Mathematics
Show how can be represented on the number line.
Number System
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Answer
Representing as the sum of squares of two natural numbers:
5 = 4 + 1 = (2)2 + (1)2
Let l be the number line. If point O represents number 0 and point A represents number 2 then draw a line segment OA = 2 units.
At A, draw AC ⊥ OA. From AC, cut off AB = 1 unit.
We observe that OAB is a right angled triangle at A. By Pythagoras theorem, we get:
(OB)2 = (OA)2 + (AB)2
(OB)2 = (2)2 + (1)2
(OB)2 = 4 + 1
(OB)2 = 5
OB = units
With O as centre and radius = OB, we draw an arc of a circle to meet the number line l at point P.
As, OP = OB = units, the point P will represent the number on the number line as shown in the figure below:
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Related Questions
State whether the following statements are true or false. Justify your answers.
(i) Every irrational number is a real number.
(ii) Every point on the number line is of the form √m , where m is a natural number.
(iii) Every real number is an irrational number.
Are the square roots of all positive integers irrational? If not, give an example of the square root of a number that is a rational number.
Write the following in decimal form and say what kind of decimal expansion each has :
(i)
(ii)
(iii)
(iv)
(v)
(vi)
You know that = . Can you predict what the decimal expansions of , , , , are, without actually doing the long division? If so, how?
[Hint : Study the remainders while finding the value of carefully.]