Standard formIB MYP Maths Extended: Revision notes

Section 1

What standard form is

Standard form (scientific notation) writes a number as $a\times10^k,\qquad 1\le a<10,\ k\text{ an integer}.$ The number $a$ must have exactly one non-zero digit before the decimal point. So $4.5\times10^7$ is in standard form, but $45\times10^6$ and $0.45\times10^8$ are not. A positive $k$ is for large numbers and a negative $k$ is for small numbers.

Key termsstandard formpower of ten

Common mistake

Leaving $a$ outside the range 1 to 10, such as $12.5\times10^7$ . Adjust it: $12.5\times10^7=1.25\times10^8$ .

Section 2

Very large numbers

To write a large number in standard form, put the decimal point after the first non-zero digit and count how many places it moved. That count is the positive exponent. Example: $150\,000\,000$ . Move the point 8 places to get $1.5$ , so $150\,000\,000=1.5\times10^8$ .

Exam tip

Count the places the point moves, not the number of zeros.

Section 3

Very small numbers

For a number between 0 and 1, the decimal point moves to the right and the exponent is negative. Count the places moved. Example: $0.000\,007=7\times10^{-6}$ (6 places). To change back, move the point the other way: $3.2\times10^{-4}=0.00032$ . Check the size: a negative exponent always means a number smaller than 1.

Key termsnegative exponent

Common mistake

Using a positive exponent for a small number. $0.00032$ is $3.2\times10^{-4}$ , not $3.2\times10^{4}$ .

Section 4

Multiplying and dividing without a calculator

Deal with the number parts and the powers of ten separately.

Multiply: multiply the $a$ values and add the exponents. $(2.5\times10^3)(5\times10^4)=12.5\times10^7=1.25\times10^8$ .
Divide: divide the $a$ values and subtract the exponents. $\dfrac{1.5\times10^8}{3\times10^5}=0.5\times10^3=5\times10^2$ . Always finish by checking that $a$ is between 1 and 10 and adjusting the power if it is not.

Exam tip

If $a$ becomes bigger than 10, increase $k$ by 1 for every move of the decimal point left. If $a$ becomes smaller than 1, decrease $k$ .

Section 5

Adding and subtracting

To add or subtract, the powers of ten must be the same. Rewrite one number first. Example: $3\times10^5+4\times10^4=3\times10^5+0.4\times10^5=3.4\times10^5$ . You can also write both as ordinary numbers: $300\,000+40\,000=340\,000=3.4\times10^5$ .

Common mistake

Adding the exponents when adding numbers. $3\times10^5+4\times10^4$ is not $7\times10^9$ .

Section 6

Using a calculator and real-life use

Calculators have an EXP or $\times10^x$ key for entering standard form: type $1.5$ , press the key, then $8$ for $1.5\times10^8$ . A display such as $1.5\mathrm{E}8$ means $1.5\times10^8$ . Write the answer in full standard form, not as E notation. Standard form is used for astronomy (distance from Earth to Sun $1.5\times10^8$ km), biology (red blood cell $7\times10^{-6}$ m) and computing ( $1$ GB $=10^9$ bytes). Give answers to a sensible number of significant figures and use whole numbers for things you count.

Key termssignificant figures

Exam tip

To compare sizes, compare the exponents first. The number with the larger exponent is bigger when both are in standard form.

That's the notes covered.

Carry on to the next subtopic.

Exam questions on Standard form

The Earth is about $150\,000\,000$ km from the Sun. Light travels at $3\times10^{5}$ km per second.
Mars is $2.28\times10^{8}$ km from the Sun. Find how many times further from the Sun Mars is than the Earth. Give your answer to 2 significant figures.2 marks
A red blood cell has a diameter of about $7\times10^{-6}$ m. A hydrogen atom has a diameter of about $1.06\times10^{-10}$ m.
A bacterium has a length of $2.5\times10^{-6}$ m. Without a calculator, find the total length of $4000$ of these bacteria placed end to end. Give your answer in standard form, in metres.2 marks
A lorry carries $2.5\times10^{3}$ kg of rice. One kilogram of rice contains about $5\times10^{4}$ grains.
Without a calculator, find the number of grains of rice on the lorry. Give your answer in standard form.3 marks

See the full worksheet

Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).