Decimals and all types of numbers
Posted: Dec 23, 2015
A decimal number is a number with a decimal point in it, like these: 1.5, 3.21, 4.173, 5.1 etc
The number to the left of the decimal is an ordinary whole number. The first number to the right of the decimal is the number of tenths (1/ 10’ s). The second is the number of hundredths (1/ 100’ s) and so on. So, for the number 5.1, this is a shorthand way of writing the mixed number 1 5/ 10. 3.27 is the same as 3 + 2/ 10 + 7/ 100.
A word on where decimals originate from:
Consider the situation where there are 5 children and you have to distribute 10 chocolates between them in such a way that all the chocolates should be distributed and each child should get an equal number of chocolates? How would you do it? Well, simple? divide 10 by 5 to get 2 chocolates per child.
Now consider what if you had to do the same thing with 9 chocolates amongst 5 children? In such a case you would not be able to give an integral number of chocolates to each person. You would give 1 chocolate each to all the 5 and the ‘remainder’ 4 would have to be divided into 5 parts. 4 out of 5 would give rise to the decimal 0.8 and hence you would give 1.8 chocolates to each child. That is how the concept of decimals enters mathematics in the first place.
Taking this concept further, you can realize that the decimal value of any fraction essentially emerges out of the remainder when the numerator of the fraction is divided by the denominator. Also, since we know that each divisor has a few defined remainders possible, there would be a limited set of decimals that each denominator gives rise to.
Thus, for example the divisor 4 gives rise to only 4 remainders (viz. 0,1,2 and 3) and hence it would give rise to exactly 4 decimal values when it divides any integer. These values are:
0 (when the remainder is 0)
.25 (when the remainder is 1)
.50 (when the remainder is 2)
.75 (when the remainder is 3)
There would be similar connotations for all integral divisors? although the key is to know the decimals that the following divisors give you:
2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 15, 16
18, 20, 24, 25, 30, 40, 50, 60, 80, 90, 120
Composite Numbers It is a natural number that has at least one divisor different from unity and itself.
Every composite number n can be factored into its prime factors. (This is sometimes called the canonical form of a number.)
In mathematical terms: n = p1m. p2n …pks, where p1, p2… pk are prime numbers called factors and m, n… k are natural numbers.
Thus, 24 = 23? 3, 84 = 7? 3? 22 etc.
This representation of a composite number is known as the standard form of a composite number. It is an extermely useful form of seeing a composite number as we shall see.
Whole Numbers The set of numbers that includes all natural numbers and the number zero are called whole numbers. Whole number are also called as Non-negative integers.
The Concept of the Number Line The number line is a straight line between negative infinity on the left to positive infinity to the right.
The distance between any two points on the number line is got by subtracting the lower value from the higher value. Alternately, we can also start with the lower number and find the required addition to reach the higher number.
For example: The distance between the points 7 and –4 will be 7 – (– 4) = 11.
Real Numbers All numbers that can be represented on the number line are called real numbers. Every real number can be approximately replaced with a terminating decimal.
The following operations of addition, subtraction, multiplication and division are valid for both whole numbers and real numbers: [For any real or whole numbers a, b and c].
(a) Commutative property of addition: a + b = b + a.
(b) Associative property of addition: (a + b) + c = a + (b + c).
(c) Commutative property of multiplication: a? b = b? a.
(d) Associative property of multiplication: (a? b)? c = a? (b? c).
(e) Distributive property of multiplication with respect to addition: (a + b) c = ac + bc.
(f) Subtraction and division are defined as the inverse operations to addition and multiplication respectively.
Thus if a + b = c, then c – b = a and if q = a/ b then b? q = a (where b? 0).
Division by zero is not possible since there is no number q for which b? q equals a non zero number a.
Once you have understood the number system practice it. The key to success in Mat 2015 however is your practice. Try to solve problem within time limit which can help you increase accuracy with speed. Remember, consistent practice and revision of different areas in Number system is key to your success in Quant section.
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