##### Standard Forms

When numbers are very small, like 0.0000362, or very large, like 358 000 000, calculations can be time consuming and it is easy to miss out some of the zeros. Standard form is used to express very small and very large numbers in a compact and efficient way. In standard form, numbers are written as a number multiplied by 10 raised to a given power.

### Standard form for large numbers

The key to standard form for large numbers is to understand what happens when you multiply by power of 10. Each time you multiply a number by 10 each digit within the number moves one place order to the left (notice that this looks like the decimal point has moved one place to the right).

3.2

$ \displaystyle 3.2\times 10=32.0$

The digits have moved one place order to the left.

$ \displaystyle 3.2\times {{10}^{2}}=3.2\times 100=320.0$

The digits have moved two places.

$ \displaystyle 3.2\times {{10}^{3}}=3.2\times 1000=3200.0$

The digits have moved three places

…..and so on. You should see a pattern forming.

### Any large number can be expressed in standard form by writing it as a number between 1 and 10 multiplied by a suitable power of 10. To do this write the appropriate number between 1 and 10 first (using the non-zero digits of the original number) and then count the number of places you need to move the first digit to the left. The number of places tells you by what power, 10 should be multiplied.

Example 1: Write 320 000 in standard form.

$ \displaystyle 3.2$

Start by finding the number 1 and 10 that has the same digits in the same order as the original number. Here, the extra 4 zero digits can be excluded because they do not change the size of your new number.

Now compare the position of the first digit in both numbers: ‘3’ has to move 5 place orders to the left to get from the new number to the original number.

$ \displaystyle 320000=3.2\times {{10}^{5}}$

The first digit, ‘3’, has moved five places. So you multiply by $ \displaystyle {{10}^{5}}$.

### Calculating using standard forms

Once you have converted large numbers into standard form, you can use the index laws to carry out calculations involving multiplication and division.

Example 2: Solve and give your answers in standard form.

a) $ \displaystyle (3\times {{10}^{5}})\times (2\times {{10}^{6}})$

Simplify by putting like terms together. Use the laws of indices where appropriate. Write the number in standard form.

$ \displaystyle (3\times {{10}^{5}})\times (2\times {{10}^{6}})=$

$ \displaystyle (3\times 2)\times ({{10}^{5}}\times {{10}^{6}})=$

$ \displaystyle 6\times {{10}^{{5+6}}}=6\times {{10}^{{11}}}$

You may be asked to convert your answer to an ordinary number. To convert $ \displaystyle 6\times {{10}^{{11}}}$ into an ordinary number, the ‘6’ needs to move 11 places to the left.

b) $ \displaystyle (2\times {{10}^{3}})\times (8\times {{10}^{7}})$

$\displaystyle (2\times {{10}^{3}})\times (8\times {{10}^{7}})$$\displaystyle =(2\times 8)\times ({{10}^{3}}\times {{10}^{7}})$$\displaystyle =16\times {{10}^{{10}}}$

The answer $ \displaystyle 16\times {{10}^{{10}}}$ is numerically correct but it is not in standard form because 16 is not between 1 and 10.You can change it to standard form by thinking of 16 as $ \displaystyle 1.6\times 10$

c) $ \displaystyle (2.8\times {{10}^{6}})\div (1.4\times {{10}^{4}})$

$ \displaystyle (2.8\times {{10}^{6}})\div (1.4\times {{10}^{4}})=$

$ \displaystyle \frac{{2.8\times {{{10}}^{6}}}}{{1.4\times {{{10}}^{4}}}}=\frac{{2.8}}{{1.4}}\times \frac{{{{{10}}^{6}}}}{{{{{10}}^{4}}}}=$

$ \displaystyle 2\times {{10}^{{6-4}}}=2\times {{10}^{2}}$

Simplify by putting like terms together. Use the laws of indices.

d) $ \displaystyle (9\times {{10}^{6}})+(3\times {{10}^{8}})$

When adding or subtracting numbers in standard form it is often easiest to rewrite them both as ordinary numbers first, then convert the answer to standard form.

$ \displaystyle 9\times {{10}^{6}}=9000000$

$ \displaystyle 3\times {{10}^{8}}=300000000$

So,$ \displaystyle (9\times {{10}^{6}})+(3\times {{10}^{8}})=$

$ \displaystyle 300000000+9000000=$

$ \displaystyle 309000000=3.09\times {{10}^{8}}$

Be careful! When you solve problems in standard form you need to check your results carefully. Always be sure to check that your final answer is in standard form. Check that all conditions are satisfied. Make sure that the number part us between 1 and 10.

### Standard form for small numbers

You have seen that the digits move place order to the left when multiplying by powers of 10.If you divide by powers of 10 move the digits in place order to the right and make the number smaller.

Consider the following pattern:

$ \displaystyle 2300$

$ \displaystyle 2300\div 10=230$

$ \displaystyle 2300\div {{10}^{2}}=2300\div 100=23$

$ \displaystyle 2300\div {{10}^{3}}=2300\div 1000=2.3$ and so on.

The digits move place order to the right(notice that this looks like the decimal point is moving to the left). We know that if a direction is taken to be positive, the opposite direction is taken to be negative. Since moving place order to the left raises 10 to the power of a positive index, it follows that moving place order to the right raises 10 to the power of a negative index.

We write negative powers to indicate that you divide, and you saw above that will small numbers, you divide by 10 to express the number in standard form.

Example 3: Write each of the following in standard form.

**a)** $ \displaystyle 0.004$

Start with a number between 1 and 10, in this case 4.Compare the position of the first digit: ‘4’ needs to move 3 place orders to the right to get from the new number to the original number. In __ Example 1__ you saw that moving 5 places to the left meant multiplying by $ \displaystyle {{10}^{5}}$, so it follows that moving 3 places to the right means multiply by $ \displaystyle {{10}^{{-3}}}$. Notice also that the first non-zero digit in 0.004 is in the 3rd place after the decimal point and that the power of 10 is -3.

Alternatively: you know that you need to divide by 10 three times, so you can change it to a fractional index and then a negative indexs.

$ \displaystyle 0.004=4\div {{10}^{3}}=$

$ \displaystyle 4\times {{10}^{{\frac{1}{3}}}}=4\times {{10}^{3}}$

**b)** $ \displaystyle 0.00000034$

$ \displaystyle 0.00000034=3.4\div {{10}^{7}}=3.4\times {{10}^{{-7}}}$

Notice that the first non-zero digit in $ \displaystyle 0.00000034$ is in the 7th place after the decimal point and that the power of 10 is -7.

**c)** $ \displaystyle (2\times {{10}^{{-3}}})\times (3\times {{10}^{{-7}}})$

$ \displaystyle (2\times {{10}^{{-3}}})\times (3\times {{10}^{{-7}}})=$

$ \displaystyle (2\times 3)\times ({{10}^{{-3}}}\times {{10}^{{-7}}})=$

$ \displaystyle 6\times {{10}^{{-3+(-7)}}}=6\times {{10}^{{-10}}}$