Understanding Integers: Properties and Operations

Study Notes

Integers

Integers are whole numbers including positive numbers, negative numbers and zero. They can have any number of digits and extend infinitely to the left (negative) and right (positive). For example, -8, -7, -6, -5...-2, -1, 0, 1, 2, ... 8, 9, 10 and further.

Properties of Integers

Commutativity

The commutative property states that when performing arithmetic operations such as addition or multiplication, we don't really care about the order of the operands. This means that a + b = b + a and a * b = b * a.

Associativity

Associativity is similar to commutativity, but it deals with grouping symbols. It states that regardless of the grouping, the result remains the same. So, (a + b) + c = a + (b + c) and (a * b) * c = a * (b * c).

Distributive Property

The distributive property is another fundamental principle of arithmetic. It states that if you multiply two integers together, you can break up the first integer into individual parts and multiply each part by the second integer separately before adding them all together. For example:

a(b + c) = ab + ac.

Operations with Integers

Addition

When adding two integers, there are three possible outcomes:

If both integers are either positive or negative, their sum will also be a member of the original set.
If one integer is positive and the other negative, the sum will always be a positive number.
If one integer is negative and the other is positive, the result will depend on the magnitude of the positive integer. For example, -x + y = y - x.

Subtraction

Subtracting one integer from another works similarly to addition. However, unlike addition, subtraction does not follow the commutative property. That is why, when performing subtraction, we need to ensure that the smaller number is subtracted from the larger one.

Multiplication

Multiplication operates with the concept of scalars. In mathematics, scaling can refer to anything we do to transform vectors using some sort of vector space, like addition and subtraction. Scalar multiplication involves changing the size of a vector without causing a change in its direction. In simpler terms, you just multiply the coordinates by a single value.

Division

Division is the inverse operation of multiplication. When dividing, the divisor is the number being divided into groups, and the quotient is the number of groups consisting of that many items. For example, if you want to divide 12 apples equally among 4 people, you would say that the quotient is 3 apples per person.

Comparing Integers

Comparing integers allows us to determine which one is bigger or smaller. There are four types of relationships between integers: equality, less than, greater than, and inequality:

Equality Two numbers are equal if they represent the same amount.
Less Than If one number is greater than the other, then we say the larger number is greater than the smaller one.
Greater Than Two numbers are equal if they represent the same amount.
Inequality Two numbers are unequal if one is greater than the other and neither is less than the other.

Description

Learn about integers, whole numbers that include positive numbers, negative numbers, and zero. Explore properties like commutativity, associativity, and the distributive property. Study operations including addition, subtraction, multiplication, and division, as well as comparisons between integers.