Standard Index Form for ESAT Mathematics

Standard form

A number written in standard index form (often simply called standard form) is expressed in the structure $a \times 10^n$. There are two strict rules for this format: first, the value of $a$ must satisfy $1 \le a < 10$. Second, the index $n$ must be an integer, which can be positive, negative, or zero.

Converting to Standard Form

To convert an ordinary number into standard form, you must determine the appropriate value for $a$ and the corresponding power of 10.

Worked Example: Express 124 in standard form

1. To find $a$, decide where the decimal point must be placed in 124 so that the resulting number is between 1 and 10. This gives $a = 1.24$.
2. Identify the relationship: $1.24 = 124 \div 100$, which is $124 \div 10^2$.
3. Rearrange this relationship: if $1.24 = 124 \div 10^2$, then $124 = 1.24 \times 10^2$.

Worked Example: Express 124,000 in standard form

1. Again, to make $a$ a number between 1 and 10 using the digits provided, we choose $a = 1.24$.
2. Determine the division needed: $1.24 = 124,000 \div 100,000 = 124,000 \div 10^5$.
3. Therefore, $124,000 = 1.24 \times 10^5$.

Worked Example: Express 0.124 in standard form

1. For the number 0.124, we still use $a = 1.24$ to keep $a$ within the required range.
2. Note that $1.24 = 0.124 \times 10 = 0.124 \times 10^1$.
3. To isolate the original number, we see $0.124 = 1.24 \div 10^1$.
4. Using index laws, dividing by $10^x$ is equivalent to multiplying by $10^{-x}$. Thus, $0.124 = 1.24 \times 10^{-1}$.

Worked Example: Express 0.0000124 in standard form

1. Once more, we select $a = 1.24$.
2. Observe that $1.24 = 0.0000124 \times 100,000 = 0.0000124 \times 10^5$.
3. Consequently, $0.0000124 = 1.24 \div 10^5 = 1.24 \times 10^{-5}$.

Ordering numbers in standard form

To place numbers written in standard form ($a \times 10^n$) in order of size, you must evaluate the components in a specific sequence: first compare the indices $n$, and then compare the multipliers $a$ for any numbers that share the same index.

Worked Example: Ordering numbers in standard form

Place the following numbers in order of size, smallest first: $6.1 \times 10^4$, $3.6 \times 10^7$, $8.135 \times 10^{-2}$, $9.6 \times 10^{-4}$, $6 \times 10^4$.

1. List the indices of 10 in ascending order: $-4, -2, 4, 4, 7$.
2. Match the numbers to these indices. The smallest is $9.6 \times 10^{-4}$, followed by $8.135 \times 10^{-2}$.
3. There are two numbers with the index 4: $6 \times 10^4$ and $6.1 \times 10^4$. Since 6 is smaller than 6.1, $6 \times 10^4$ comes first.
4. The final number is $3.6 \times 10^7$.

The final ordered list is: $9.6 \times 10^{-4}, 8.135 \times 10^{-2}, 6 \times 10^4, 6.1 \times 10^4, 3.6 \times 10^7$.

Calculating with numbers in standard form

Calculations in standard form utilize the standard index laws. Given $p = 4 \times 10^5$ and $q = 8 \times 10^{-3}$, we can perform the following operations:

Multiplication ($pq$)

To multiply, multiply the coefficients and then add the indices of the powers of 10. $(4 \times 10^5) \times (8 \times 10^{-3}) = (4 \times 8) \times (10^5 \times 10^{-3}) = 32 \times 10^{5 + (-3)} = 32 \times 10^2$. Because 32 is not between 1 and 10, we must adjust it: $32 = 3.2 \times 10^1$. Therefore, $3.2 \times 10^1 \times 10^2 = 3.2 \times 10^3$.

Division ($p/q$)

To divide, divide the coefficients and subtract the indices. $(4 \times 10^5) \div (8 \times 10^{-3}) = (4 \div 8) \times (10^{5 - (-3)}) = 0.5 \times 10^8$. Since 0.5 is not between 1 and 10, we adjust: $0.5 = 5 \times 10^{-1}$. Thus, $5 \times 10^{-1} \times 10^8 = 5 \times 10^7$.

Raising to a power ($q^2$)

$(8 \times 10^{-3})^2 = (8 \times 8) \times (10^{-3} \times 10^{-3}) = 64 \times 10^{-6}$. Adjusting to standard form: $64 = 6.4 \times 10^1$, so $6.4 \times 10^1 \times 10^{-6} = 6.4 \times 10^{-5}$.

Addition ($p+q$)

Consider the sum $6 \times 10^4 + 8 \times 10^3$. There are two common methods for addition and subtraction.

Method 1: Convert to ordinary numbers Convert both values into ordinary decimal form first: $60,000 + 8,000 = 68,000$. Then convert back to standard form: $6.8 \times 10^4$.

Method 2: Use common factors Factor out a common power of 10, usually the smaller power: $6 \times 10^4 + 8 \times 10^3 = 10^3(6 \times 10^1 + 8) = 10^3(60 + 8) = 68 \times 10^3$. Adjusting to standard form gives $6.8 \times 10^4$.