What is "Standard Form"? Show
that depends on what you are dealing with! I have gathered some common "Standard Form"s here for you..
Note: Standard Form is not the "correct form", just a handy agreed-upon style. You may find some other form to be more useful. Standard Form of a Decimal NumberIn Britain this is another name for Scientific Notation, where you write down a number this way: In this example, 5326.6 is written as 5.3266 × 103, because 5326.6 = 5.3266 × 1000 = 5.3266 × 103 In other countries it means "not in expanded form" (see Composing and Decomposing Numbers):
Standard Form of an EquationThe "Standard Form" of an equation is: (some expression) = 0 In other words, "= 0" is on the right, and everything else is on the left. Standard Form of a PolynomialThe "Standard Form" for writing down a polynomial is to put the terms with the highest degree first (like the "2" in x2 if there is one variable).
The highest degree is 6, so that goes first, then 3, 2 and then the constant last: x6 + 4x3 + 3x2 − 7 Standard Form of a Linear EquationThe "Standard Form" for writing down a Linear Equation is Ax + By = C A shouldn't be negative, A and B shouldn't both be zero, and A, B and C should be integers.
Bring 3x to the left: −3x + y = 2 Multiply all by −1: 3x − y = −2 Note: A = 3, B = −1, C = −2 This form: Ax + By + C = 0 is sometimes called "Standard Form", but is more properly called the "General Form". Standard Form of a Quadratic EquationThe "Standard Form" for writing down a Quadratic Equation is (a not equal to zero)
Expand "x(x-1)": x2 − x = 3 Bring 3 to left: x2 − x − 3 = 0 Note: a = 1, b = −1, c = −3 Copyright © 2017 MathsIsFun.com
Let be an integer-valued -ary quadratic form, i.e., a polynomial with integer coefficients which satisfies for real . Then can be represented by
where
is a positive symmetric matrix (Duke 1997). If has positive entries, then is called an integer-matrix form. Conway et al. (1997) have proven that, if a positive integer-matrix quadratic form represents each of 1, 2, 3, 5, 6, 7, 10, 14, and 15, then it represents all positive integers.Fifteen Theorem Conway, J. H.; Guy, R. K.; Schneeberger, W. A.; and Sloane, N. J. A. "The Primary Pretenders." Acta Arith. 78, 307-313, 1997.Duke, W. "Some Old Problems and New Results about Quadratic Forms." Not. Amer. Math. Soc. 44, 190-196, 1997.Integer-Matrix Form Weisstein, Eric W. "Integer-Matrix Form." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Integer-MatrixForm.html Subject classifications
Integers include positive numbers, negative numbers, and zero. 'Integer' is a Latin word which means 'whole' or 'intact'. This means integers do not include fractions or decimals. Let us learn more about integers, the definition of integers, and the properties of integers in this article.
Integers include all whole numbers and negative numbers. This means if we include negative numbers along with whole numbers, we form a set of integers. Definition of IntegersAn integer is a number with no decimal or fractional part and it includes negative and positive numbers, including zero. A few examples of integers are: -5, 0, 1, 5, 8, 97, and 3,043. A set of integers, which is represented as Z, includes:
Z = {... -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, ...} Observe the figure given below to understand the definition of integers. Integers on a Number LineA number line is a visual representation of numbers on a straight line. This line is used for the comparison of numbers that are placed at equal intervals on an infinite line that extends on both sides, horizontally. Just like other numbers, the set of integers can also be represented on a number line. Graphing Integers on a Number LinePositive and negative integers can be visually represented on a number line. Integers on a number line help in performing arithmetic operations. The basic points to keep in mind while placing integers on a number line are as follows:
Integer OperationsThe four basic arithmetic operations associated with integers are:
There are some rules for performing these operations of integers. Before we start learning these methods of integer operations, we need to remember a few things.
Addition of IntegersAdding integers is the process of finding the sum of two or more integers where the value might increase or decrease depending on the integer being positive or negative. The different rules and the possible cases for the addition of integers are given in the following section. Rules of Integers in AdditionWhile adding two integers, we use the following rules:
Example: Add the given integers: 2 + (-5) Solution: Here, the absolute values of 2 and (-5) are 2 and 5 respectively. Their difference (larger number - smaller number) is 5 - 2 = 3 Now, among 2 and 5, 5 is the larger number and its original sign “-”. Hence, the result gets a negative sign, "-”. Therefore, 2 + (-5) = -3 Example: Add the given integers: (-2) + 5 Solution: Here, the absolute values of (-2) and 5 are 2 and 5 respectively. Their difference (larger number - smaller number) is 5 - 2 = 3. Now, among 2 and 5, 5 is the larger number and its original sign “+”. Hence, the result will be a positive value. Therefore,(-2) + 5 = 3 We can also solve the above problem using a number line. The rules for the addition of integers on the number line are as follows.
Example: Find the value of 5 + (-10) using a number line. Solution: In the given problem, the first number is 5 which is positive. So, we start from 0 and move 5 units to the right side. The next number in the given problem is -10, which is negative. We move 10 units to the left side from 5. Finally, we reach at -5. Therefore, the value of 5 + (-10) = -5 Subtraction of IntegersSubtracting integers is the process of finding the difference between two or more integers where the final value might increase or decrease depending on the integer being positive or negative. The different rules and the possible cases for the subtraction of integers are given in the following section. Rules of Integers in SubtractionIn order to carry out the subtraction of two integers, we use the following rules:
Example: Subtract the given integers: 7 - 10 Solution: 7 - 10 can be written as (+ 7) - (+)10
Multiplication of IntegersFor the multiplication of integers, we use the following rules given in the table. The different rules and the possible cases for the multiplication of integers are given in the following section. Rules of Integers in MultiplicationIn order to carry out the multiplication of two integers, we use the following rules:
Example: Multiply (-6) × 3 Solution: Using the rules of multiplication of integers, when we multiply a positive and negative integer, the product has a negative sign. Therefore, (-6) × 3 = -18 Division of IntegersDivision of integers means equal grouping or dividing an integer into a specific number of groups. For the division of integers, we use the rules given in the following table. The different rules and the possible cases for the division of integers are given in the following section Rules of Integers in DivisionIn order to carry out the division of two integers, we use the following rules.
Example: Divide (-15) ÷ 3 Solution: Using the rules of division of integers, when we divide a negative integer by a positive integer, the quotient has a negative sign. Therefore, (-15) ÷ 3 = -5 Integers WorksheetsDownload integers worksheets, including addition and subtraction of integers, adding and subtracting multiple integers, and multiplication and division of integers. Properties of IntegersThe main properties of Integers are as follows:
Closure Property The closure property states that the set is closed for any particular mathematical operation. Z is closed under addition, subtraction, multiplication, and division of integers. For any two integers, a and b:
Associative Property According to the associative property, changing the grouping of two integers does not change the result of the operation. The associative property applies to the addition and multiplication of two integers. For any two integers, a and b:
Commutative Property According to the commutative property, changing the position of the operands in an operation does not affect the result. The addition and multiplication of integers follow the commutative property. For any two integers, a and b:
Distributive Property The distributive property states that for any expression of the form a (b + c), which means a × (b + c), operand a can be distributed among the operands b and c as: (a × b) + (a × c) that is, a × (b + c) = (a × b) + (a × c) Additive Inverse Property The additive inverse property states that the addition operation between any integer and its negative value will give the result as zero (0). For any integer, a: a + (-a) = 0 Multiplicative Inverse Property The multiplicative inverse property states that the multiplication operation between any integer and its reciprocal will give the result as one (1). For any integer, a: a × 1/a = 1 Identity Property Integers follow the identity property for addition and multiplication operations. The additive identity property states that when zero is added to an integer, it results in the integer itself. This means, a + 0 = a Similarly, the multiplicative identity states that when 1 is multiplied to any integer, it results in the integer itself. This means, a × 1 = a ☛ Related Articles
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FAQs on IntegersAn integer is a number that includes negative and positive numbers, including zero. It does not include any decimal or fractional part. A few examples of integers are: -5, 0, 1, 5, 8, 97, and 3,043 What are the Different Types of Integers?There are generally three types of integers:
Can a Negative Number be an Integer?Yes, a negative number can also be an integer, given that it should not have a decimal or fractional part. For example: Negative numbers: -2, -234, -71, etc., are all integers. What are Consecutive Integers?The integers that follow each other in order are called consecutive integers. For example: Numbers 2,3,4, and 5 are four consecutive integers. What is the Rule for Adding a Positive and Negative Integer?According to the rules of integers in addition, when one integer is positive and the other is negative, we find the difference of the absolute values of the numbers and then give the sign of the larger number to the result. For example, if we need to add 6 + (-4), we will find the difference of the absolute values of the given integers, that is, the difference between 6 and 4 is 2, and the result will have a positive sign because the larger number (6) has a positive sign. What are the Properties of Integers?Various arithmetic operations can be performed on integers, like addition, subtraction, multiplication and division. The major properties of integers associated with these different operations are:
What are the Applications of Integers?The application of positive and negative numbers in the real world is different. While positive numbers are commonly used everywhere, the negative integers are used in measuring temperature which can also have a negative value, that is, the temperature of a city can be -4°C or -10°C. The negative and positive numbers and zero in the scale denote different temperature readings. Bank credit and debit statements also use integers to represent the negative or positive values in transactions. |