Introduction for inequalities and number theory:
Number theory is the branch of the pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study. Number theory may be subdivided into the several fields, according to the methods used and the type of questions investigated. These are the somewhat older terms, which are no longer as popular as they once were. Inequalities are the statement about the relative size or order of two objects or about whether they are the same or not.
Number Theory and Inequalities:-examples for Number Theory:
Ex 1:
Let a sequence can be defined by a1 = 1, a2 = 1, an = an–1 + an–2 for n > 2. Find the sequence.
Sol:
a1 = 1, a2 = 1an = an–1 + an–2 for n > 2
a3 = a2 + a1 = 1 + 1 = 2
a4 = a3 + a2 = 2 + 1 = 3
a5 = a4 + a3 = 3 + 2 = 5
? The sequence is: 1, 1, 2, 3, 5,...
Ex 2:
Find the common difference and the next three terms of the A.P. 1, 4, 7, ...
Sol:
The common difference = 4 – 1 = 3
The next three terms are 7 + 3 = 10, 10 + 3 = 13
13 + 3 = 16
Integers:
Ex1:
The numbers 0, 1, -1, 2, -2 … are called integers
1, 2, 3 … are called positive integers
-1, -2, -3… are called negative integers. The collection of all integers is denoted by the letter Z. Thus Z = {…, -3, -2, -1, 0, 1, 2, 3…}.
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Number Theory and Inequalities:-examples for Inequalities:
Solve the inequalities - x2 + 3x - 2 > 0
Sol:
- x2 + 3x - 2 > 0 ? - (x2 - 3x + 2) > 0
? x2 - 3x + 2 < 0
? (x - 1) (x - 2) < 0
On equating the factors to zero, we get x = 1, x = 2 are the roots of the quadratic equation. Plotting these roots on number line and making positive and negative alternatively from the right most part we get the corresponding number line as given below.
The three intervals are (- 8, 1), (1, 2) and (2, 8). Since the sign of (x - 1) (x - 2) is negative, select the interval in which (x - 1) (x - 2) is negative. ? x ? (1, 2)
Practice Problem for Inequalities Theory:
Solve : 4x2 - 25 = 0
Answer: Thus the solution set is (- 8,- 5/2 ] ? [5/2 , 8)
Number theory is the branch of the pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study. Number theory may be subdivided into the several fields, according to the methods used and the type of questions investigated. These are the somewhat older terms, which are no longer as popular as they once were. Inequalities are the statement about the relative size or order of two objects or about whether they are the same or not.
Number Theory and Inequalities:-examples for Number Theory:
Ex 1:
Let a sequence can be defined by a1 = 1, a2 = 1, an = an–1 + an–2 for n > 2. Find the sequence.
Sol:
a1 = 1, a2 = 1an = an–1 + an–2 for n > 2
a3 = a2 + a1 = 1 + 1 = 2
a4 = a3 + a2 = 2 + 1 = 3
a5 = a4 + a3 = 3 + 2 = 5
? The sequence is: 1, 1, 2, 3, 5,...
Ex 2:
Find the common difference and the next three terms of the A.P. 1, 4, 7, ...
Sol:
The common difference = 4 – 1 = 3
The next three terms are 7 + 3 = 10, 10 + 3 = 13
13 + 3 = 16
Integers:
Ex1:
The numbers 0, 1, -1, 2, -2 … are called integers
1, 2, 3 … are called positive integers
-1, -2, -3… are called negative integers. The collection of all integers is denoted by the letter Z. Thus Z = {…, -3, -2, -1, 0, 1, 2, 3…}.
I am planning to write more post on math word problems for 2nd grade, math problem solver algebra. Keep checking my blog.
Number Theory and Inequalities:-examples for Inequalities:
Solve the inequalities - x2 + 3x - 2 > 0
Sol:
- x2 + 3x - 2 > 0 ? - (x2 - 3x + 2) > 0
? x2 - 3x + 2 < 0
? (x - 1) (x - 2) < 0
On equating the factors to zero, we get x = 1, x = 2 are the roots of the quadratic equation. Plotting these roots on number line and making positive and negative alternatively from the right most part we get the corresponding number line as given below.
The three intervals are (- 8, 1), (1, 2) and (2, 8). Since the sign of (x - 1) (x - 2) is negative, select the interval in which (x - 1) (x - 2) is negative. ? x ? (1, 2)
Practice Problem for Inequalities Theory:
Solve : 4x2 - 25 = 0
Answer: Thus the solution set is (- 8,- 5/2 ] ? [5/2 , 8)
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