$$\DeclareMathOperator{cosec}{cosec}$$

# Number

## Furthermore

Numbers in standard form, also known as scientific notation, are expressed as $$a \times 10^n$$, where $$1 \leq |a| < 10$$ and $$n$$ is an integer. This notation is particularly useful when dealing with very large or very small numbers, as it allows for a concise representation.

When performing operations with numbers in standard form, it is crucial to follow the rules of arithmetic carefully. Here are some examples to illustrate the operations:

When adding numbers in standard form, it is essential to have the same exponent. If the exponents are different, adjust them appropriately before performing the addition.

$$(3 \times 10^4) + (5 \times 10^3) = (3 \times 10^4) + (0.5 \times 10^4) \\ = 3.5 \times 10^4$$

Subtraction:

Similar to addition, when subtracting numbers in standard form, ensure that the exponents are the same before performing the subtraction.

$$(7 \times 10^6) - (2 \times 10^5) = (7 \times 10^6) - (0.02 \times 10^6) \\ = 6.98 \times 10^6$$

Multiplication:

When multiplying numbers in standard form, multiply the coefficients (the numbers in front of the power of 10) and then add the exponents of the powers of 10.

$$(12 \times 10^3) \times (4 \times 10^5) = 48 \times 10^8$$

Remember to always express your final answer in standard form, ensuring that the coefficient is a number between 1 and 10 (including 1 but excluding 10). Adjust the power of 10 to compensate any changes you have made to the coefficient.

$$= 4.8 \times 10^9$$

Division:

For division, divide the coefficients and then subtract the exponent in the denominator from the exponent in the numerator.

$$\frac{{3 \times 10^9}}{{6 \times 10^2}} = 0.5 \times 10^7$$

Again remember to always express your final answer in standard form, ensuring that the coefficient is a number between 1 and 10 (including 1 but excluding 10). Adjust the power of 10 to compensate any changes you have made to the coefficient.

$$= 5 \times 10^6$$

This video on Scientific Notation is from Revision Village and is aimed at students taking the IB Maths Standard level course.

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