Addition
Addition
Hello Science Bee. In this lesson, we are going to learn about addition. In simple terms, addition is a process of adding something to something else. Addition is the most basic of all arithmetic skills and having a good foundation in addition is an important first step towards becoming a Math guru. This is what we will learn here.
What is addition?
Addition (often signified by the plus symbol "+") is one of the four basic operations of arithmetic; the others are subtraction, multiplication and division. The addition of two whole numbers is the total amount of those quantities combined.
For example, if there is a combination of three apples and two apples together, making a total of five apples. This observation is equivalent to the mathematical expression "3 + 2 = 5" i.e., "3 add 2 is equal to 5".
Video: Intro to Addition
Types of addition
Besides counting items, addition can also be defined on other types of numbers, such as integers, real numbers, and complex numbers. This is part of arithmetic, a branch of mathematics. In algebra, another area of mathematics, addition can be performed on abstract objects such as vectors and matrices.
Commutative law of addition
Figure: 4 + 2 = 2 + 4 with blocks
Addition is commutative: one can change the order of the terms in a sum, and the result is the same. Symbolically, if a and b are any two numbers, then
a + b = b + a.
The fact that addition is commutative is known as the "commutative law of addition".
Some other binary operations are commutative, such as multiplication, but many others are not, such as subtraction and division.
Associative law of addition
Figure: 2 + (1 + 3) = (2 + 1) + 3 with segmented rods
Addition is associative: when adding three or more numbers, the order of operations does not matter.
As an example, should the expression a + b + c be defined to mean (a + b) + c or a + (b + c)? That addition is associative tells us that the choice of definition is irrelevant. For any three numbers a, b, and c, it is true that (a + b) + c = a + (b + c). For example, (1 + 2) + 3 = 3 + 3 = 6 = 1 + 5 = 1 + (2 + 3).
When addition is used together with other operations, the order of operations becomes important. In the standard order of operations, addition is a lower priority than exponentiation, nth roots, multiplication and division, but is given equal priority to subtraction.
What is an identity element?
Figure: 5 + 0 = 5 with bags of dots
When adding zero to any number, the quantity does not change; zero is the identity element for addition, also known as the additive identity.
In symbols, for any a, a + 0 = 0 + a = a.
Successor of an integer
Within the context of integers, addition of one also plays a special role: for any integer a, the integer (a + 1) is the least integer greater than a, also known as the successor of a. For instance, 3 is the successor of 2 and 7 is the successor of 6. Because of this succession, the value of a + b can also be seen as the bth successor of a, making addition iterated succession. For examples, 6 + 2 is 8, because 8 is the successor of 7, which is the successor of 6, making 8 the 2nd successor of 6.
Addition of rational numbers (fractions)
Addition of rational numbers can be computed using the least common denominator, but a conceptually simpler definition involves only integer addition and multiplication:
As an example, the sum
.
.Addition of fractions is much simpler when the denominators are the same; in this case, one can simply add the numerators while leaving the denominator the same
: ,
,so .
.The commutativity and associativity of rational addition is an easy consequence of the laws of integer arithmetic.
ADVANCED READING
Addition of complex numbers
Figure: Addition of two complex numbers can be done geometrically by constructing a parallelogram.
Complex numbers are added by adding the real and imaginary parts of the summands.
That is to say
Using the visualization of complex numbers in the complex plane, the addition has the following geometric interpretation: the sum of two complex numbers A and B, interpreted as points of the complex plane, is the point X obtained by building a parallelogram three of whose vertices are O, A and B. Equivalently, X is the point such that the triangles with vertices O, A, B, and X, B, A, are congruent.
Vector addition
In linear algebra, a vector space is an algebraic structure that allows for adding any two vectors and for scaling vectors. A familiar vector space is the set of all ordered pairs of real numbers; the ordered pair (a,b) is interpreted as a vector from the origin in the Euclidean plane to the point (a,b) in the plane. The sum of two vectors is obtained by adding their individual coordinates:
(a,b) + (c,d) = (a+c,b+d).
This addition operation is central to classical mechanics, in which vectors are interpreted as forces.
Matrix addition
Matrix addition is defined for two matrices of the same dimensions. The sum of two m × n (pronounced "m by n") matrices A and B, denoted by A + B, is again an m × n matrix computed by adding corresponding elements:
For example:
Let's Review
- Addition is ______: one can change the order of the terms in a sum, and the result is the same.
- Addition is ______: when adding three or more numbers, the order of operations does not matter.
- When adding zero to any number, the quantity does not change; zero is the ______ element for addition, also known as the ______ ______.
- Complex numbers are added by adding the ______ and ______ parts of the summands.
Answer
- Addition is commutative: one can change the order of the terms in a sum, and the result is the same.
- Addition is associative: when adding three or more numbers, the order of operations does not matter.
- When adding zero to any number, the quantity does not change; zero is the identity element for addition, also known as the additive identity.
- Complex numbers are added by adding the real and imaginary parts of the summands.
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