Transform the following into direct narration :
1. Father told me to let the kitten lie in his bed .
2. Mohan told Karuna not to let the dog enter the kitchen .
3. She suggested to her friends that they should enjoy boating in the river .(remember that sometimes we use suggested in place of proposed)
4. I said that he might work ever so hard ,he could not win the scholarship (we begin this type of sentence also wit `Let him ´.
Answers
Answer:
Number Systems: Naturals, Integers, Rationals, Irrationals, Reals, and Beyond
The Natural Numbers
The natural (or counting) numbers are 1,2,3,4,5, etc. There are infinitely many natural numbers. The set of natural numbers, {1,2,3,4,5,...}, is sometimes written N for short.
The whole numbers are the natural numbers together with 0.
(Note: a few textbooks disagree and say the natural numbers include 0.)
The sum of any two natural numbers is also a natural number (for example, 4+2000=2004), and the product of any two natural numbers is a natural number (4×2000=8000). This is not true for subtraction and division, though.
The Integers
The integers are the set of real numbers consisting of the natural numbers, their additive inverses and zero.
{...,−5,−4,−3,−2,−1,0,1,2,3,4,5,...}
The set of integers is sometimes written J or Z for short.
The sum, product, and difference of any two integers is also an integer. But this is not true for division... just try 1÷2.
The Rational Numbers
The rational numbers are those numbers which can be expressed as a ratio between two integers. For example, the fractions 13 and −11118 are both rational numbers. All the integers are included in the rational numbers, since any integer z can be written as the ratio z1.
All decimals which terminate are rational numbers (since 8.27 can be written as 827100.) Decimals which have a repeating pattern after some point are also rationals: for example,
0.0833333....=112.
The set of rational numbers is closed under all four basic operations, that is, given any two rational numbers, their sum, difference, product, and quotient is also a rational number (as long as we don't divide by 0).
The Irrational Numbers
An irrational number is a number that cannot be written as a ratio (or fraction). In decimal form, it never ends or repeats. The ancient Greeks discovered that not all numbers are rational; there are equations that cannot be solved using ratios of integers.
The first such equation to be studied was 2=x2. What number times itself equals 2?
2√ is about 1.414, because 1.4142=1.999396, which is close to 2. But you'll never hit exactly by squaring a fraction (or terminating decimal). The square root of 2 is an irrational number, meaning its decimal equivalent goes on forever, with no repeating pattern: