adding and Subtracting Integers

You may recall having approached adding and subtracting integers using the number line from earlier in your academic life. Adding a positive integer results in moving to the right on the number line. Adding a negative integer results in moving to the left. Subtracting a positive integer results in a move to the left on the number line. But subtracting a negative integer results in a move to the right.

This leads to a few adding and subtracting rules, such as:

Rule 1: Subtracting a negative is the same as adding a positive.

Rule 2: Adding two negative integers always results in a negative integer.

Rule 3: Adding two positive integers always results in a positive integer.

Rule 4: The sign when adding integers with opposite signs is the same as the integer with the larger absolute value.

These rules are good to keep in the back of your mind, as they can serve as a quick error check when you use a calculator.

Adding Integers

Try it.

Use your calculator to calculate 4 + (−7). Explain how the answer agrees with what was expected.

Solution

Using a calculator, we find that 4 + (−7) = −3. Since we are adding integers with opposite signs, the sign of the answer matches the sign of the integer with the larger absolute value which |−7|=7.

Subtracting Positive Integers

Try it.

Use your calculator to calculate 18 − 9. Explain how the answer agrees with what was expected.

Solution

Using a calculator, we find that 18 − 9 = 9. Since 18 was larger than 9, we expected the difference to be positive.

Subtracting with Negative Integers

Try it.

Use your calculator to calculate 27 − (−13). Explain how the answer agrees with what was expected.

Solution

Using a calculator, we find that 27 – (−13) = 40. Since we’re subtracting a negative number, it is the same as adding a positive, so this is the same as 27 + 13 = 40.

Adding Integers with Opposite Signs

Try it.

Use your calculator to calculate (−13) + 90. Explain how the answer agrees with what was expected.

Solution

Using a calculator, we find that (−13) + 90 = 77. Since we are adding integers with opposite signs, the sign of the answer matches the sign of the integer with the larger absolute value, which is positive since 90 is positive.

Calculating Net Worth

Try it.

Jennifer is owed $50 from her friend Janice, but owes her friend Pat $87. What is Jennifer’s net worth?

Solution

Net worth is the amount that one is owed minus the amount one owes. Jennifer is owed $50 but owes $87. So, her net worth is $50 – $87 = −$37. The negative indicates that Jennifer owes more than she is owed.

Condensed — the full section is in OpenStax Contemporary Mathematics.

Multiplying and Dividing Integers

Similar to addition and subtraction, the signs of the integers impact the results when multiplying and dividing integers. The rules are fairly straightforward, but again rely on the direction on the number line. There are only two rules.

Rule 1: When multiplying or dividing two integers with the same sign, the result is positive.

Rule 2: When multiplying or dividing two integers with opposite signs, the result is negative.

Just as before, these rules can serve as a quick error check when using a calculator.

Multiplying Positive Integers

Try it.

Use your calculator to calculate 4 × 8. Explain how the answer agrees with what was expected.

Solution

Entering 4 × 8 into your calculator, the result is 32. This agrees with our expectation. The numbers have the same signs, so the result is positive.

Multiplying Integers with Different Signs

Try it.

Use your calculator to calculate 9 × (−10). Explain how the answer agrees with what was expected.

Solution

Entering 9 × (−10) into your calculator, the result is −90. This agrees with our expectation. The numbers have opposite signs, so the result is negative.

Dividing Integers with Different Signs

Try it.

Use your calculator to calculate 400/(−25). Explain how the answer agrees with what was expected.

Solution

Entering 400/(−25) into your calculator, the result is −16. This agrees with our expectation. The numbers have opposite signs, so the result is negative.

Dividing Negative Integers

Try it.

Use your calculator to calculate −750/(−3). Explain how the answer agrees with what was expected.

Solution

Entering −750/(−3) into your calculator, the result is 250. This agrees with our expectation. The numbers have the same signs, so the result is positive.

At the end of a season, a team may wish to buy their coach an end-of season gift. It makes sense to share the cost equally among the members. To do so, the team would need to find the average (or mean) cost per member. The average (or mean) of a set of numbers is the sum of the numbers divided by the number values that are being averaged.

Condensed — the full section is in OpenStax Contemporary Mathematics.

Key Concepts

A set of numbers that can be built from the natural numbers are the integers, which consist of the natural numbers, zero (0), and the negatives of the natural numbers.
Integers are often graphed on a number line, which helps display the relative positions and values of those numbers.
The number line can be used to visualize when one integer is larger than or smaller than another integer.
Arithmetic operations with integers are similar to the operations with natural numbers, except that the sign (positive or negative) of the numbers will determine the sign (positive or negative) of the result.

Practice (17)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

Which of the following are integers and which are not?

−3	Is an integer, as it is the negative of a counting number
$\sqrt{24}$	This is not written as an integer. Entering the square root of 24 in a calculator, such as desmos, the result is 4.899 (rounded off). Since this is not an integer, then $\sqrt{24}$ is not an integer.
36/4	Since 36 divided by 4 is 9, and 9 is an integer, then 36/4 is an integer.
45	Is an integer, as it is a counting number
63.9	Is not an integer, because it is not a counting number and not the negative of a counting number.
2/7	Dividing 2 by 7 results in a number less than 1, but greater than 0, so is between two consecutive integers. So, 2/7 is not an integer.
−16.0	Is an integer, since the decimal part is 0

Graph the following on the number line:
1. 1
2. −4
3. 3
发送答案
Determine which of −6 and 4 is larger using a number line, and express that using both the greater than and the less than notations.

发送答案

To illustrate this, we use a number line ().
Since −6 is to the left of 4, then −6 is less than 4. We can write this as −6 < 4. Another way of expressing this is that 4 is greater than −6. So we can also write $4>-6$.
Determine which of −6 and −2 is larger, and express that using both the greater than and the less than notations.

发送答案

To illustrate this, we use a number line ().
Since −6 is to the left of −2, then −6 is less than −2. We can write this as $-6<-2$.
Another way of expressing this is that −2 is greater than −6. So we can also write $-2>-6$.
Determine which of 27 and 410 is larger, and express that using both the greater than and the less than notations.

发送答案

When thinking about quantity, 410 is more than 27. So, 410 is greater than 27 and 27 is less than 410. We can write this as $410>27$ or as $27<410$.
Calculate |19|.

发送答案

Since the number inside the absolute value symbol is positive, the absolute value is just the number itself. So |19| = 19.
Calculate |−435|.

发送答案

Since the number inside the absolute value is negative, the absolute value removes the negative sign. So |−435| = 435.
Use your calculator to calculate 4 + (−7). Explain how the answer agrees with what was expected.

发送答案

Using a calculator, we find that 4 + (−7) = −3. Since we are adding integers with opposite signs, the sign of the answer matches the sign of the integer with the larger absolute value which |−7|=7.
Use your calculator to calculate 18 − 9. Explain how the answer agrees with what was expected.

发送答案

Using a calculator, we find that 18 − 9 = 9. Since 18 was larger than 9, we expected the difference to be positive.
Use your calculator to calculate 27 − (−13). Explain how the answer agrees with what was expected.

发送答案

Using a calculator, we find that 27 – (−13) = 40. Since we’re subtracting a negative number, it is the same as adding a positive, so this is the same as 27 + 13 = 40.
Use your calculator to calculate (−13) + 90. Explain how the answer agrees with what was expected.

发送答案

Using a calculator, we find that (−13) + 90 = 77. Since we are adding integers with opposite signs, the sign of the answer matches the sign of the integer with the larger absolute value, which is positive since 90 is positive.
Jennifer is owed $50 from her friend Janice, but owes her friend Pat $87. What is Jennifer’s net worth?

发送答案

Net worth is the amount that one is owed minus the amount one owes. Jennifer is owed $50 but owes $87. So, her net worth is $50 – $87 = −$37. The negative indicates that Jennifer owes more than she is owed.
Use your calculator to calculate 4 × 8. Explain how the answer agrees with what was expected.

发送答案

Entering 4 × 8 into your calculator, the result is 32. This agrees with our expectation. The numbers have the same signs, so the result is positive.
Use your calculator to calculate 9 × (−10). Explain how the answer agrees with what was expected.

发送答案

Entering 9 × (−10) into your calculator, the result is −90. This agrees with our expectation. The numbers have opposite signs, so the result is negative.
Use your calculator to calculate 400/(−25). Explain how the answer agrees with what was expected.

发送答案

Entering 400/(−25) into your calculator, the result is −16. This agrees with our expectation. The numbers have opposite signs, so the result is negative.
Use your calculator to calculate −750/(−3). Explain how the answer agrees with what was expected.

发送答案

Entering −750/(−3) into your calculator, the result is 250. This agrees with our expectation. The numbers have the same signs, so the result is positive.
The daily low temperatures in Barrie, Ontario, for the week of February 14, 2021, were −20°, −12°, −15°, −23°, −17°, −13°, and −19° degrees Celsius. What was the average daily temperature for the week of February 14, 2021, in Barrie?

发送答案

Step 1: To find the average daily temperature, we first need to add the temperatures.
(−20) + (−12) + (−15) + (−23) + (−17) + (−13) + (−19) = −119
Step 2: That sum will then be divided by 7 since we are averaging over seven days, giving −119/7 = −17. So, the average daily temperature in Barrie, Ontario the week of February 14, 2021, was −17° Celsius.

Symbols used here

\sqrt{x},\ \sqrt[n]{x}

square root, n-th root
The non-negative number whose square (n-th power) is x.

\log_b x,\ \ln x

logarithm, natural log
The exponent b must be raised to for x; ln uses base e.

\sum_{k=1}^{n} a_k

summation
Add a_k for k = 1 up to n.

\prod_{k=1}^{n} a_k

product
Multiply a_k for k = 1 up to n.

\mathbb{N},\ \mathbb{Z},\ \mathbb{Q},\ \mathbb{R},\ \mathbb{C}

number sets
Naturals, integers, rationals, reals, complex numbers.

a \equiv b \pmod n

congruent modulo n
n divides a − b; a and b have the same remainder.

a \mid b,\ \gcd(a,b)

divides, greatest common divisor
b is a multiple of a; the largest number dividing both.

\varphi(n),\ \pi(x)

Euler's totient, prime-counting function
Count of 1..n coprime to n; number of primes up to x.

\mathbb{Z}/n\mathbb{Z},\ \mathbb{Z}_n

integers modulo n
The remainders 0…n−1 with clock arithmetic.

a \bmod n

remainder
What is left after dividing a by n.

How to: The Integers

Define and identify numbers that are integers.
Graph integers on a number line.
Compare integers.
Compute the absolute value of an integer.
Add and subtract integers.
Multiply and divide integers.
1
−4

Questions people ask

Why are primes so important?

Every integer factors into primes in exactly one way, so primes are the atoms of multiplication. Cryptography relies on that factoring being easy to state and hard to do.

How do I tell whether a big number is prime?

Trial division up to the square root works for small numbers. For large ones, probabilistic tests (Miller–Rabin) give an answer that is wrong with negligible probability, and deterministic tests (AKS) exist but are slower.

试试你自己试试

Parts of this page are adapted from OpenStax Contemporary Mathematics (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

−3	Is an integer, as it is the negative of a counting number
\(\sqrt{24}\)	This is not written as an integer. Entering the square root of 24 in a calculator, such as desmos, the result is 4.899 (rounded off). Since this is not an integer, then \(\sqrt{24}\) is not an integer.
36/4	Since 36 divided by 4 is 9, and 9 is an integer, then 36/4 is an integer.
45	Is an integer, as it is a counting number
63.9	Is not an integer, because it is not a counting number and not the negative of a counting number.
2/7	Dividing 2 by 7 results in a number less than 1, but greater than 0, so is between two consecutive integers. So, 2/7 is not an integer.
−16.0	Is an integer, since the decimal part is 0

The Integers

Defining and Identifying Integers

Graphing Integers on a Number Line

Comparing Integers

The Absolute Value of an Integer

Adding and Subtracting Integers

Multiplying and Dividing Integers

Key Concepts

Practice (17)

Symbols used here

How to: The Integers

Questions people ask

试试你自己试试

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