Integers: Count Operations, and Example Problems

Integer Numbers – When discussing mathematics, it must be very synonymous with numbers. Because of course mathematics is a study that studies calculation problems that require numbers as the main subject of the lesson.

There are many types of numbers. There are complex numbers, real numbers, imaginary numbers, rational numbers, irrational numbers, integers, fractional numbers, whole numbers, real numbers and many others.

Of all the numbers that can be studied in mathematics lessons, there is one number that is quite common, namely, integers. Integers are divided into two types, positive and negative integers. Positive integers can also be referred to as natural numbers or sets of positive values. Negative integers are the set of integers that have a negative value.

Number is a mathematical concept that assigns a total value to the thing being counted. That is why numbers are used in measuring and counting. Well, a number has a symbol or symbol. These symbols are called numbers.

For more details in understanding what integers are, in this discussion we have summarized various information about integers, how to calculate them, and examples of questions that all of you Sinaumed’s friends can listen to.

You can see further discussion about integers below!

Definition of Integer

What is an integer? Integers are sets or groups whose values ​​are integers. Integer itself consists of positive integers and negative integers. The set of integers is denoted by the letter Z. The letter Z comes from the word zahlen (German) which means number.

Sinaumed’s friends already know about integers, right? Yes! Integers consist of zero and positive integers. So integers consist of original integers, zero and negative integers. Therefore, decimals and fractions are not included in the set of integers.

Natural numbers, or positive integers, consist of the numbers 1, 2, 3, etc. Natural numbers are divided into odd, even, prime and composite. Odd numbers are the set of natural numbers whose value is not divisible by two. Conversely, even numbers are the set of natural numbers whose value is divisible by two.

Conversely, a prime number is a natural number greater than 1 which is only divisible by one or itself. Natural numbers greater than 1 which are not prime numbers are called composite numbers. Integers are numbers that consist of positive integers, zero, and negative integers.

From this we can conclude that there are two forms of integers, namely positive integers and negative integers. As the name suggests, positive integers are the positive numbers to the right of zero on the number line. For example 1,2,3,4,5,6,7,8,9 etc. In contrast, negative integers are negative numbers to the left of zero on the number line. For example -1,-2,-3,-4,-5,-6,-7,-8,-9 etc.

Integers can be thought of as discrete points, equally spaced along the number line.

Integer Calculation Operations

Arithmetic operations are required to calculate integers. In mathematics, arithmetic operations are defined as the process of working with a number, namely in the form of addition, subtraction, multiplication, division, and so on.

  • sum

If you add integers of the same suit, you get a number of the same suit. That is, if you add positive integers, the result is a positive integer. The same goes for negative numbers. But if the addition occurs in positive and negative numbers. Then the type is determined by the type of integer with the largest value.

There are three ways to add integers:

  • Adding positive integers with positive integers results in positive integers. For example: 8 + 9 = 17.
  • Add negative integers to negative integers to get negative integers. For example: (-13) + (-8) = -21
  • Adding negative integers with positive integers or vice versa gives the result:
  • A negative integer if the negative integer is greater than the positive integer. For example: (-8) + 6 = -2.
  • A positive integer if the negative integer is less than the positive integer. For example: (-8) + 10 = 2.
  • A negative integer if it is equal to a positive integer. For example: (-8) + 8 = 0.
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The properties of addition in integer arithmetic include:

  • Commutative property → a + b = b + a.
  • Compound attribute → (a + b) + c = a + (b + c).
  • Zero property (0) → a + 0 = 0 + a.
  • Mutual property of numbers → a + (-a) = 0.
  • Subtraction

In integer subtraction, if the minus sign “-” in the integer meets the subtraction symbol, the result of the calculation will be added up. The computation of reduced integers can be divided into:

Subtraction of positive integers by positive integers, the result is:

  • A positive integer if the number of numbers being subtracted is greater than the number being subtracted. For example: 6 – 5 = 1
  • A negative integer if the number of positive integers subtracted is less than the sum of the positive integers subtracted. For example: 8 – 9 = -1.
  • Zero if the sum of the positive integers subtracted equals the sum of the positive integers subtracted. For example: 9 – 9 = 0.

Subtracting a negative integer from a negative integer gives:

  • A positive integer if the sum of the minus integers being subtracted is less than the sum of the negative integers being subtracted. For example: (-6)-(-8)=2.
  • A negative integer if the number of negative integers being subtracted is greater than the number of negative integers being subtracted. For example: (-8) – (-5) = -3.
  • Zero if the sum of the negative integers being subtracted is equal to the sum of the negative integers being subtracted. For example: (-7) – (-7) = 0.

Subtracting a negative integer by a positive integer always results in a negative integer. For example: (-5) – 5 = -10

Subtract a positive integer by a negative integer, the result is always a positive integer. For example: 6 – (-7) = 13

Properties of subtraction in integer arithmetic include:

  • a – b = (a+c) – (b+c).
  • a(b + c) = (ab)–c.
  • (a+b)-c=a+(bc).
  • Multiplication

If two positive numbers are added, the result is a positive integer. Whereas multiplication involving two negative integers will result in a positive integer. However, if a positive integer and a negative integer are multiplied, the result is a negative integer.

How to calculate the multiplication of integers can be seen as follows:

  • Multiplying a positive integer by a positive integer gives a positive integer. For example: 8×5=40.
  • Multiply a positive integer by a negative integer, or vice versa, the result is a negative integer. For example: 6 x -3 = -18.
  • Multiplying a negative integer by a negative number gives a positive integer. For example: -7 x -4 = 28.
  • Multiply the integer by zero, the result is zero. For example: 0x0=0.

The properties of multiplication in integer arithmetic include:

  • Commutative property → axb = bxa.
  • Associative property → ax (bxc) = (axb) x c. The distributive property of multiplication over addition → ax (b + c) = (axb) + (axc).
  • The distributive property of multiplication over subtraction → ax (b – c) = (axb) – (axc).
  • Distribution

Regardless of whether it is a positive or negative number, if two integers of the same type are divided, the result will be a positive integer. However, if you divide a positive integer by a negative integer, the result is a negative integer. This concept is basically the same as the multiplication arithmetic operation.

How to calculate the division of integers can be seen as follows:

  • Divide a positive integer by a positive integer to get a positive integer. For example: (8): (2) = (4).
  • Dividing positive integers by negative integers or vice versa to get negative integers. For example: (6): (-3) = (-2).
  • Multiplying a negative integer by a negative number gives a positive integer. For example: (-8):(4) = (-2).

Properties of division in integer arithmetic include:

  • The nature of the distribution of division over addition → (a + b) : c = (a : c) + (b : c).
  • The nature of the distribution of division against subtraction → (a – b) : c = (a : c) – (b : c).

How to Compare Integers

To compare integers, you must first know the order of the integers. Sorting integers means sorting integers from the smallest value to the largest value or vice versa. Based on the order of the numbers, the further to the right of the number, the higher the value. While the more to the left of the number, the smaller the value.

Now, after knowing the order, we can compare these integers. Comparing integers means determining the value of an integer if it is greater than, equal to, or less than another integer. The symbols used to compare integers are:

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Assuming a and b belong to the set of integers, then

– If a is greater than b, then a > b

– If a equals b, then a = b

– If a is less than b, then a < b

Example :

5 > -1

-4 < 2

3 = 3

Examples of Integers

After knowing the meaning of integers, you should also know examples of integers.

Integers are divided into three, namely negative integers (-), zero (0) and positive integers (+), which are explained below:

  • Negative integer (-)

Is the number to the left of the number (0) on the number line. Examples of negative integers: -1, -2, -3, -4, etc.

  • Zero (0)

These are independent numbers and are right in the middle of the sequence.

  • Positive integer (+)

Is the number to the right of 0 on the number line. Examples of positive integers (+):

1, 2, 3, 4, etc.

Examples of Integer Problems

  • Calculate the following deductions:

Problem : 321-(-125)

Results:

321-(-125)= 321+ 125 = 446

Problem: 216-326

Results:

216-326-(326-216)= -110

Problem: -222-(-111)

Results:

-222-(-111)=-222+111–(222-111) = -111

  • Calculate the following multiplication and division:

Problem: 245x(-123)

Results:

-245 × (-123) = 30.135

Problem: -234×25

Results:

234 x 25 = -5.850

  • Calculate the following example questions:

Problem: 47×77+ 47×23

Results:

47 × 77 + 47 × 23 = 47 × (77 + 23)

= 47 x 100

= 4,700

Problem: 26×891+ 26×109

Results:

26×891 + 26×109 = 26x (891 + 109)

= 26 x 1000

= 26,000

  • Simplify the following problem:

Problem: 23x22x25

Results:

23x22x25 = 23+2+5 = 1024

  • Other Example Questions:

What is the result of -9 x 17?

Results:

-153

Explanation:

The result -9 x 17 can be determined by the distributive property of multiplication:

ax (b + c) = (axb) + (axc)

So :

-9×17 = -9x (10=7)

= (-9×10) + (-9×7)

= -90 + (-63)

= -153

Here are some other sample problems on the topic of integers:

QUESTION:

  • The result of 5 + [6:(-3 is ?

Answer:

5+[6:(3)] ​​= 5 + (2) = 5-2 = 3

  • Mrs. Salwa has 92 mangoes. All mangoes were distributed almost equally among 28 neighbours. The number of mangoes each neighbor received was approximately… fruit (use the best estimate)

Answer:

Math phrases Word problems: 92: 28 best guess: 92 -> 90 28 -> 30 Most accurate prediction: 92: 28 = 90: 30 = 3 So the number of mangoes each hamlet receives is 3.

  • Rizki has IDR 20,000 in cash. He used the money to buy 2.5 kg of rice. It turns out that the price of rice per kg is IDR 10,000. Considering that the distance between Rizki’s family and the rice shop was very far, Rizki finally decided to take out debt first because of the shortage.

Answer:

Rizki’s money = IDR 20,000 Price of 2.5 kg of rice = IDR 10,000 × 2.5 = IDR 25,000 Debt = price of rice – Rizki’s money Debt = IDR 25,000 – IDR 20,000 = IDR 5,000 or can be written -5000

  • Pay attention to the following numbers:

-15, -17, -21, -9, -51. What is the correct order of these numbers if sorted from smallest to largest?

Answer:

If sorted from the smallest it will be -51, -21, -17, -15, -9.

  • At first the temperature of a room is 18°C, after noon the temperature rises 5°C. And at night, the temperature in the room drops 7°C. So the room is now… ºC.

Answer:

18ºC + 5ºC – 7ºC = 23ºC – 7ºC = 16ºC

  • The result of (−18 + 30): (−3−1) is?

Answer:

(−18 + 30): (−3 − 1) = 12: (4) = 3

  • Pak Raeng has 36 sheets of colored paper. All sheets of colored paper are divided equally among the three children. Each child receives colored paper.

Answer:

Number of sheets = 36 sheets Number of children = 3 people The number of sheets of paper each child receives is: 3 = 12. So each child gets 12 sheets of colored paper.

Conclusion

That’s a brief discussion of Integer Numbers. Not only discussing the meaning of integers, but also discussing integer arithmetic operations, how to compare integers, and examples of problems that can be listened to properly.

Knowing about the meaning of integers is a new additional knowledge in understanding the basics of studying mathematics for those of you who are interested in these mathematics subjects.