Properties of Integers

Positive Integers

The whole number is called as positive integer and the positive integer is represented by the symbol ‘+’. The positive integers are involved in graph theory. The arithmetic operations are performed in positive integers.

Positive integers for addition and subtraction:
The positive integers are combined with negative integers also. So based on that some rules are followed for addition and subtraction operation.

( + ) + ( + ) = ( + )

( + ) + ( - ) or ( - ) + ( + ) = The resultant value is based on large number sign. Consider the numbers 4 and -2 and perform the addition operation as 4 + (-2) = 2.

Positive integers multiplication:

( + ) x ( + ) = ( + )

( + ) x ( - ) = ( - )

( - ) x ( + ) = ( - )

Positive integers division:
The sign of numerator value is deciding the resultant value sign.

( + ) / ( + ) = ( + )

( - ) / ( - ) = ( + )

Negative Integers

The definition of negative integer, is one of the most important topic in mathematics. Negative integer is present before the zero value in the number line. The symbol used for representing the negative integer is known as the " - ". Most of the mathematical problems are also can be done using the negative integer. In this article, we are going to see about the negative integer with the example problems.

Explanation to negative integers definition :

• Addition problem for the definition of negative integer.
• Subtraction problem for the definition of negative integer.
• Multiplication problem for the definition of negative integer.
• Division problem for the definition of negative integer.

Integer Rules

Integer rules are the procedures to be followed for doing problems in integers. For doing arithmetic operations on integers, rules for integers are very useful. Based on the sign of integers the rules differ. The integer rules are applicable for arithmetic operations with integers such as addition, subtraction, multiplication and division.

The integers rules follows for :

·         Adding integers

·         Subtracting integers

·         Multiplying integers

·         Dividing integers

Adding integers

To add a positive number and a negative number we can subtract the smaller number from the greater number without taking the sign into account and to the result, give the sign of the greater number.
A Positive number + a positive number = a positive number
A Negative number + a negative number = a negative number

Solved Examples

Question 1: Add the given two positive number 10, 23.
Solution:

Add 10 + 23 = 33

So, the answer is 33

Question 2: Add the given two negative number -10, -23.
Solution:

Given two numbers are negative numbers.

So the addition rules is applied to it. (-10) + (-23)

The answer is the -33.

Subtracting Integers

In whole numbers, we know that addition and subtraction are inverse operations.
The rules are similar to addition of integers but one different thing here we can do the subtraction.
A Positive number + a positive number = a positive number
A Negative number + a negative number = a negative number

Solved Examples

Question 1: Subtract -10 from -25.
Solution:

 Given two numbers are negative numbers -25 and -10,

So using the subtraction rules we get, (-25) - (-10) This is written as -25 + 10 Subtracting the above two numbers.

The answer is -15 (subtracting the larger value to the smaller value and then put the larger value sign).

Question 2: Subtract -63 from -5
Solution:

Given two numbers are both negative So by the subtraction rules (-5) - (-63)

This is written as -5 + 63 Subtracting the above two numbers.

The answer is 58 (subtracting the larger value to the smaller value and then put the larger value sign).

Multiplying Integers

We know that multiplication is repeated process of addition. The rules are,

A Positive number × a positive number = a positive number
A Positive number × a negative number = a negative number
A Negative number × a positive number = a negative number
A Negative number ×a negative number = a positive number

Solved Examples

Question 1: Multiply 2 with -3.
Solution:

Here 2 is a positive number  and 3 is a negative number

When we multiply a positive with negative the resultant will be negative 2 x (-3) = -6

Question 2: Multiply -5 and -4
Solution:

 Here -5  and -4 are both negative number.

When we multiply a negative with negative the resultant will be positive (-5) x (-4) = 20

Dividing Integers

Division is a repeated process of subtraction. The rules are,

A Positive number / a positive number = a positive number
A Positive number / a negative number = a negative number
A Negative number / a positive number = a negative number
A Negative number / a negative number = a positive number

Below you could see examples for integer division

Solved Examples

Question 1: Divide 12 with 4
Solution:

 Here 12 is a positive number 4 is a positive number.

When we divide positive with a positive resultant will be positive 124124 = 3

Question 2: Divide -50 by 5
Solution:

Here -50 is a negative number 5 is a positive number.

When we divide negative by positive the resultant will be negative −505−505 = -10

Integer Word Problems

Integer word problems it is has the series (with the continuous of expression) we have to fit that expression in the mathematical expression so we called that kind of expression as the word problems.

Solved Examples

Question 1: Find the sum of the two consecutive integer number is 717, find the integer?
Solution:

The numbers are consecutive so we take one integer as x and the another integer as x+1(consecutive so we take x+1)In the question they are given the sum is 717
So here we add these two integer and then find the value of x
X+x+1=717.
Subtracting 1 on both sides we get
2x =717-1 =716
Then divide the 2 on both sides we get the answer of x (integer)
X = 71627162 =358
So the one number is 358 and the number is 359(x+1) both the answer are the integer.

Question 2: The morning temperature of the New York was -17 ⁰F if the temperature will be dropped as 11⁰F, calculate the temperature of the New York now?
Solution:

Here the morning temperature as given as -17
And the drooped temperature will be given as 11
We have to calculate the temperature of the city now,
 morning temperature - dropped temperature
 -17-11 = -28 ⁰F

Reference:

http://math.tutorvista.com/number-system/positive-integers.html

http://math.tutorvista.com/number-system/negative-integers.html

http://math.tutorvista.com/number-system/dividing-integers.html

Fundamental Operations

The fundamental operations in mathematics are addition, subtraction, multiplication and division. There are corresponding symbols for each. The plus sign (+) is for addition. The minus sign (-) is for subtraction. The symbols “x”, “*” and “•” signify multiplication. The obelus (÷) and forward slash (/) are used for division.

 Addition combines two or more numbers to get their sum or total, while subtraction finds the difference between two quantities. Multiplication is repeated addition; one of the numbers in a multiplication equation indicates how many times the other number needs to be added to itself. Division is the inverse of multiplication.

Examples on fundamental operations in simplifying mathematical expressions on different types of questions on integers are discussed here step by step. 

The following examples will help us to understand the precedence of operations of addition, subtraction, multiplication and division.

1. Simplify: 24 - 4 ÷ 2 x 3

Solution:

24 - 4 ÷ 2 x 3

[Here order is expressed in short as ‘DMAS’ where ‘D’ stands for division, ‘M’ for multiplication, ‘A’ for addition and, ‘S’ for subtraction]

= 24 - 2 x 3 [Performing division - 4 ÷ 2 = -2]

= 24 - 6 [Performing multiplication 2 x 3 = 6]

= 18. [Performing subtraction 24 – 6 = 18]

Answer: 18.

2. Simplify: 48 ÷ 8 x 3 + 2

Solution:

48 ÷ 8 x 3 + 2

[Here order is expressed in short as ‘DMAS’ where ‘D’ stands for division, ‘M’ for multiplication, ‘A’ for addition and, ‘S’ for subtraction]

= 6 x 3 + 2 [Performing division 48 ÷ 8 = 6]

= 18 + 2 [Performing multiplication 6 x 3 = 18]

= 20. [Performing addition 18 + 2]

Answer: 20.

3. Simplify: (-20) + (-8) ÷ (-2) x 3

Solution:

(-20) + (-8) ÷ (-2) x 3

= (-20) + 4 x 3 [Performing division (-8) ÷ (-2) = 8 ÷ 2 = 4]

= (-20) + 12 [Performing multiplication 4 x 3 = 12]

= - 8. [Performing subtraction -20 + 12 = -8]

Answer: -8.



4. Simplify: (-5) - (-48) ÷ (-16) + (-2) x 6

Solution:

(-5) - (-48) ÷ (- 16) + (-2) x 6

= (-5) - 3 + (-2) x 6 [Performing division (-48) ÷ (- 16) = 48 ÷ 16 = 3]

= (-5) - 3 + (-12) [Performing multiplication (-2) x 6 = -12]

= -5 - 3 -12

= -8 - 12. [Performing addition -5 - 3 = -8]

= -20 [Performing addition -8 - 12 = -20]

Answer: -20.

5. Simplify: 52 - (2 x 6) + 17

Solution:

52 - (2 x 6) + 17

= 52 – 12 + 17

= 52 + 17 - 12

= 57

Answer: 57


In simplification these are the basic examples on fundamental operations used in the expression.

Rational expressions

Rational expressions and rational equations can be useful tools for representing real life situations and for finding answers to real problems. In particular, they are quite good for describing distance-speed-time questions, and modeling multi-person work problems.

Solving Work Problems

Work problems often ask us to calculate how long it will take different people working at different speeds to finish a task.  The algebraic models of such situations often involve rational equations derived from the work formula, W = rt.  The amount of work done (W) is the product of the rate of work (r) and the time spent working (t). The work formula has 3 versions:

Some work problems have multiple machines or people working on a project together for the same amount of time but at different rates. In that case, we can add their individual work rates together to get a total work rate. Let’s look at an example:

How can equations and inequalities be used in real life situations?

One example is the Squeeze theorem where we can squeeze a complicated function between two simpler functions & use results on the simpler functions to prove bounds on the more complicated function. This applies to all sorts of maths including infinite sums, even finding the area of a circle.

See The Beautifully Simple Method Archimedes Used To Find The First Digits Of Pi

I usually try to think of inequalities physically / geometrically / spartially as either squeezing things together, or as constraints / boundaries.

It’s surprising what we can do when we combine constraints

The game of Sudoku is really just a set of constraints. Each number must occur in each row, column and square exactly once, and each cell must contain one number. We can actually set up a system of inequalities (multiple inequalities that must be met at the same time) to represent sudoku, and feed this to an inequality solver. See Linear programmingand Integer programming & https://pypi.python.org/pypi/PuLP

(so this is very useful because, we can create the inequalities we know about, at let the computer algorithms created by smart people figure out solutions to the inequalities - i.e. some of these problems are more complex that sudoku and coding solutions for them is hard, but if we can feed the inequalities into a program created by many PHD students, (magic black box) - then it can give us an answer.

In two dimensions, inequalities look like lines which (if straight lines), define convex polygons. (x>0, x<1, y>0, y<1 would be a 1x1 square for example). (so if we’re thinking about inequalities in 2D, any point within the polygon would satisfy the inequality. Example could be, what are the pokemon that are strong and common. One axis could be strength of the pokemon, the other axis could be how many pokemon of that type exist, and the polygon would be the region that is above a certain strength and certain population count threshold)

In higher dimensions we have convex polytopes defined by hyperplanes but we think about them in the same way as we do with polygons. (these polygons or polytopes are convex because the inequality lines or hyperplanes are straight (There are probably ways of dealing with concave regions but it would probably be a lot more complicated))

If you think of dimensions as the axis of variables on a graph, we’re not restricted to x and y. Variables could be anything we could measure. And restrictions can be any form of heuristic. E.g. I’m baking a chocolate cake using milk, butter and chocolate, but I want a restriction on the amount of fat. The ingredients could be seen as variables (e.g. x, y, x), and the fat restriction could be seen as total_fat = a*x + b*y + c*x <= max_fat_restriction. (in this example, y could represent the measures of butter, and ‘b’ could represent the amount of fat in each measure of butter)

If we were making burgers, the dimensions could be the following ingredients:
chicken, beef, lettuce, tomato, salt, Pepper, Chilli, bread, potato..

We can find our burgers in polytopes. Each hyperplane (face) of the polytope restricts a certain heuristic of ingredients.. though that might be a bit of a mouthful..

What are the heuristics?
Price, Spiciness, Weight, Volume, Carbs (can't have too many carbs!), Meatiness, Variety.

How do inequalities on heuristics create inequalities on ingredients?

- The heuristics can be represented by servings of ingredients. Thus restrictions on heuristics will create restrictions on ingredients!

For example, the serving of each ingredient has a price.
We say Price = Chicken Servings * Price Per serve of chicken + Beef Servings * Price per service of Beef...
So if we want our burger at under 10$, we are creating a hyperplane that will restrict us to only the affordable burgers.

Spiciness = Spiciness of Pepper * Pepper Servings + Spiciness of Chilli * Chilli servings.

Again we can create inequalities on spiciness depending on if we want mild, medium or hot!

We might not want beef and chicken in the same burger, we can say servings of beef + servings of chicken < 2. (note, what if we wanted the possibility of 3 servings of chicken, then we need to create a new variable and use integer programming - which is beyond the scope of this answer).

There are software packages where we can easily list our variables (ingredients) and our equations (hyperplane restrictions). The software package will then figure out the region and allow you to ask questions about heuristics in the region such as "given the constraints, what is the spiciest burger" or "what is the most profitable burger" etc.

Think about real world problems such as supply chain management - we want to get supplies from suppliers around the world to various factories at certain times adhering to certain demands, volatilities, buffers etc. Supplies (ingredients) have costs, transport costs etc. We could create inequalities to match the demand, requirements etc and then minimise costs across supplies and shipping etc. It would be much to difficult to do by hand.

The hard part is figuring out how to formulate a real world problem with inequalities. What are the dimensions or variables? what are the inequalities on the combinations of these variables?

Reference:

https://en.wikipedia.org/wiki/Squeeze_theorem

http://www.businessinsider.com.au/archimedes-pi-estimation-2014-3?r=US&IR=T

https://en.wikipedia.org/wiki/Linear_programming

https://pypi.python.org/pypi/PuLP