Standard form

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What is standard form?

Standard form is a way of writing very large or very small numbers using powers of 10.
A number in standard form is written as A × 10ⁿ, where A is a number with 1 ≤ A < 10 and n is an integer.
In the expression A × 10ⁿ, A is called the coefficient (or significand) and n is called the exponent.
Standard form is also called scientific notation or standard index form.
It is useful because it avoids writing long strings of zeros.
On a scientific calculator, standard form is usually shown in SCI mode.
The exponent n can be positive, negative, or zero.
If the number is negative, a minus sign is placed before A, for example −3.5 × 10².

Writing large numbers in standard form

To write a large number in standard form, move the decimal point left until you have a number between 1 and 10.
The number of places you moved the decimal point becomes the positive exponent.
For example, 350 = 3.5 × 10² because the decimal point moves 2 places left.
Another example: 6,022,000,000,000,000,000,000,000 = 6.022 × 10²³.
If there is no decimal point shown, it is at the end of the number.
The exponent tells you how many places the decimal point has moved.

Writing small numbers in standard form

To write a small number (between 0 and 1) in standard form, move the decimal point right until you have a number between 1 and 10.
The number of places you moved the decimal point becomes the negative exponent.
For example, 0.5 = 5 × 10^-1 because the decimal point moves 1 place right.
Another example: 0.00000000000000000000000000000000000016 = 1.6 × 10^-35.
A negative exponent means the original number is less than 1.
The more places you move the decimal point, the larger the absolute value of the negative exponent.

Converting from standard form to ordinary numbers

To convert from standard form to an ordinary number, look at the exponent.
If the exponent is positive, move the decimal point in A that many places to the right.
If the exponent is negative, move the decimal point in A that many places to the left.
For example, 3.5 × 10² = 350.
For example, 5 × 10^-1 = 0.5.
If the exponent is 0, the number is just A, for example 7 × 10⁰ = 7.

Ordering numbers in standard form

Numbers in standard form can be compared easily by looking at their exponents.
A number with a larger exponent is larger than a number with a smaller exponent (when the numbers are positive).
If the exponents are the same, compare the coefficients A.
For example, 3 × 10⁴ > 2 × 10⁴ because 3 > 2.
For example, 2 × 10⁵ > 9 × 10⁴ because 5 > 4.
For negative numbers, the rules reverse: a number with a larger exponent is more negative (smaller).
Subtracting exponents gives an estimate of the number of orders of magnitude between two numbers.

Multiplying numbers in standard form

To multiply two numbers in standard form, multiply the coefficients and add the exponents.
Formula: (A × 10^m) × (B × 10ⁿ) = (A × B) × 10^m+n.
For example, (3 × 10⁴) × (2 × 10⁵) = (3 × 2) × 10⁴⁺⁵ = 6 × 10⁹.
If the product of the coefficients is not between 1 and 10, adjust it and change the exponent.
For example, (5 × 10³) × (4 × 10²) = 20 × 10⁵ = 2 × 10⁶.
Always check that the final coefficient is between 1 and 10.

Dividing numbers in standard form

To divide two numbers in standard form, divide the coefficients and subtract the exponents.
Formula: (A × 10^m) ÷ (B × 10ⁿ) = (A ÷ B) × 10^m−n.
For example, (8 × 10⁶) ÷ (2 × 10³) = (8 ÷ 2) × 10⁶⁻³ = 4 × 10³.
If the quotient of the coefficients is not between 1 and 10, adjust it and change the exponent.
For example, (3 × 10⁴) ÷ (6 × 10²) = 0.5 × 10² = 5 × 10¹.
Always check that the final coefficient is between 1 and 10.

Standard form in science contexts

Standard form is used in science to write very large distances, such as distances in space.
For example, the distance from the Earth to the Sun is about 1.5 × 10¹¹ metres.
It is also used for very small sizes, such as the sizes of cells.
For example, a typical cell might be about 1 × 10^-5 metres across.
Scientists, mathematicians, and engineers use standard form to simplify calculations with extreme numbers.
Standard form helps avoid errors when counting zeros.

Calculator and computer notation

Scientific calculators often display numbers in standard form using SCI mode.
Computers and calculators may use E notation, where E means 'times ten raised to the power of'.
For example, 6.022 × 10²³ is written as 6.022E23 or 6.022e23.
For example, 1.6 × 10^-35 is written as 1.6E-35 or 1.6e-35.
Sometimes the positive power is shown with a plus sign, like 6.022E+23.
E notation is common in computer output but is discouraged in published documents by some style guides.

Key points to remember

Standard form is A × 10ⁿ with 1 ≤ A < 10 and n an integer.
Positive exponent means a large number (≥ 10).
Negative exponent means a small number (between 0 and 1).
To convert to ordinary form, move the decimal point according to the exponent.
To multiply, multiply coefficients and add exponents.
To divide, divide coefficients and subtract exponents.
Always adjust the coefficient to be between 1 and 10 if needed.

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1.Which of these numbers is written correctly in standard form?
Easy
- A3.5 × 10²
- B35 × 10¹
- C350 × 10⁰
- D0.35 × 10³
2.Write 0.5 in standard form.
Easy
- A5 × 10^-1
- B5 × 10¹
- C0.5 × 10⁰
- D1 × 10^-2
3.What is 6.022 × 10²³ written in E notation?
Medium
- A6.022E23
- B6.022E-23
- C6022E23
- D6.022E+24
4.Which of the following are written in correct standard form? (select all that apply)
Medium
- A2.5 × 10³
- B10 × 10⁴
- C1.0 × 10⁰
- D0.5 × 10²
- E9.99 × 10^-5
5.In standard form, the exponent must be an integer.
Easy
True or false?
6.Match each ordinary number to its standard form.
Medium
- 0.0002
- 2000
- 0.02
- 200
- 2 × 10^-4
- 2 × 10³
- 2 × 10^-2
7.Arrange these numbers in standard form in ascending order (smallest first).
Medium
- 3.5 × 10²
- 3.5 × 10^-1
- 3.5 × 10⁰
- 3.5 × 10³
8.Arrange these values in standard form in ascending order (smallest first).
Medium
- 3.1 × 10^-2
- 9.4 × 10^-3
- 1.2 × 10⁰
- 5.0 × 10^-1

Thêm câu hỏi

9.Which of these numbers are written correctly in standard form? (select all that apply)
Medium
- A4.7 × 10⁵
- B0.8 × 10^-3
- C1.0 × 10⁰
- D12.3 × 10²
- E9.99 × 10^-7
10.What is 0.00304 written in standard form?
Medium
- A3.04 × 10^-3
- B3.04 × 10³
- C30.4 × 10^-4
- D0.304 × 10^-2
11.Which of these is the standard form of 7 million?
Medium
- A7 × 10⁶
- B7 × 10⁷
- C0.7 × 10⁷
- D70 × 10⁵
12.In standard form, the exponent n must be a whole number.
Easy
True or false?
13.The number 0.5 can be written as 5 × 10^-1 in standard form.
Medium
True or false?
14.Which of these statements are true for numbers written in standard form? (select all that apply)
Hard
- AA number with a larger exponent is larger than one with a smaller exponent.
- BThe coefficient A can be any number greater than 0.
- CThe exponent n can be negative.
- DThe coefficient A is always an integer.
- ESubtracting the exponents gives an estimate of the number of orders of magnitude between two numbers.
15.Which of the following is the standard form of 0.00062?
Medium
- A6.2 × 10^-4
- B6.2 × 10^-3
- C62 × 10^-5
- D0.62 × 10^-3
16.What is 3.4 × 10⁵ written as an ordinary number?
Medium
- A340000
- B34000
- C3400000
- D340
17.Calculate (2 × 10³) × (4 × 10²). Give your answer in standard form.
Medium
- A8 × 10⁵
- B8 × 10⁶
- C6 × 10⁵
- D8 × 10¹
18.Calculate (8 × 10⁵) ÷ (2 × 10²). Give your answer in standard form.
Medium
- A4 × 10³
- B4 × 10⁷
- C4 × 10^2.5
- D6 × 10³
19.Which of the following calculations are correct? (select all that apply)
Hard
- A(3 × 10⁴) × (2 × 10³) = 6 × 10⁷
- B(5 × 10⁶) ÷ (2 × 10²) = 2.5 × 10⁴
- C(4 × 10²) × (3 × 10⁵) = 12 × 10⁷
- D(6 × 10⁸) ÷ (3 × 10²) = 2 × 10⁴
- E(2 × 10³) × (5 × 10⁴) = 10 × 10⁷
20.When multiplying two numbers in standard form, you add the exponents.
Easy
True or false?
21.Match each calculation to its correct result in standard form.
Medium
- (2 × 10³) × (3 × 10²)
- (8 × 10⁵) ÷ (4 × 10²)
- (5 × 10⁴) × (2 × 10^-1)
- 6 × 10⁵
- 2 × 10³
- 1 × 10⁴
22.A star is 4.2 × 10¹³ km from Earth. What is this distance written as an ordinary number?
Medium
- A42000000000000
- B4200000000000
- C420000000000000
- D42000000000
23.A rectangle has length 4 × 10³ m and width 2 × 10² m. What is its area in standard form?
Medium
- A8 × 10⁵ m²
- B8 × 10⁶ m²
- C6 × 10⁵ m²
- D8 × 10⁴ m²
24.A car travels 1.2 × 10⁵ m in 4 × 10³ s. What is its average speed in m/s, in standard form?
Medium
- A3 × 10¹ m/s
- B3 × 10² m/s
- C0.3 × 10² m/s
- D4.8 × 10⁸ m/s
25.A cylinder has radius 2 × 10^-1 m and height 5 × 10¹ m. Using π ≈ 3.14, what is its volume in m³, in standard form (to 2 significant figures)?
Medium
- A6.3 × 10⁰ m³
- B6.3 × 10¹ m³
- C1.3 × 10¹ m³
- D6.3 × 10^-1 m³
26.A rectangle has area 6 × 10⁴ m² and width 3 × 10² m. What is its length in standard form?
Hard
- A2 × 10² m
- B2 × 10⁶ m
- C1.8 × 10⁷ m
- D2 × 10¹ m
27.A triangle has base 6 × 10³ m and height 4 × 10² m. Using A = \frac{1}{2} \times b \times h, what is its area in standard form?
Medium
- A1.2 × 10⁶ m²
- B2.4 × 10⁶ m²
- C1.2 × 10⁵ m²
- D1.2 × 10⁷ m²
28.A rectangular garden has length 7 × 10² m and width 3 × 10² m. What is its perimeter in standard form?
Medium
- A2 × 10³ m
- B2.1 × 10³ m
- C2 × 10² m
- D2.1 × 10⁵ m
29.In the standard form expression A × 10ⁿ, what is the number A called?
Easy
- Athe exponent
- Bthe significand
- Cthe base
- Dthe power
30.In the standard form expression A × 10ⁿ, what is the number n called?
Easy
- Athe significand
- Bthe mantissa
- Cthe exponent
- Dthe coefficient
31.Which display mode on a scientific calculator is usually used to show numbers in scientific notation?
Easy
- AFIX
- BSCI
- CENG
- DNORM
32.In engineering notation, the exponent n is restricted to which values?
Easy
- Aany integer
- Bmultiples of 2
- Cmultiples of 3
- Donly positive integers
33.In engineering notation, the significand m lies in which range?
Easy
- A1 ≤ |m| < 10
- B1 ≤ |m| < 100
- C1 ≤ |m| < 1000
- D0 < |m| < 1
34.Which of these numbers is written in engineering notation?
Easy
- A1.25 × 10^-8
- B12.5 × 10^-9
- C125 × 10^-10
- D0.125 × 10^-7
35.In E notation, how is 1.6 × 10^-35 written?
Easy
- A1.6E35
- B1.6E-35
- C1.6E+35
36.In standard form A × 10ⁿ, what must the value of A satisfy?
Easy
- AA < 1
- BA = 10
- C1 ≤ A < 10
- DA > 10
37.Which of these numbers is written in standard form?
Medium
- A0.5 × 10³
- B5 × 10²
- C50 × 10¹
- D500 × 10⁰
38.In standard form, what is 10⁰ equal to?
Medium
- A0
- B1
- C10
- D100
39.When a number in standard form is converted to engineering notation, the exponent is adjusted so that it is a multiple of 3. Which of these is the engineering notation form of 2.5 × 10^-4?
Medium
- A2.5 × 10^-4
- B25 × 10^-5
- C250 × 10^-6
- D0.25 × 10^-3
40.In standard form, what is 10^-3 as an ordinary number?
Medium
- A0.001
- B0.01
- C0.1
- D1000
41.Complete the sentence about standard form.
Easy
In standard form, a number is written as A × 10ⁿ where 1 ≤ A < ____ and n is an ____.
42.What name is given to the number A in the standard form expression A × 10ⁿ?
Easy
43.Write 350 in standard form. Give your answer as a number in the form A × 10ⁿ, but enter only the value of A.
Medium
44.Write 0.00045 in standard form. Give your answer as a number in the form A × 10ⁿ, but enter only the value of n.
Medium
45.Write 0.0000081 in standard form. Enter only the value of A (the number between 1 and 10).
Medium
46.Write 45 600 000 in standard form. Enter only the value of n (the exponent).
Medium
47.Write 0.00000074 in standard form. Enter only the value of A (the number between 1 and 10).
Medium
48.Complete the sentence about converting a number to standard form.
Medium
To write 45000 in standard form, the decimal point is moved ____ places to the left, giving 4.5 × 10^____.
49.A red blood cell has a diameter of 0.0000075 m. Write this diameter in metres in standard form. Enter only the value of A (the number between 1 and 10).
Easy
50.A grain of sand has a mass of 0.00062 g. Write this mass in grams in standard form. Enter only the value of n (the exponent).
Medium
51.What name is given to the number A in the standard form expression A × 10ⁿ? Give a single word.
Medium
52.What name is given to the number n in the standard form expression A × 10ⁿ? Give a single word.
Medium
53.Explain why 12.5 × 10^-9 m is written in engineering notation rather than standard form.
Medium
54.Explain how to multiply two numbers written in standard form, giving the result in standard form.
Hard
55.Explain how to divide two numbers written in standard form, giving the result in standard form.
Hard
56.Write 350 in standard form. Give your answer as a number in the form A × 10ⁿ, but enter only the value of A.
Medium
- A5.5
- B4.5
- C2.5
- D3.5
57.Write 0.00045 in standard form. Give your answer as a number in the form A × 10ⁿ, but enter only the value of n.
Medium
- A-3
- B-4
- C-2
- D-5
58.Write 0.0000081 in standard form. Enter only the value of A (the number between 1 and 10).
Medium
- A10.1
- B9.1
- C7.1
- D8.1
59.Write 45 600 000 in standard form. Enter only the value of n (the exponent).
Medium
- A9
- B6
- C8
- D7
60.Write 0.00000074 in standard form. Enter only the value of A (the number between 1 and 10).
Medium
- A9.4
- B7.4
- C6.4
- D8.4
61.A red blood cell has a diameter of 0.0000075 m. Write this diameter in metres in standard form. Enter only the value of A (the number between 1 and 10).
Easy
- A6.5 m
- B7.5 m
- C8.5 m
- D9.5 m
62.A grain of sand has a mass of 0.00062 g. Write this mass in grams in standard form. Enter only the value of n (the exponent).
Medium
- A-2 g
- B-5 g
- C-4 g
- D-3 g

Ghi chú ôn tập

## Standard Form (KS3 Number) **Standard form** writes very large or very small numbers compactly as: \(A \times 10^{n}\) where **1 ≤ A < 10** and **n** is an integer. - A is the **coefficient** (also called the significand or mantissa). - n is the **exponent** (the power of 10). - A must be at least 1 but less than 10. ### Converting ordinary numbers → standard form - Move the decimal point until exactly one non-zero digit is in front of it — that gives A. - Count how many places you moved: this is n. - Moving left → positive n. Moving right → negative n. Examples: - 350 = 3.5 × 10^2 - 0.5 = 5 × 10^{-1} ### Converting standard form → ordinary numbers - If n is positive, move the decimal point n places to the right (makes the number larger). - If n is negative, move it n places to the left (makes the number smaller). ### Ordering numbers in standard form - Compare the exponents first: the bigger exponent means the bigger number. - If exponents are equal, compare the A values. ### Multiplying and dividing - Multiply the A values, then add the exponents: \((A \times 10^{m}) \times (B \times 10^{n}) = (A \times B) \times 10^{m+n}\) - Divide the A values, then subtract the exponents: \(\frac{A \times 10^{m}}{B \times 10^{n}} = \frac{A}{B} \times 10^{m-n}\) - Always re-check that the final A is still between 1 and 10; adjust the exponent if needed. ### Science contexts Standard form is used for distances in space and the sizes of cells — numbers too long to write out in full. ### Calculator note On scientific calculators this is the **SCI** display mode. Computer output often uses **E notation**: 6.022E23 means 6.022 × 10^23.

Standard form: A way of writing numbers as A × 10^n where 1 ≤ A < 10 and n is an integer. Also called scientific notation or standard index form.
Coefficient (significand / mantissa): The number A in A × 10^n. In standard form it must be at least 1 and less than 10.
Exponent: The integer n in A × 10^n, showing the power of ten.
Normalized notation: Scientific notation where the exponent is chosen so the coefficient is at least 1 and less than 10 — the form called standard form in the UK.
Engineering notation: Like standard form but the exponent is restricted to multiples of 3, so the coefficient is between 1 and 1000.
E notation: A calculator/computer shorthand where E or e means 'times ten raised to the power of', e.g. 6.022E23 = 6.022 × 10^23.

Thẻ ghi nhớ

What is the general form of a number written in standard form?

A number in standard form is written as A × 10ⁿ, where 1 ≤ A < 10 and n is an integer.

In standard form A × 10ⁿ, what is the number A called?

A is called the coefficient (or significand). It must be at least 1 and less than 10.

In standard form A × 10ⁿ, what is n called and what type of number is it?

n is called the exponent (or index). It is an integer, which can be positive, negative or zero.

Write 350 in standard form.

350 = 3.5 × 10²

Write 0.5 in standard form.

0.5 = 5 × 10^-1

How do you convert a large number like 6 000 000 into standard form?

Move the decimal point left until you have a number between 1 and 10, then count the moves as a positive power of 10. 6 000 000 = 6 × 10⁶.

How do you convert a small number like 0.0004 into standard form?

Move the decimal point right until you have a number between 1 and 10, then count the moves as a negative power of 10. 0.0004 = 4 × 10^-4.

Convert 2.7 × 10³ to an ordinary number.

2.7 × 10³ = 2700

Convert 8.1 × 10^-2 to an ordinary number.

8.1 × 10^-2 = 0.081

When ordering numbers in standard form, what should you compare first?

Compare the exponents first. The number with the larger exponent is larger (for positive numbers). If exponents are equal, compare the coefficients.

Which is larger: 3.2 × 10⁴ or 9.8 × 10³?

3.2 × 10⁴ is larger because its exponent (4) is greater than the exponent (3) of 9.8 × 10³.

How do you multiply two numbers in standard form, e.g. (2 × 10³) × (4 × 10⁵)?

Multiply the coefficients and add the exponents: (2 × 4) × 10³⁺⁵ = 8 × 10⁸. Adjust if the new coefficient is not between 1 and 10.

How do you divide two numbers in standard form, e.g. (6 × 10⁷) ÷ (2 × 10²)?

Divide the coefficients and subtract the exponents: (6 ÷ 2) × 10^7-2 = 3 × 10⁵. Adjust if the new coefficient is not between 1 and 10.

What must you do if, after multiplying or dividing, the coefficient is not between 1 and 10?

Adjust the coefficient and exponent. For example, 12 × 10⁵ becomes 1.2 × 10⁶ (move the decimal point one place left and increase the exponent by 1).

Why is standard form useful in science?

It makes very large and very small numbers easier to write, read and compare, such as distances in space or the sizes of cells.

What does the 'E' notation on a calculator mean, e.g. 6.022E23?

E means 'times ten raised to the power of'. So 6.022E23 means 6.022 × 10²³.

What is the exponent of 10 when a number is written without a power of 10, e.g. 7?

The exponent is 0, because 10⁰ = 1. So 7 = 7 × 10⁰.

Is 35 × 10² in standard form? Explain.

No, because the coefficient 35 is not between 1 and 10. It should be written as 3.5 × 10³.

Write 0.0000006 in standard form.

6 × 10^-7

Write 4.5 × 10⁰ as an ordinary number.

4.5

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