53,000,000 = 53 x 10⁶
0.002 = 2 x 10^-3
0.0004 = 4 x 10^-4

53,000,000 x 0.002 = 53 x 10⁶ x 2 x 10^-3

Law of multiplication of indices

a^m x aⁿ = a^{m + m}

53 x 10⁶ x 2 x 10^-3 = 53 x 2 x 10^6-3
53 x 10⁶ x 2 x 10^-3 = 106 x 10³

$\frac{\mathrm{53,000,000\; x\; 0.002}}{0.0004}$ = $\frac{\mathrm{106\; x}{10}^3}{\mathrm{4\; x}{10}^{-4}}$

Law of division of indices

$\frac{{\mathrm{a}}^{\mathrm{m}}}{{\mathrm{a}}^{\mathrm{n}}}$ = a^{m - n}

$\frac{\mathrm{106\; x}{10}^3}{\mathrm{4\; x}{10}^{-4}}$ = 26.5 x 10^3-(-4)

$\frac{\mathrm{106\; x}{10}^3}{\mathrm{4\; x}{10}^{-4}}$ = 26.5 x 10³⁺⁴

$\frac{\mathrm{106\; x}{10}^3}{\mathrm{4\; x}{10}^{-4}}$ = 26.5 x 10⁷

$\frac{\mathrm{106\; x}{10}^3}{\mathrm{4\; x}{10}^{-4}}$ = 2.65 x 10¹ x 10⁷

$\frac{\mathrm{106\; x}{10}^3}{\mathrm{4\; x}{10}^{-4}}$ = 2.65 x 10¹⁺⁷

$\frac{\mathrm{106\; x}{10}^3}{\mathrm{4\; x}{10}^{-4}}$ = 2.65 x 10⁸

Note: For standard form, the decimal point must be after the first non-zero digit.

Change the decimal numbers to the form a x 10ⁿ, where a is a whole number.

0.42 = 0.⌒4⌒2 = 42 x 10^-2

2.5 = 2.⌒5 = 25 x 10^-1

0.5 = 0.⌒5 = 5 x 10^-1

2.05 = 2.⌒0⌒5 = 205 x 10^-2

Note: the power is negative since the decimal is moved to the right.

0.42 ÷ 2.5 = $\frac{\mathrm{42\; x}{10}^{-2}}{\mathrm{25\; x}{10}^{-1}}$

a^m ÷ aⁿ = $\frac{{\mathrm{a}}^{\mathrm{m}}}{{\mathrm{a}}^{\mathrm{n}}}$ = a^{m - n}

0.42 ÷ 2.5 = $\frac{42}{25}$ x 10^-2-(-1)

0.42 ÷ 2.5 = $\frac{42}{25}$ x 10^{-2 + 1}

0.42 ÷ 2.5 = $\frac{42}{25}$ x 10^-1

0.5 x 2.05 = 5 x 10^-1 x 205 x 10^-2

a^m x aⁿ = a^{m + m}

0.5 x 2.05 = 5 x 205 x 10^{-1 + -2}

0.5 x 2.05 = 5 x 205 x 10^-3

$\frac{\mathrm{0.42\; ÷\; 2.5}}{\mathrm{0.5\; x\; 2.05}}$ = $\frac{42}{25}$ x 10^-1 ÷ (5 x 205 x 10^-3)

Every number is being divided by 1

5 x 205 x 10^-3 = $\frac{\mathrm{5\; x\; 205\; x}{10}^{-3}}{1}$

$\frac{\mathrm{0.42\; ÷\; 2.5}}{\mathrm{0.5\; x\; 2.05}}$ = $\frac{42}{25}$ x 10^-1 ÷ $\frac{\mathrm{5\; x\; 205\; x}{10}^{-3}}{1}$

Reciprocate the fraction at the right of the ÷ and change ÷ to multiplication (x)

Note: reciprocate means the numerator becomes the denominator and the denominator becomes the numerator.

$\frac{\mathrm{0.42\; ÷\; 2.5}}{\mathrm{0.5\; x\; 2.05}}$ = $\frac{42}{25}$ x 10^-1 x $\frac{1}{\mathrm{5\; x\; 205\; x}{10}^{-3}}$

$\frac{\mathrm{0.42\; ÷\; 2.5}}{\mathrm{0.5\; x\; 2.05}}$ = $\frac{\mathrm{42\; x\; 1}}{\mathrm{25\; x\; 5\; x\; 205}}$ x 10^-1-(-3)

$\frac{\mathrm{0.42\; ÷\; 2.5}}{\mathrm{0.5\; x\; 2.05}}$ = $\frac{42}{25625}$ x 10^{-1 + 3}

$\frac{\mathrm{0.42\; ÷\; 2.5}}{\mathrm{0.5\; x\; 2.05}}$ = 0.0016390 x 10²

0.0016390 = 1.639 x 10^-3 (Standard form)

$\frac{\mathrm{0.42\; ÷\; 2.5}}{\mathrm{0.5\; x\; 2.05}}$ = 1.639 x 10^-3 x 10²

$\frac{\mathrm{0.42\; ÷\; 2.5}}{\mathrm{0.5\; x\; 2.05}}$ = 1.639 x 10^{-3 + 2}

$\frac{\mathrm{0.42\; ÷\; 2.5}}{\mathrm{0.5\; x\; 2.05}}$ = 1.639 x 10^-1

Notes:

A standard form should be in the form a.bcde x 10ⁿ

The decimal point must be after the first non-zero digit.

The n is the number of times the decimal point has to be moved to be right after the first non-zero digit.

If it is moved to the left, the n is positive and if to the right, n is negative.

Significant figures, any of the digits of a number beginning with the digit farthest to the left that is not zero and ending with the last digit farthest to the right that is either not zero or that is a zero but is considered to be exact.

The significant figures are 9, 4, 5 and 3 and the third significant figure is 5. Since the next figure to the third is less than 5, 1 is not added to the third significant figure. The fourth significant figure becomes 0 hence the significant figure is 9450 x 10^-6

9450 x 10^-6 = 0.⌒0⌒0⌒9⌒4⌒5⌒0.0

Significant figures, any of the digits of a number beginning with the digit farthest to the left that is not zero and ending with the last digit farthest to the right that is either not zero or that is a zero but is considered to be exact.

0.0063075

The leading zeros (0) are not significant.

The significant figures are 6, 3, 0, 7 and 5

The third significant falls on 0 but the next significant to it is more than 4 hence 1 is added to the 0 (0 + 1) making 1.

0.0063075 = 0.00631 (three significant figures)

$\frac{9}{28}$ - $\frac{2}{7}$ = $\frac{\mathrm{1\; x\; 9\; -\; 4\; x\; 2}}{28}$

$\frac{9}{28}$ - $\frac{2}{7}$ = $\frac{\mathrm{9\; -\; 8}}{28}$

$\frac{9}{28}$ - $\frac{2}{7}$ = $\frac{1}{28}$

$\frac{9}{28}$ - $\frac{2}{7}$ = 0.0357142857142857 ≈ 0.036 (Two significant figures)

Significant figures

Significant figures (or significant digits) are the number of digits important to determine the accuracy and precision of measurement, such as length, mass, or volume.

Significant digits in math convey the value of a number with accuracy. They are considered substantial figures that contribute to the precision of a number.

Rules to determine the number of significant figures in a number

1. Leading zeros are not significant
2. Non-zeros are significant
3. Zeros in between non-zero digits are significant
4. Trailing zeros to the right of the decimal point are significant

Examples

How to Round Off Significant Figures

To round off significant figures, we have to omit one or more digits from the right side of the number until we reach the number of significant digits that we want to round it off to.

First, we have to look at the digit on the right end of the number (to the right of the digit we want to round it off to).

If the digit is lower than 5, the number is rounded off to the lower number.

If the digit is greater than or equal to 5, the number is rounded up to the higher number.

If the number to be rounded off is a whole number, the remaining significant figures are replaced by 0.

Rounding off 0.0357142857142857

The significant figures are 3,5,7,1,4,2,8,5,7,1,4,2,8,5,7.

The second significant digit is 5 and the digit next to it (7) is greater than 5 hence we have to round up the 5 to 6 by simply adding 1 to the 5.

0.0357142857142857 = 0.036 (Two significant figures).

53,000,000 = 53 x 10⁶
0.002 = 2 x 10^-3
0.0004 = 4 x 10^-4

53,000,000 x 0.002 = 53 x 10⁶ x 2 x 10^-3

Law of multiplication of indices

a^m x aⁿ = a^{m + m}

53 x 10⁶ x 2 x 10^-3 = 53 x 2 x 10^6-3
53 x 10⁶ x 2 x 10^-3 = 106 x 10³

$\frac{\mathrm{53,000,000\; x\; 0.002}}{0.0004}$ = $\frac{\mathrm{106\; x}{10}^3}{\mathrm{4\; x}{10}^{-4}}$

Law of division of indices

$\frac{{\mathrm{a}}^{\mathrm{m}}}{{\mathrm{a}}^{\mathrm{n}}}$ = a^{m - n}

$\frac{\mathrm{106\; x}{10}^3}{\mathrm{4\; x}{10}^{-4}}$ = 26.5 x 10^3-(-4)

$\frac{\mathrm{106\; x}{10}^3}{\mathrm{4\; x}{10}^{-4}}$ = 26.5 x 10³⁺⁴

$\frac{\mathrm{106\; x}{10}^3}{\mathrm{4\; x}{10}^{-4}}$ = 26.5 x 10⁷

$\frac{\mathrm{106\; x}{10}^3}{\mathrm{4\; x}{10}^{-4}}$ = 2.65 x 10¹ x 10⁷

$\frac{\mathrm{106\; x}{10}^3}{\mathrm{4\; x}{10}^{-4}}$ = 2.65 x 10¹⁺⁷

$\frac{\mathrm{106\; x}{10}^3}{\mathrm{4\; x}{10}^{-4}}$ = 2.65 x 10⁸

Note: For standard form, the decimal point must be after the first non-zero digit.

Significant figures, any of the digits of a number beginning with the digit farthest to the left that is not zero and ending with the last digit farthest to the right that is either not zero or that is a zero but is considered to be exact.

The significant figures in 0.00798516 are 7,9,8,5,1 and 6.

The third significant figure is 8.

If the significant figure next to the position of the significant figure you seek is 5 or more, you add 1 to the last significant figures you seek.

The next significant to the third (8) is 5, hence 1 is added to the 8, making 9.

Significant figures, any of the digits of a number beginning with the digit farthest to the left that is not zero and ending with the last digit farthest to the right that is either not zero or that is a zero but is considered to be exact.

The significant figures are 3597 and the third significant figure is 9. The number next to the third significant is more than 4, hence 1 is added to the third significant figure resulting in 0.00360.

Change the decimal numbers to the form a x 10ⁿ, where a is a whole number.

0.42 = 0.⌒4⌒2 = 42 x 10^-2

2.5 = 2.⌒5 = 25 x 10^-1

0.5 = 0.⌒5 = 5 x 10^-1

2.05 = 2.⌒0⌒5 = 205 x 10^-2

Note: the power is negative since the decimal is moved to the right.

0.42 ÷ 2.5 = $\frac{\mathrm{42\; x}{10}^{-2}}{\mathrm{25\; x}{10}^{-1}}$

a^m ÷ aⁿ = $\frac{{\mathrm{a}}^{\mathrm{m}}}{{\mathrm{a}}^{\mathrm{n}}}$ = a^{m - n}

0.42 ÷ 2.5 = $\frac{42}{25}$ x 10^-2-(-1)

0.42 ÷ 2.5 = $\frac{42}{25}$ x 10^{-2 + 1}

0.42 ÷ 2.5 = $\frac{42}{25}$ x 10^-1

0.5 x 2.05 = 5 x 10^-1 x 205 x 10^-2

a^m x aⁿ = a^{m + m}

0.5 x 2.05 = 5 x 205 x 10^{-1 + -2}

0.5 x 2.05 = 5 x 205 x 10^-3

$\frac{\mathrm{0.42\; ÷\; 2.5}}{\mathrm{0.5\; x\; 2.05}}$ = $\frac{42}{25}$ x 10^-1 ÷ (5 x 205 x 10^-3)

Every number is being divided by 1

5 x 205 x 10^-3 = $\frac{\mathrm{5\; x\; 205\; x}{10}^{-3}}{1}$

$\frac{\mathrm{0.42\; ÷\; 2.5}}{\mathrm{0.5\; x\; 2.05}}$ = $\frac{42}{25}$ x 10^-1 ÷ $\frac{\mathrm{5\; x\; 205\; x}{10}^{-3}}{1}$

Reciprocate the fraction at the right of the ÷ and change ÷ to multiplication (x)

Note: reciprocate means the numerator becomes the denominator and the denominator becomes the numerator.

$\frac{\mathrm{0.42\; ÷\; 2.5}}{\mathrm{0.5\; x\; 2.05}}$ = $\frac{42}{25}$ x 10^-1 x $\frac{1}{\mathrm{5\; x\; 205\; x}{10}^{-3}}$

$\frac{\mathrm{0.42\; ÷\; 2.5}}{\mathrm{0.5\; x\; 2.05}}$ = $\frac{\mathrm{42\; x\; 1}}{\mathrm{25\; x\; 5\; x\; 205}}$ x 10^-1-(-3)

$\frac{\mathrm{0.42\; ÷\; 2.5}}{\mathrm{0.5\; x\; 2.05}}$ = $\frac{42}{25625}$ x 10^{-1 + 3}

$\frac{\mathrm{0.42\; ÷\; 2.5}}{\mathrm{0.5\; x\; 2.05}}$ = 0.0016390 x 10²

0.0016390 = 1.639 x 10^-3 (Standard form)

$\frac{\mathrm{0.42\; ÷\; 2.5}}{\mathrm{0.5\; x\; 2.05}}$ = 1.639 x 10^-3 x 10²

$\frac{\mathrm{0.42\; ÷\; 2.5}}{\mathrm{0.5\; x\; 2.05}}$ = 1.639 x 10^{-3 + 2}

$\frac{\mathrm{0.42\; ÷\; 2.5}}{\mathrm{0.5\; x\; 2.05}}$ = 1.639 x 10^-1

Notes:

A standard form should be in the form a.bcde x 10ⁿ

The decimal point must be after the first non-zero digit.

The n is the number of times the decimal point has to be moved to be right after the first non-zero digit.

If it is moved to the left, the n is positive and if to the right, n is negative.