STANDARD FORM

What is Standard Form?

Did you know that Earth is 4,543,000,000 years old? We know you skipped past all those zeroes, so we will give you a simpler way of saying it: Earth is 4.543 billion years old. Do you see how reading numbers a certain way makes them easier to understand? That’s why all the mathematicians in the world decided to agree on some rules on writing mathematical concepts so that it is convenient for everyone to read, write, and work with. This particular way is called the standard form.

All of the things you see in math, like numbers or fractions or equations or expressions, have a standard form defined for them. We can think of the standard form as the most common way of representing a mathematical element.

Any number that we can write as a decimal number, between 1.0 and 10.0, multiplied by a power of 10, is said to be in standard form.

Standard form is a way of writing down very large or very small numbers easily. 103 = 1000, so 4 × 103 = 4000 . So 4000 can be written as 4 × 10³ . This idea can be used to write even larger numbers down easily in standard form.

Small numbers can also be written in standard form. However, instead of the index being positive (in the above example, the index was 3), it will be negative. The rules when writing a number in standard form is that first you write down a number between 1 and 10, then you write × 10(to the power of a number).

Steps for writing standard form

The steps to write the standard form of a number are as follows:

Step 1: Move the decimal point to the first non-zero number. Count as you move.

Step 2: Exclude all zeros at the end of the decimal number (NOTE: Zeros between non-zero numbers should be maintained)

Step 3: Multiply the new decimal number by 10 to the power the number of counts (10^{No. of counts})

Step 4: If you have to move the decimal point backward, add negative (-) to the number of counts (10^{- No. of counts})

NOTE: Every whole number has a decimal point (.0) after that number which is not written. E.g. 1 = 1.0, 2 = 2.0, 3 = 3.0, 4 = 4.0, 5 = 5.0, 6 = 6.0, 7 = 7.0, 12345 = 12345.0, etc.

Illustration

Example 1

Write 67300000000 in standard form by following the steps above.

Step 1

Move the decimal point to the first non-zero number. Count as you move.

67300000000 = 67300000000.0 (There is a decimal point after every whole number which is not written).

The first non-zero number is 6, moving the decimal point in (67300000000.0) to 6, is 10 counts.

6.⌒7⌒3⌒0⌒0⌒0⌒0⌒0⌒0⌒0⌒0.0

Step 2

Exclude all zeros at the end of the decimal number (NOTE: Zeros between non-zero numbers should be maintained)

New decimal number = 6.73 (All the zeros after 3 are excluded).

Step 3

Multiply the new decimal number by 10 to the power the number of counts (10^{No. of counts})

6.73 x 10¹⁰

Example 2

Write 1231002300 in standard form

Step 1

Move the decimal point to the first non-zero number. Count as you move.

1231002300 = 1231002300.0 (There is a decimal point after every whole number which is not written).

The first non-zero number is 1, moving the decimal point in (1231002300.0) to 1, is 9 counts.

1.⌒2⌒3⌒1⌒0⌒0⌒2⌒3⌒0⌒0.0

Step 2

Exclude all zeros at the end of the decimal number (NOTE: Zeros between non-zero numbers should be maintained)

New decimal number = 1.2310023 (All the zeros after 3 are excluded).

NOTE: The zeros between 1 and 2 are maintained because they are between non-zero numbers.

Step 3

Multiply the new decimal number by 10 to the power the number of counts (10^{No. of counts})

1.2310023 x 10⁹

Example 3

Write 523.1023 in standard form

Step 1

Move the decimal point to the first non-zero number. Count as you move.

The first non-zero number is 5, moving the decimal point in (523.1023) to 5, is 2 counts.

5.⌒2⌒3.1023

Step 2

Multiply the new decimal number by 10 to the power the number of counts (10^{No. of counts})

5.231023 x 10²

Example 4

0.0000017

Step 1

Move the decimal point to the first non-zero number. Count as you move.

The first non-zero number is 1, moving the decimal point in (0.0000017) to 1, is 6 counts.

0.⌒0⌒0⌒0⌒0⌒0⌒1.7

Step 2

Multiply the new decimal number by 10 to the power the number of counts (10^{No. of counts})

Step 3

If you have to move the decimal point backward, add negative (-) to the number of counts (10^{- No. of counts})

1.7 x 10^-6

APPLICATION OF KNOWLEDGE

Question 1

The distance between two towns is 12875 km. Express this distance in standard form.

A. 1.2875 x 10³ km B. 1.2875 x 10⁴ km C. 12.875 x 10³ km D. 128.75 x 10² km

[B.E.C.E 1990 Section A Question 14]

Solution

Question 2

Write 4687.02 in standard form.

A. 46.8702 x 10³ B. 46.8702 x 10⁴ C. 4.68702 x 10⁵ D. 4.68702 x 10³

[B.E.C.E 1991 Section A Question 5]

Solution

Question 3

Express 0.0043216 in standard form.

A. 4.3216 x 10^-4 B. 4.3216 x 10^-3 C. 4.3216 x 10 D. 4.3216 x 10³

[B.E.C.E 1994 Section A Question 20]

Solution

Question 4

Write in standard form 1342.

A. 0.1432 x 10^-3 B. 0.1432 x 10^-4 C. 13.42 x 10² D. 1.342 x 10³

[B.E.C.E 1999 Section A Question 4]

Solution

Question 5

Express 2345 in standard form.

A. 2.345 x 10¹ B. 2.345 x 10² C. 2.345 x 10³ D. 2.345 x 10⁴

[B.E.C.E 2001 Section A Question 18]

Solution

Question 6

Express 2474.5 in standard form.

A. 2.4745 x 10² B. 2.4745 x 10³ C. 2.4745 x 10^-2 D. 2.4745 x 10^-3

[B.E.C.E 2002A Section A Question 4]

Solution

Question 7

Write 83000 in standard form.

A. 8.3 x 10^-4 B. 8.3 x 10^-3 C. 8.3 x 10³ D. 8.3 x 10⁴

[B.E.C.E 2003 Section A Question 18]

Solution

Question 8

Express 962 in standard form.

A. 96.2 x 10 B. 9.62 x 10² C. 0.962 x 10³ D. 0.0962 x 10²

[B.E.C.E 2006 Section A Question 16]

Solution

Question 9

Express 0.000344 in standard form.

A. 3.44 x 10^-6 B. 3.44 x 10^-5 C. 3.44 x 10^-4 D. 3.44 x 10^-3

[B.E.C.E 2019 Section A Question 2]

Solution

Question 10

Write 1930.54 in standard form.

A. 1.93054 X 10³ B. 1.93054 X 10^-3 C. 1.93054 X 10^-2 D. 1.93054 X 10²

[B.E.C.E 2022 Section A Question 22]

Solution