Standard form in Mathematics is a way of representing numbers (mostly large) compactly so that it is easy to comprehend the magnitude. A large expanded number is compressed in decimals and powers of 10. Standard forms are also used for polynomials, linear equations, rational numbers, etc. There are certain sets of rules to write the standard forms of decimals, rational numbers, polynomials, linear equations, and quadratic equations. The standard form may also be referred to as “scientific notation” because it has the maximum usage in scientific calculations. If the numbers are too small, negative indices of 10 can be used and if the numbers are large, positive indices of 10 can be used.
Let a/b be a rational number.
a/b may be considered in standard form only when b is a positive integer and the fraction is in its simplest form.
(a and b have no common divisor except 1)
How to convert a rational number in its standard form?
Verify if the denominator is more than zero. If it is negative, the numerator and the denominator must be multiplied with -1 to make the denominator positive.
Find the greatest common divisor (GCD) and reduce the fraction to the simplest form.
The resulting fraction must be the rational number in standard form.
Decimal numbers carry the base 10 and have 10 different numbers (0,1,2,3,4,5,6,7,8,9) for representation. The decimal point is denoted by a small dot between the numbers separating the ones and tenths place. The ten different digits take up different places in the system signifying different position values. Let us take several 5489.412 to demonstrate the position values of the decimal number system.
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In the above example, 5 is in the thousands place, 4 in the hundreds place, 8 in the tens place, 9 in one's place, 4 in the tenths place, 1 in the hundredths place, 2 in the thousandths place. Hence the number can be written as 5 thousand, 4 hundred, 8 tens, 9 ones, 4 tenths, 1 hundredth, and 2 thousandths.
Decimals can be represented in standard form or scientific form as:
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Decimals can be written in expanded form and further in exponential form:
Let us consider the number 256.5. The number in the expanded form will be:
256.5 = (2 x 100) + (5 x 10) + (6 x 1) + (5 x (\[\frac{1}{10}\]))
Now the number in exponential form would be:
256.5 = (2 x 10\[^{2}\]) + (5 x 10\[^{1}\]) + (6 x 10\[^{0}\]) + (5 x 10\[^{-1}\])
Hence, this is how decimal numbers in the exponential form are written.
This way of representing numbers in decimals and the powers of 10 were introduced when we started dealing with large numbers. With the advancement in science and technology, larger numbers popped into our calculations and the easiest way to tackle them was to introduce this standard form.
Write the first digit - 6
Introduce a decimal and write the remaining digits except for the zeroes at the end - 6.24
Count the number of digits after the initial digit and multiply 10 to the power of that number - 6.24 x 10\[^{7}\]
Answer: 6.24 x 10\[^{7}\]
Write the two numbers in standard form
Multiply the decimals and powers of 10 separately.
Take the result in standard form using the steps discussed in the previous problem.
85000 = 8.5 x 10\[^{4}\]
2000 = 2 x 10\[^{3}\]
∴ 85000 x 2000 = (8.5 x 2) x (10\[^{4+3}\]) = 17 x 10\[^{7}\] = 1.7 x 10\[^{8}\]
Hence the resulting product is 1.7 x 10\[^{8}\]
1. What is the Importance of Representing Numbers in Standard Form?
Ans: With the development of science and technology, we come across new data almost every day which needs to be processed efficiently. For small numbers within the range of 0-10000, it is relatively comfortable for us to evaluate results. However, with large numbers or very small numbers, it sometimes becomes difficult for us to process the data. Here is when the standard form comes into play.
The standard form helps us represent large expanded numbers with a large number of digits as compact numbers in the form of decimals and powers of 10. Standard forms are not only used in large numbers but are also used for linear equations, quadratic equations, and polynomials. Unnecessary calculation errors can be avoided using the standard form and it also reduces the effort and time taken to solve a particular problem with the given data.
2.What are Some Applications of the Standard Form?
Ans: There are numerous applications of the standard form in science and advanced mathematics. Scientists use the standard form in their research work when they work with the speed of light, the distance between the planets and galaxies, and other calculations related to big values. The size of microscopic entities like atoms, molecules, electrons, etc. is measured in the standard form. Even the size of bacteria and other microorganisms are measured in the standard for ease of interpretation.
For example, the distance between the sun and the earth is measured as 1.4967 × 10⁹ km, the mass of the electron is written as 9.1093 × 10⁻³¹ kg, the radius of the earth is 6.371 × 10⁶m.
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