Calvin Industries ▼
Services and Companies ▼
▼ Connect With Us
MATHEMATICS | PORTFOLIOS
Understanding Integer Operations
Written By: Calvin Musk · CEO at Calvin Industries
Department : Calvin Industries News Network - Department of Intelligence and Media
Date : Friday, November 25th, 2022
Introduction
What are integers?
When we are talking about integers, we are talking about whole numbers. This does not include fractions or decimals. We are talking about whole numbers- in both positive and negative values, and understanding how to apply certain operations to these whole numbers. Integers also include the number 0. The word "integer" comes from the Latin integer, meaning whole or untouched. We can use certain visuals, models, mental math, or a basic understanding of integer chips to easily model integers and operations. Often times when given a large sequence of numbers, we have to follow the rules of BEDMAS, often times known as PEDMAS. Integers can be seen in various different scenarios in the real world- like temperature, where we have an integer below 0 as our temperature. It can be seen in money, altitude, magnitude, scores in different games, and uses simple arithmetic operations. Before we begin to discuss about integers themselves, it is critical to understand the properties of BEDMAS (or PEDMAS) to better understand how to properly solve expressions and equations. In this unit, we will be understanding PEDMAS, what integers are and how to use operations within, and how to use BEDMAS on negative and positive values on both positive and negative integers and fractions.
What is order of operations?
BEDMAS, OR PEDMAS is the universal order to solve equations. This rule is enforced so we all have a proper understanding of the same guidelines, and so we can better understand and solve equations that each equal the same amount. Imagine if BEDMAS didn't exist- people would be doing operations in all sorts of random orders. In terms, there would be a lot of conflict and miss-understanding of the math world. To combat this issue, mathematician, Achilles Reselfelt, invented BEDMAS- the universal rule to solving equations. BEDMAS says that from left to right, we must solve the equation in a specific order by doing the specific operations first :
B - Brackets, or sometimes known as parenthesis is ( ), and enclose a group of terms of numbers in one.
E - Exponents, aka that tiny number on the top right next to a number. Exponents is when you time a number by itself by x amount of times.
D - Divison is reverse multiplication- its when we split them into a specified amount of equal groups and see's how many times a number can fit into x.
M - Multiplication is basically grabbing a specific amount of groups and duplicating it by x. It doesn't matter if we do multiplication or divsion first.
A - Addition is adding and combining numbers to create a sum. Multiplication basically is taking this concept and repeating it. (repeated addition)
S - Finally, we have subtraction. Subtraction is removing an amount from a number and finding the difference between it. It doesn't matter if you do addition, or subtraction first.
Example equation ) 9 – 24 ÷ 8 × 2 + 3
Look for brackets, and there are none. Look for exponents, and there are none. Look for division, and we have the 2 factors (24 and 8) divided by each other. BEDMAS says that we must divide first. 8 can go into 24 a total of 3 times, meaning 24 / 8 is 3. We can reword the equation to 9 - 3 x 2 + 3. BEDMAS now says the multiplication is right after division. 3 times 2 is equal to 6. We can reword the equation to 9 - 6 + 3. With addition/subtraction, or multiplication/divison, it really doesn't matter which order you go. Just solve the equation by solving it from left to right. 9 - 6 equals to 3. 3 + 3 equals to 6, which is our final answer. Integers is the same idea- but instead of using just positive values, we can see more of values like (-24), or (-3) instead of positive integers.
How do we model integer operations?
There are 2 ways to model integer operations.
Integer Chips
Integer chips are great at modeling and representing integers. There are 2 types of integer chips- yellow, and red. Yellow represents addition, and red represents subtraction. We always assume that we start at 0 unless told otherwise. We can use these integer chips to model and better represent our values and numbers. We can model multiplication since we know multiplication is repeated addition- basically taking the same number, and duplicating it into groups of a specified number. We can use this idea to form a group of integers, and duplicate it by X amount. Using division, we can take the same idea and concept- since we know division is about seeing how many times our second factor can fit into our first factor. Taking advantage of this, we can use it to find a sum, difference, product, or quotient based off of our equation/expression. Use the example :
View Example ) 62 + -14 ÷ 2 – (-8)
Though not the best represented the best through this expression, we have some mix of operations- and positive and negative values. We have some exponents, and 2 negative values -14 and -8. Usually, we would follow the brackets in BEDMAS, but, these sets of brackets do not represent an operation. They represent our negative value, and therefore, there are technically no brackets in this equation. But, we do have exponents, which we do not have to adjust much considering how exponents is needed to be done first anyway. So, how can we model this equation using integer chips?
Number Lines
Number lines are also great at modeling integer operations! We always assume that we begin at 0, unless we're told otherwise. We can simply move left to represent negative values, or move right to represent positive/addition values. Now that we have gotten a proper introduction and basic understanding to integers, lets explore operations, and how to use it with integers and whole numbers.
Multiplication Operations
This visual shows how to use integer chips to model the expression 5 x (-4) using repeated subtraction and the idea of opposite integers. We have 5 groups of (-4), which amounts of -20. Through the visual, we can see the magic behind multiplying a positive with a negative- with how one value is positive, and how one value is a negative.
There are numerous different uses for multiplication, otherwise known as repeated addition. The concept "of repeated addition" comes from how 9 + 9 + 9 is the same as 9 times 3- we are grabbing a group of a specific number, and duplicating it (multiplying it) by a specific number to get our product. In integers, multiplication uses the same idea and concept- but, sometimes, we are using repeated subtraction rather than addition. For instance, the example of the left 5 x (-4) means to subtract 4- by a total of 5 times. We have 5 groups of 4 negative integer chips, and if we were to model it using integer chips, the total would amount to -20. Multiplication involving integers is quite simple when we have a basic understanding of the following rules.
A positive times a positive will ALWAYS equal a positive.
A positive times a negative will ALWAYS equal a negative.
A negative times a negative will ALWAYS equal a negative.
The moral of the story? That negative dominates. When we multiply 2 factors, no matter which factor is negative or positive- we will always get the same product in terms of number. Its just that the product may be positive or negative. For instance- 5 x (-4), 5 x 4, and (-5) x (-4) all equal 20. But, they all differ in whether or not its a negative or positive value.
We can represent this question through the expression 2 x (-50) + 1 x (-70). We are going backwards 50 by a total of 2 times (going 50 km/h for 2 hours), and we are driving for 70 km/h for one hour. If we add them together, our product and sum will amount to the distance that the car traveled to get to the current state of the intersection. We can use substitution if the world problem ever changes, however. If the distance changes, we can simply swap out -50, or -70 with a x value, and use substitution to find an answer according to the word problem/
This data shows the rapidly falling stocks at Calvin Industries, and a projection for what the stocks will look like in the up-coming years based on statistics released by the cooperation. Your goal is to identify the stocks, and use it to solve the following word problems that relates to multiplying integers. Use your understanding of graphs, fractions, and integer operations to solve the following word problem- answer provided if needed.
Question ) In 2020, the stocks at Calvin Industries reached around 600 1/3 in terms of unit sales (in thousands). Calvin Industries believes that stocks will be rapidly decreasing by 4/9 of that amount over the next 5 years (2022-2027), which would put Calvin Industries in danger. If stocks at Calvin Industries rapidly fell around 4/9 per month beginning from 2020, estimate the position that Calvin Industries would be in in terms of unit sales in the up-coming 50 months.
Solution ) Identify how many years 50 months is. 50 divided by 12 is around 4.1 years, or rounding 12 to 10 to do 50 divided by 10 is 5 years. Times -4/9 by 50 to get a product of -22 11/50 (4 times 50 = 200/9, 200 divided by 9, rounded to 10, is around -20 11/50). With this answer, we can say that this is the position that Calvin Industries' stocks will be in 5 years time beginning from 2020.
Question ) You multiply two integers and get a product. You increase both integers by 4 and get a product that is exactly 4 greater than your first product. What could you have multiplied each time? You could have multiplied 0 by -3- which results in 0. If you increase both factors by 4, we get 4 multiplied by 1, which is 4, and that is exactly 4 greater than our original product of 0.
Question ) The product of two numbers is 50 less than one of its factors. What could the factors be?
a ) 5 and -10 - The answer is not this as 5 times -10 is -50, and the product will need to be 0 in order for it to work out.
b ) -2 and 25 - The answer cannot be this as doubling 25 gives us a difference of 25, not 50.
c ) 2 and 50 - The answer cannot be this as our product will be greater than both of these factors.
d ) 5 and -9 - The answer is this as the product, -45, is 50 less than 5.
Why does multiplying a negative by a negative bring us a positive?
-45 divided by -5 is equal to positive 9. When we multiply or divide any 2 negative numbers, we will always get a positive. But, why? Because multiplication and division is linked to repeated addition and subtraction, we can use that as an example. Lets use the example -9 times -5, which gets us the answer +45. Imagine we have 9 negative nine integer chips as one group, and we remove 5 of those. We keep removing 5 integer chips, and pretend that there is still integer chips once we exceed the value 9. If we have -9 dollars of debt, and we are removing -5 of those in groups of 9, then our value will turn into a positive. Assume you're playing a game with black and red chips. At the end of the game, you will receive one dollar (+1) for each black chip you have. You must pay one dollar for each red chip you possess (-1). Now, these chips are packaged in bags of five, and let's say you have several bags of black chips and several bags of red chips at some point during the game. If you receive three bags of black chips, you will earn $15. (3)(5)=15. If someone steals three bags of black chips from you, you will lose $15. (-3)(5)=-15. If you are given three bags of red chips, you will lose $15. (3)(-5)=-15. If someone steals three of your bags of red chips, you gain $15. (-3)(-5)=15. The key concept is that negative numbers represent changes rather than amounts. It makes no sense to say you have four slices of bread. However, it makes sense to say that you ate 4 slices of bread and thus the change in the number of slices you have is -4.
Division Operations
This visual shows how to medal 49 divided by (-7) via integer chips. Basically, division means how many groups we can make to equal our first factor- like reverse multiplying. We can solve this by seeing how many groups of 7 it takes to make a total of 49, and we need 7 groups of 7 integer chips to amount to 49. We can use the idea that a negative divided by a positive, or a positive divided by a negative always results in a negative quotient. While it may be difficult to visualise dividing a positive number by a negative number, it is equivalent to dividing a negative number by a positive number using the transitive principle. For example, you could multiply both sides by -1. And obviously, a negative number divided by any positive number must remain negative.
Division is known by multiple different terms. Division often times is associated with repeated subtraction, reverse multiplication, even distribution of groups, and seeing how many times our second factor can fit into our first factor, aka multiplication. If we have 49 divided by 7, we are seeing how many times 7 can fit into 49- which 7 can fit into 49 exactly 7 times, without any decimal points. In dividing both positive and negative values, division is the same concept in integer operations. Before we can begin to discuss more about division and related problems, we must first discuss about the rules in terms of dividing integers.
Negative / Positive will always get us a negative quotient.
Positive / Negative will always get us a negative quotient.
Positive / Positive will always get us a positive quotient.
Negative / Negative will always get us a positive quotient.
Dividing integers is the same as regular division. 49 divided by 7, 49 divided by (-7), (-49) divided by (-7), etc. They all equal 7. But, its just that some value will equal positive, while some will be negative. If you've mastered regular division, then technically, you've mastered dividing intergers already.
View example :
-27 ÷ +3 =
Negative 27 divided by positive 3 equals negative 9- meaning 3 groups of negative 9 integer chips will equal to negative 27.
Why does diving a negative and a positive give us a positive?
Think of it like this. Division sometimes is associated with repeated subtraction, like how multiplication is like repeated addition. Using this rule, -45 / 5 will be used as our example. We have 5 integer chips- alone, without any additional groups. We need to find how many groups of 5 integer chips it will get us to the product of -45. However, because its in the negative area, we must transform that into negative as well. If we have 5 groups of -5 integer chips, then its like repeated subtraction. 5 - 5 - 5 - 5 - 5 - 5 -5 - 5 - 5 will get us the product -45.
Question ) You divide two integers that are 36 apart. The quotient is −8. What could the integers be? To solve this equation, we can simply subtract 4 from 36, which leaves us with 32. 32 and 8 is both a multiple of 4, and is 32 apart- but, -32 divided by 4 is only -8. To combat this, we can change our value from -32 divided by 8, to -32 divided by 4 to get our answer of -8, and both factors are exactly 36 apart.
Question ) The quotient of two integers is 14 less than one of the integers. What could the integers be?
a) −28 ÷ 2 ) We can automatically assume this one is correct because half of -28 is -14, and therefore, the difference shows by the value ( divided by 2)
b) −45 ÷ 9 ) we can automatically rule this out because -45 divided by 9 gets us a factor that is much bigger than the original value of 45, not closer to 14
c) −14 ÷ 2 ) We can automatically rule this out because finding the half of 14 automatically means our value will be more than 14 less
d) −7 ÷ 2 ) we can automatically rule this one out because our quotient cannot exceed
The number line above represents our question of : A car driven west from a point 340 km east of you just arrived after a 4 hour trip. What was the cars speed? How could you express it as an integer division situation?. What the question is asking is- how fast was the car driving if the car has traveled 340 km east following a 4 hour trip? We can represent this problem with the equation : (-4x)= -(340), meaning we are multiplying 4 by a value to get us the number 340. We can represent both of these numbers by turning them into negative integers since 4 hours ago, from 340 km east. Division basically is reverse multiplication, and we can transform our equation into 340 / 4 = x. Once we solve this equation, we can know how fast our car is going because we are trying to figure out how many times 4 can go into 340, and each value will represent an hour of the total journey.
Without using a calculator, we can easily find the answer of our number. The divisibility rules for 4 states that the last 2 digits must be divisible by 4. 0 is divisible by any number, and 4 is divisible by itself. We know that our answer will not have any decimals. We can skip count by the multiples of 40- each representing the value of 4 times 10. 40 (4x10), 80 (4x20), 120 (4x30), 160 (4x40), 200 (4x50), 240 (4x60), 280 (4x-70), 320 *(4x80), and the next value will be at 360, exceeding our goal of 340. Returning back to 320, we know that 4x5 equals 20- therefore, using our factors 80+5, we get an answer of 85 which will be an answer. Using the division rule- negative divided by a negative equals a positive, we can assure that our answer is positive. We can fact check this by doing 85 times 4- meaning we are driving 85 kmh for 4 hours, and that will get us 340 km.
If you multiply two integers and then increase one of them by 1 and decrease the other by 1, the product will increase by 6 if the original two integers are either both even or both odd. This is because when you increase an even number by 1, it becomes odd, and when you decrease an odd number by 1, it becomes even. So, if the two original integers are both even, the product of the increased and decreased numbers will be the product of two odd numbers, which will be 6 more than the original product. Similarly, if the two original integers are both odd, the product of the increased and decreased numbers will be the product of two even numbers, which will also be 6 more than the original product.
