Standard form in Math is mentioned for basically decimal numbers, equations, polynomials, linear equation, etc. Its correct definition could be explained better in terms of decimal numbers and following certain rules. As we know, the decimal numbers are the simplified form of fractions. Some fractions give decimal numbers which have numbers after decimal at thousandths, hundredths or tenths place. But there are some fractions, which gives a big decimal number. To represent such big numbers, we use such simpler forms, which is also stated as Scientific notation.
Basically, we can say, it is the representation of rational numbers in standard form. Rational numbers are numbers which can be represented in the form of p/q, where p and q are both integers. For example, 1/10, 4/5, 8/9, etc.
Standard Form of a Rational Number
A rational number ‘p/q’ is said to be in the standard form if the denominator q is positive and both the integers a and b have no common divisor other than 1.
Steps to convert a rational number into the standard form:
- Write the given rational number.
- Check if the denominator is positive or negative. If it is negative, then we need to multiply both numerator and denominator by -1, to make the denominator positive.
- Now, find out the greatest common divisor (GCD) of numerator and denominator, which could be cancelled.
- Divide the numerator and denominator by GCD.
- The obtained number is the Standard form of the given rational number.
Representation of Decimal Numbers
Decimal numbers employ 10 as the base and require 10 different numerals and a dot for the representation of its numbers. In this system, the digits used in denoting the number take different place values depending upon their position. For example, the number 645.221 can be written as 6 hundreds 4 tens 5 ones 2 tenths 2 hundredth and 1 thousandth.
In this section, we shall learn about the representation of decimal numbers in the exponential form.
Decimal numbers in Exponential Form
Let us consider a 6 digit number 657891. In the expanded form this number can be written as:
657891 = 6 × 100000 + 5 × 10000 + 7 × 1000 + 8 × 100 + 9 × 10 + 1
We can express this number in the exponential form as well, as we know that 10000=104, 10 = 100 and so on.
657891 = 6 × 105 + 5 × 104 + 7 × 103 + 8 × 102 + 9 × 101 + 1 × 100
Example 1: Represent 567.21 in exponential form.
567.21 = (5 × 100) + (6 × 10) + 7 + (2 × 1/10) + (1 × 1/100)
Writing it in the exponential form:
567.21 = 5 × 102 + 6 × 101 + 7 × 100 + 2 × 10-1 + 1 × 10-2
Expressing Large Numbers
Any number can be expressed as decimal numbers between 1 and 10 multiplied by a power of 10. This form of representation is termed as the standard form. We can understand this concept with the following examples.
500 = 5 × 102
567 = 5.67 × 102
56.78 = 5.678 × 101
The advancement in science and technology has introduced us with numbers as big as the diameter of the Earth and as small as the size of a human cell. In order to represent these numbers, we use the exponential form to make their reading and writing more convenient.
Example 2: Represent the distance between the Earth and the Sun in exponential form.
Solution: The distance of the Earth from the Sun is 1496000000 km.
1496000000Km = 1.496 × 109 Km
Example 3: Express the size of blood cells in the standard form.
Solution: The average size of blood cells in a human body is 0.000015 m
0.000015 = 1.5 × 10-5 m
Thus, we can say that the numbers when written in the standard form are much easier to read, understand and compare than when they are written in their actual form.
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