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Integers

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Integers

Integers are whole numbers that are either positive, zero, or negative, and do not contain decimals.

Facts about integers

Integers can be generated from the set of counting numbers and the subtraction operation. For example, when you subtract a larger natural number from a smaller one, you have a negative number. When a natural number is also subtracted from itself, you have zero.

The result of adding, subtracting, or multiplying integers is always an integer. This cannot be true with dividing integers. Dividing 5 by 2 will give you 2.5, which isn't an integer.

Positive integers are known as natural numbers. An important characteristic of natural numbers can be seen in the equation a + x = b. This only has a solution if b> a, as a and x can only be positive and their addition will produce a larger number. In the realm of integers, the equation a + x = b will always have an answer.

A good way to represent integers on a number line is shown in the figure below.

Integers, An image of an integer number line, StudySmarterAn image of an integer number line

A set of integers is denoted by Z, which is written as Z = {…, -4, -3, -2, -1, 0, 1, 2, 3, 4, ...}. Z here has a property that shows that it has an infinite number of elements (...- 4, -3, -2, -1, 0,) in figure 1 in the set.

Real-life example of integers

Integers help to capture values in every field.

  • In weather forecasting, they can be used to show the temperature in different regions. Where temperatures can go below zero in Fahrenheit and Celsius scales, integers will be negative

  • Integers are used to represent values in every transaction we make, from banks to everyday cash machines.

Consecutive integers

Consecutive integers are integer numbers that follow each other in a sequence without gaps. They represent an unbroken sequence of numbers where one follows the other by the addition of one. If we had x as an integer, then x + 1 and x + 2 will be the two consecutive integers. These numbers are in ascending order, and some examples are:

  • -5, -4, -3, -2, -1, 0, 1, 2

  • 200, 201, 202, 203, 204, 205

  • -1, 0, 1, 2, 3, 4, 5

  • -13, -12, -11, -10, -9, -8, -7

Assuming you had to solve a mathematical equation and you know the sum of two consecutive integers is 97. What are the two integers?

answer:

Let's assume that the first integer is x. We know from the description of a consecutive integer that the second must be x + 1. We can write an equation for this.

x + (x + 1) = 97

2x + 1 = 97

2x = 97 - 1

x = 48

This means the first integer is 48. And the second will be 48 + 1, which is 49.

Odd consecutive integers

These are odd integers that follow each other yet differ by two. When x is an odd integer, then consecutive odd integers are x + 2, x + 4, x + 6. Examples are:

  • {5, 7, 9, 11, 13...}

  • {-7, -5, -3, -1, 1..}

Even consecutive integers

These are also even integers that follow each other yet differ by two. When x is an even integer, then consecutive even integers are x + 2, x + 4, x + 6. Examples are:

  • {2, 4, 6, 8, 10, 12..}

  • {-10, -8, 6, 4..}

Integer rules for mathematical operations

It's useful to learn the rules for integers in mathematical operations.

Addition

  • Adding two positive integers will always give you a positive integer.

  • Adding two negative integers will always give you a negative integer.

  • Adding one positive and one negative integer will give you:

    • A positive number if the positive integer is greater

    • A negative number if the negative integer is greater

Multiplication

  • The product of a positive integer and a negative integer will always give you a negative integer.

  • The product of two positive integers will always be a positive valued integer.

  • The product of two negative integers will always be a positive integer.

Division

  • Dividing two positive integers will always give you a positive value.

  • Dividing two negative integers will give a positive value.

  • Dividing a negative integer by a positive integer will give you a negative value, and the opposite applies too.

Adding and subtracting integers

Let's take a few examples to get familiar with these operations.

Sam owes his friend Frank $5. He goes to borrow an additional $3, how much will he owe in all?.

Answer:

This is quite simple. We add both and know he owes $8.

However, this can be expressed mathematically as - $5 + (- $3) = - $8. This can in turn be written as: $5 - $3 = - $8

This can be confusing – using a number line makes it much easier.

Integers, An image of a number line expressing integer additions, StudySmarterAn image of a number line expressing integer additions

Number line expressing integer additions

Using your first figure as a reference point, move three steps back on the integer number line. Whilst positive values move right (forward), negative ones move left (backward). And with our example, we have -8 as our answer again.

Let's say Sam eventually pays back $4 out of the $8 he owes. How much is left to pay?

Answer:

This is another simple calculation. Intuitively, we know that the answer is $4.

However, we can write this mathematically as - $ 8 + $ 4 = - $ 4, as well as draw a number line again.

Integers, An image of a number line expressing integer additions, StudySmarter

Using your first figure as a reference, move four steps forward on the integer number line. This shows that -4 is our answer.

You might be presented with an equation like -3 - (-6) = x.

Answer:

When two negative signs meet as they do in this equation, they both become positive.

So we can have -3 + 6 = x

x = 3

Multiplying and dividing integers

Let's look at examples that prove the rule of multiplication.

What is the product of -3 and 7?

Remember – the product of a positive and a negative integer will be a negative one.

What is the product of 5 and 4?

As we mentioned the product of two positive integers, will be a positive one, in this case 20.

What is the product of -6 and -8?

Divide

Remember, dividing two positive integers will give you a positive integer.

Divide

Integers - Key takeaways

  • Integers are whole numbers that are either positive, zero, or negative.
  • The result of adding, subtracting, or multiplying integers is always an integer.
  • Consecutive integers are integer numbers that follow each other in a sequence or in order without gaps.
  • A set of integers is denoted by Z.
  • You cannot always have an integer when two integers are divided.

Frequently Asked Questions about Integers

Integers are whole numbers that can be positive or negative.

In integer division, the fraction is discarded.

Consecutive integers are integer numbers that follow each other in a sequence or in order without gaps.

Whole numbers are a set of natural numbers that start with zero while integers are a set of positive and negative natural numbers including zero.

-3, 2, 4, 0, 6

Final Integers Quiz

Question

What are integers?

Show answer

Answer

 Integers are numbers without fractional components.

Show question

Question

Which of these are examples of an integer?


Show answer

Answer

0 and 2/2 


Show question

Question

How do you identify integers?

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Answer

They are positive and negative non-fractional natural numbers, including zero.


Show question

Question

Are all natural numbers integers?


Show answer

Answer

Yes

Show question

Question

Integer numbers that follow each other in a sequence or in order without gaps are called…


Show answer

Answer

​Consecutive

Show question

Question

Are these two examples of consecutive integers?

  • {-3, -2, -1, 0, 1, 2..}

  • {10, 11, 12, 13, 14..}

Show answer

Answer

Yes

Show question

Question

A set of integers is denoted by ...

Show answer

Answer

Z

Show question

Question

What are odd consecutive integers?


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Answer

These are a set of integers such that each integer in the set differs from the previous integer by a difference of 2 and each integer is an odd number

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Question

What is the difference between whole numbers and integers?


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Answer

Whole numbers are counting numbers including 0, while integers are positive and negative whole numbers.

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Question

Which of these is a rule to consider when adding and subtracting integers?

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Answer

Adding two positive integers will always give you a positive integer.

Show question

Question

What is the product of -4 and 7?


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Answer


Show question

Question

Which of these is an integer multiplication rule?


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Answer

The product of two positive integers will always be a positive integer.

Show question

Question

What is integer division?


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Answer

Integer division is the division where the fractional part is discarded.

Show question

Question

What is the solution to 


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Answer


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Question

What is the product of -5 and -6?


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Answer


Show question

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