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Inventing Operations

Reference > Mathematics > Number Theory > Inventing Operations
 

An operation can be thought of as a process for combining one or more numbers to produce a new result. For example, addition is an operation. The addition operation receives two numbers, and combines them to produce a new result. For example, in the equation 3 + 5 = 8; the two numbers three and five are combined to produce a new value (eight). The thre and the five are called the operands.

Multiplication is also an operation; 3 x 5 = 15. The multiplication operation takes the two numbers 3 and 5 and turns them into a single number (15).

These two operations each take two numbers, and use them to produce a single result. Operations which take two numbers are called binary operations. The prefix bi-, of course, indicates two, because there are two operands.

Not all operations are binary operations; there is a type of operation called a unary operation, which only takes one operand. An example of a unary operation is the factorial (represented by an exclamation mark). 

5! = 5 x 4 x 3 x 2 x 1 = 120
4! = 4 x 3 x 2 x 1 = 24
3! = 3 x 2 x 1 = 6

And so forth. You only need one operand to do a factorial.

In addition to the "normal" operations you'll see in math classes, sometimes math problems might include more unusual operations. These might be special operators needed to do a specific job, or they might be something that your math teacher invented for the sole purpose of helping you manipulate numbers and variables in new ways.

If we invent an operator, we need to have two things:

  • a symbol to represent the operation
  • a rule that tells us what to do with the operand(s)

Often we'll invent binary operations - operations that involve two numbers. So let's say we want to use the symbol "¿" for our operation. Now we just need a rule for combining our two operands. Maybe we'll square both operands and multiply them together. This is how we write our operation definition:

a ¿ b = a2 + b2.

Now we can use our operation in a math problem:

Example #1: If a ¿ b = a2 + b2, find the value of 6 ¿ 8.

Solution #1: 6 ¿ 8 = 62 + 82 = 36 + 64 = 100.

Let's try another one.

Example #2: If a ¤ b =
a - 1
b + 1
, and 21 ¤ (x ¤ 2) = 5, find x.

Solution #2: We deal with parentheses first:
21 ¤
x - 1
2 + 1
= 5
21 - 1
x - 1
3
+ 1
= 5
20
x + 2
3
= 5
60
x + 2
= 5
x = 10

We can also create unary operations. Maybe something like this:

a*  =
a
a + 1

We only need one operand for this. If we put the number five in the equation, we get:

5* =
5
6

Example #3: If a♦ = 2a + 1, find x such that x♦♦ = 27.

Solution #3: In this case, we have to do the operation ♦ twice.

x♦ = 2x + 1
x♦♦ = 2(2x + 1) + 1 = 27. 4x + 2 + 1 = 27; 4x = 24; x = 6.

Questions

1.
Is division a binary or unary operation?
2.
Is the square root a binary or unary operation?
3.
Is exponentiation a binary or unary operation?
4.
If a * b = 3a + 2b, find the value of 10 * 3
5.
If a * b =
a + b
ab
, find the value of 2 * 3
6.
If a * b =
a + b
ab
, and x * 3 =
1
2
, find the value of x.
7.
If a* =
x - 1
, find 677**
8.
If a * b = ab - 5, and 3 * (x * 2) = 52, find x.
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