Writing Mathematical Expression

In mathematics, we can divide expression in two ways. That is -

1) Numerical Expression
2) Algebraic Expression

Here, by the word 'expression' or 'mathematical expression' we would like to define it as mathematical phrase that contains number,operator,variable or symbol etc.

Numerical Expression
A numerical expression contains numbers and operations. For example:

5+ 3
35÷7
53·4
94 - 31
102 - 9·3

All are example of numerical expression because it contains only number and operator.
Word problem: Robin plays football for 1.5 hours in the morning and 1 hour in the afternoon. He does this for a total of 30 days. Write a numerical expression with parentheses for the total number of hours Robin played football.

30 · (1.5 + 1)

Algebraic Expression
An algebraic expression contains numbers,operations and variables . For example:

5x + 3
9y - 8
6 + 3d
x/9

Word problem: Marco's mom gave him 'd' dollars for his allowance for a week. He also earned $14.55 for his newspaper route that week. How much money does Marco have? Write your answer as an expression.

d+14.55

Variable

Suppose you are going to buy a bicycle and you know it's price is 90$ and you also heard that there is a discount offer going on and you don't know what is the amount of discount. Then what will be the total price of that bicycle?

Total Price = 90$- Deduction 

So, if the discount is 10%(When you go to shop you see that 10% discount is going on) then the price is 

Total Price = 90$ - 10%

              = 90$ - 9$

       = 81$

So, the final price is 81$.

This unknown quantity of discount can be said as 'Variable' in mathematical context and a variable is just a symbol that can represent different values in an expression. So, in following expression Deduction is variable.

Total Price = 90$- Deduction 

It is representing different varying values, that's why called it variable.

In mathematics, variables are usually denoted by letters such as x,y,z,u,v etc.

So, the above expression can be written as 

Total Price = 90$ - X (Here X will be 10%)

Following are the example of variables:

x + 6

z + 10 

23y

x,z,y are example of variables. 

Continuous Variable:
If a variable x  can successively take all the values from a given number 'a' to another given number 'b', then x is called a continuous variable; otherwise, it is called a discrete variable.
The domain or interval of x in this case is denoted by a <= x <= b 

If a is omitted from the domain, it is indicated as a < x <= b  and it is said to be open at the left end.
The domain a <= x <= b is said to be closed and a < x < b is  open at both end.

Polynomial

A polynomial is a mathematical expression that have variables, coefficients and exponent. Polynomials are sums of these variables, coefficients and exponents. But polynomial does not contain a negative exponent.

An example of polynomial equation is:

x² - x - 1 = 0

In the word polynomial, poly  means 'many' and nomial means 'name' but here it will be used as term. So polynomial means "many terms". 

A polynomial in one variable with constant coefficients is given by

a_nxⁿ + a_n-1x^n-1 +.............. +a₂x² +a₁x +a₀

where a₀,  … ,a_n are constants and x is the variables.

Examples of polynomial:

• 6x 
• x - 4 
• 3x²  - 5x + 4
• -4y² - (³/₄)x 
• 2x⁵ – 5x³ – 10x + 9
• 6

6 could be considered as polynomial with 1 term.  Technically, the term polynomial should only refer to sums of many terms, but the term is used to refer to anything from one term to the sum of a zillion terms. However, the shorter polynomials do have their own names.

A one-term polynomial, such as 5x or 16x², may also be called a "monomial".

A two-term polynomial, such as 3x + y or x² – 4, may also be called a "binomial".

A three-term polynomial, such as 2a + b + c or 3x + 5y² – 3, may also be called a "trinomial".

In the polynomial expression, there should not be any negative exponent  (i.e x ^–2), the variable cannot be in the denominator (i.e 1/x), or any square root of variable. That's why below expression cannot be considered as polynomials:

4xy ^–2  -  because dividing by a variable is not allowed
3/(x+3) -  because dividing by a variable is not allowed
1/x - same as above. Variable is in denominator
√x - because square root is not allowed

Degree of terms:

The degree of a term is the sum of the exponents of all the variables in that term.

3x²  - 5x + 4

The above expression has 3 terms. First term has degree 2, second term has degree 1 and third term has no degree or zero degree or can be said constant term.

3yx² - 2x

Here, the first term has two variables and its degree is 3 because variable 'y' has power 1 and variable 'x' has power 2. So, 1+2=3. The second term has degree 1.

Naming of degree:

Degree	Name	Example
0	Constant	7
1	Linear	x + 7
2	Quadratic	x2− x + 2
3	Cubic	x3 − x2 + 5
4	Quartic	6x4 − x3 + x − 2
5	Quintic	x5− 3x3 + x2 + 8

This naming table is collected from Math is Fun.

The general form of a polynomial shows the terms of all possible degree.  Here, for example, is the general form of a polynomial of the third degree:  

3x³ + 2x² + 5x + 8

An equation that is in polynomial form is called Polynomial Equation.  Polynomials is used widely in the areas of mathematics and science.

Linear Equation

A Linear Equation is a mathematical expression.  It is an algebraic equation where one variable depends on other. It is generally appears in below form:

y = 6x + 2 

The above equation is in the form of y = mx + b, where m and b are constant.

It is an equation that forms a straight line on a graph. In this particular equation, the constant m determines the slope or gradient of that line, and the constant term b determines the point at which the line crosses the y-axis, that is also known as the y-intercept. So, in the equation y = 6x + 2, slope m is 6 and y-intercept b is 2.

Linear equation is simple in manner and  each term in linear equation will not have 

• Exponents (or powers). For example, x²
• Multiplication or division of  each other. For example: "x" times "y" or xy; "x" divided by "y" or x/y
• A root sign or square root sign (sqrt). For example: √x or the "square root x"; sqrt (x) 

The following equations are  all linear equations:

i) y = 2x+1             ii)  9x = 6+3y               iii) y/4 = 9 - x

Linear equation will be in other form also. For example

i) y = 4                ii) 8x + 4 = 0                iii) y - 3 = ¼(x - 2)

The third example is in point-slope form [ y-y1 = m(x-x1) ]

 

Linear equation graph

(Photo courtesy: hotmath.com)

Solving a linear equation:

i)           4x + 8 = 20
             4x = 12
             x = 12/4
             x = 3

ii)        ( (x-2)/3 ) + 1 =  2x/7
           (x + 1)/3  =  2x/7
           7 (x + 1)  =  6x
           7x +7  =  6x
           7x - 6x  =  - 7
            x  =  - 7

What is Coefficient in mathematics?

The coefficients are the numbers by which another number or symbol is multiplied. In a mathematical equation, a coefficient is a constant by which a variable is multiplied. Consider the following equations:

 7x² + 6x + 8

Here, x is a variable, and "7" and "6" are coefficients of x² and x respectively. Now consider another expression,

 7a² - 6xy + y + 8

Here is total 4 terms. In first term 7a²,  7 is coefficients of a². In the second term -6xy,  -6 is coefficients of xy and the third term y has no written coefficient but it is considered that it has 1 as coefficient. The last term 8  is a constant.

Take a look at following equation:

ax ³ + by ² = z

Here, a is coefficient of x ³ and b is coefficient of  y ². Variables are generally declared by the letter x,y,z etc.

Euler Number

Euler's number e is a mathematical constant and it is named after the Swiss mathematician Leonhard Euler. The pronunciation of Euler is 'Oiler' in English. The number e is also known as Napier's constant because the number  appeared on John Napier's work on logarithms that was published in 1618.   Leonhard Euler was not inventor of that constant but he introduced the letter e as the base for natural logarithms and Euler's choice of the symbol e is said to have been retained in his honor. Euler probably used this symbol in 1927.

The number e is a famous irrational number, and is one of the most important numbers in mathematics. It is also called as Transcendental number. The numerical value of e is :

2.71828182845904523536 (and more ...)

There are many ways of calculating the value of e.  

n	(1 + 1/n)n
1	2.00000
2	2.25000
5	2.48832
10	2.59374
100	2.70481
1,000	2.71692
10,000	2.71815
100,000	2.71827

Its value is approximately 2.718281828459045... and has been calculated to 869,894,101 decimal places by Sebastian Wedeniwski.

Another way we can calculate the value of e and it is below:

e=1/0! + 1/1! + 1/2! + 1/3! + 1/4! + 1/5! + 1/6! + 1/7! + ...

Here  "!" means factorial.

The first few terms add up to: 1 + 1 + 1/2 + 1/6 + 1/24 + 1/120 +...... = 2.718055556

The sum of the values is 2.7182818284590452353602875 which is "e."

Permutation and Combination

Suppose there are 3 balls: Red, Green and Blue. In shortcut we can identify them as R,G and B. Now, how many was we can arrange these balls? Definitely in six ways.

RGB, RBG, GRB,GBR, BRG, BGR

Six ways. It can be found by 3! which is 3 x 2 x  1= 6.

But if we are told to arrange these 3 balls in such a way that we will pick only two at a time, then how many ways we can arrange or sort it. Let's see,

RG, RB, GB, GR, BG, BR 

This is also in six ways and this arrangement or sorting is called Permutations.

Let's think about this 3 balls again. How many ways we can pick 2 balls by ignoring their orders. 

RG, RB, BG 

or

GR, BR, GB

Only 3 ways. Both time we ignored their internal ordering or arrangement because RG and GR is same, only difference is order. So, the number of ways we picked 2 balls from 3 balls is called Combination.

Permutations:

The ordered arrangements of a set of distinct objects is called Permutations. This ordered arrangement could be of some elements. So, an ordered arrangement of r elements from n distinct object is called  r-permutation. It is denoted as ⁿP_r, nPr or P(n,r).

The number of r permutation of a set with n distinct element is

P(n,r)= n.(n-1).(n-2).....(n-r+1)

The first element of the permutation can be chosen in n ways because there are n elements at first. After that there will be (n-1) element will be left so, the second permutation can be done n-1 ways. Similarly, the third will be n-2 ways and so on. The permutation of rth element can be done in (n-r+1) ways.

P(n,r) = n.(n-1).(n-2).....(n-r+1)

Because we know that  n! =  n.(n-1).(n-2).....(n-r+1). (n-r)!

Example: 
How many different ways there are to select 3 winners(1st,2nd,3rd) from 100 different people?
Selecting 3 winners(1st,2nd,3rd) from 100 different people means selecting 3 ordered elements from a set of 100 different elements. Here 3 permutation of a set of 100 elements is P(100,3) = 100 x 99 x 98 = 970200

Combinations:
Unordered selection of some element from n distinct object is called Combinations. So, selecting r element from n distinct object by ignoring their order(no repetition) is called r combination. It is denoted as ⁿC_r, nCr or C(n,r).

The number of r combinations of a set with n elements, where n is a non-negative integer and r is an integer with 0 <= r <= n, equals

 

 

The formula can also be re-written as follows:

Example: 
How many ways are there to select 5 players from a 10 member tennis team?
The answer can given by the number of 5 combinations of a set with 10 elements. So the combination is:

C(10,5) = 10! / 5!5! = 252