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7.3 | Standard Form |

In these revision notes for Standard Form, we cover the following key points:

- What is the standard form? Why it is used?
- What is the relationship between standard form and other forms of expressing numbers?
- Why standard form is more suitable than the decomposed form when making operations with numbers?
- How to write decimals in standard form and vice-versa?
- How to add and subtract numbers in standard form? Why this is much easier than adding or subtracting in the decomposed form?
- How to multiply and divide numbers in standard form?
- How to write very big and very small numbers in standard form?
- How to deal with powers of numbers written in standard form?

Writing numbers in the decomposed form is very useful to understand what value each digit has. However, this method is too long. It is better to have the numbers written in a shorter way while preserving some features of expanded or decomposed form. Thus, we write only the power of the leftmost digit while the original number is written in two parts: one decimal part that gives all digits of the original number and another part that gives the powers of ten. This method of writing numbers is known as the **standard form**. It therefore represents a form of writing numbers in terms of powers of ten.

Any number N written in standard form has the general structure

N = A × 10^{m}

where A is a number between 1 and 10 (without reaching the value 10) while m is an integer that represents the highest power of the decomposed form of that number.

If a number W in the ordinary form has n digits, when written in the standard form it becomes

W = A × 10^{n - 1} + B × 10^{n - 2} + ⋯ + (N-1) × 10^{1} + N × 10^{0}

= A.BC…N × 10^{n - 1}

= A.BC…N × 10

where A, B, … , N are the coefficients representing the place values of the original number.

We can extend the reasoning used for numbers written in standard form beyond the decimal point to also include the decimal numbers. A decimal that is written in the form

N = abcd.efgh

(where abcd is the whole part while efgh is the decimal part) can be expressed in the decomposed form as

N = a × 10^{3} + b × 10^{2} + c × 10^{1} + d × 10^{0} + e × 10^{-1} + f × 10^{-2} + g × 10^{-3} + h × 10^{-4}

or

N = a × 1,000 + b × 100 + c × 10 + d × 1 + e × *1**/**10* + f × *1**/**100* + g × *1**/**1,000* + h × *1**/**10,000*

When written in the standard form, this number becomes

N = a.bcdefgh × 10^{3}

If we add or subtract two numbers in the decomposed form, we must be careful to make the proper arrangements if any coefficient becomes less than 1 or more than 10. This is like adding or subtracting like terms in the numerical or algebraic expressions.

To perform the operations of addition and subtraction, we must first convert all numbers at the same power of ten. Then, the numerical parts before the powers of ten are added or subtracted depending on the situation while the powers of ten are written only once.

On the other hand, in subtraction we often need to borrow values from the next term on the right when dealing with numbers in either the decomposed or standard form. At this point, we must do the operations but while taking into account the borrowing method used in subtraction of ordinary numbers. Again, all corrections are made from right to left.

In multiplication and division of numbers in standard form, it is better to not involve the expanded form because of the difficulties in handling such numbers.

When multiplying two numbers in the standard form, the numerical parts before the powers of ten multiply while indices that show the powers of ten add to each other. In symbols, we have

(A × 10^{m} ) ∙ (B × 10^{n} ) = (A ∙ B) × 10^{m + n}

All corrections derived from the fact that A · B may be greater than 10 are made after writing the result of operation in the above form.

We normally use the standard form for convenience - to express either very large or very small numbers.

The general rules applied when converting very small or very large ordinary numbers into standard form are as follows:

- If a whole number A = abcdef…n is very big, the decimal point shifts n - 1 positions due left and the index of ten also becomes n - 1, where n is the number of digits of the original number. Hence, the number becomes A = a.bcdef…n × 10
^{n-1}. For example, in the number A = 327,416, we have n = 6. Thus, we obtain for the standard form of this number: A = 3.27416 × 10^{5}. - If a big number contains a decimal part (if it is not whole therefore), the above rule is applied by starting from the decimal point. However, the digits after the decimal point are all written when the number is converted into the standard form. For example, if A = 30,681.27 is converted to standard form (n = 5), it becomes A = 3.068127 × 10
^{4}. - If a very small decimal number B = 0.0000abcd needs to be converted into standard form, we shift the decimal point from its actual position to after the first non-zero digit (here after a). As for the index, we count how many positions the decimal point has shifted and this number is written as a negative index. In our case, we have a shift by five positions due right, so B = a.bcd × 10
^{-5}.

If a number written in the standard form is raised to a certain power, we raise in that power only the numerical part before the power of ten, while the powers of ten multiply. In symbols, we have

(A × 10^{n} )^{m} = A^{m} × 10^{n ∙ m}

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