Decimal Numbers Standard Form

Standard Form Math Definition

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.

The Standard Form of a Rational Number

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.

Representation of Decimal Numbers

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. 

Solved Examples

1. Write 62400000 in 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\[^{7}\]

2. Multiply 85000 and 2000 and Return the Result in Standard Form.

• 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\[^{8}\]