Standard Index Form for the TMUA

Introduction to Standard Form

Standard index form, or simply standard form, is used to write numbers as $a \times 10^n$. There are two fundamental constraints on this notation: the coefficient $a$ must satisfy $1 \le a < 10$, and the index $n$ must be an integer. This means the number $a$ can be 1 but cannot be 10. If $n$ is positive, the number is large: if $n$ is negative, the number is small (between 0 and 1).

Converting to Standard Form

To convert a number into standard form, follow these steps: first, decide where the decimal point must go to produce a value $a$ between 1 and 10. Then, determine the power of 10 required to relate the original number to $a$.

Example: Large Numbers

To express 124 in standard form, first find $a$. Moving the decimal point in 124 gives $a = 1.24$. Since $1.24 = 124 \div 100$ or $124 \div 10^2$, we can write $124 = 1.24 \times 10^2$.

For a larger number like 124,000, we again find $a = 1.24$. Here, $1.24 = 124,000 \div 100,000$, which is $124,000 \div 10^5$. Thus, $124,000 = 1.24 \times 10^5$.

Example: Small Numbers

To express 0.124 in standard form, we again identify $a = 1.24$. Note that $1.24 = 0.124 \times 10$, so $0.124 = 1.24 \div 10^1$. This is written as $1.24 \times 10^{-1}$. Remember that dividing by $10^x$ is the same as multiplying by $10^{-x}$.

To express 0.0000124 in standard form, we identify $a = 1.24$. Since $1.24 = 0.0000124 \times 100,000$, then $0.0000124 = 1.24 \div 10^5$, which is written as $1.24 \times 10^{-5}$.

Ordering Numbers in Standard Form

To order a set of numbers written in standard form, first look at the indices $n$. A larger $n$ always indicates a larger number. If two numbers have the same index, compare their coefficients $a$.

Example: Ordering Task

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

1. Identify the indices in order: $-4$, $-2$, $4$, $4$, $7$.
2. Identify the smallest index first: $9.6 \times 10^{-4}$ followed by $8.135 \times 10^{-2}$.
3. Compare the numbers with index 4: between $6 \times 10^4$ and $6.1 \times 10^4$, the smaller coefficient is 6.
4. The largest index is 7: $3.6 \times 10^7$.

The final 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 Standard Form

Calculations follow the standard index laws. When performing multiplication or division, treat the coefficients and the powers of 10 separately, then adjust the final answer into standard form if necessary.

Example: Multiplication and Division

Given $p = 4 \times 10^5$ and $q = 8 \times 10^{-3}$:

1. Multiplication ($pq$): Multiply the coefficients $(4 \times 8 = 32)$ and add the indices $(10^5 \times 10^{-3} = 10^2)$. This gives $32 \times 10^2$. This is not in standard form because $32 > 10$. Convert $32$ to $3.2 \times 10^1$, giving $3.2 \times 10^1 \times 10^2 = 3.2 \times 10^3$.

2. Division ($p/q$): Divide the coefficients $(4 \div 8 = 0.5)$ and subtract the indices $(10^5 \div 10^{-3} = 10^{5 - (-3)} = 10^8)$. This gives $0.5 \times 10^8$. Since $0.5 < 1$, convert it to $5 \times 10^{-1}$. The result is $5 \times 10^{-1} \times 10^8 = 5 \times 10^7$.

3. Powers ($q^2$): Square the coefficient $(8^2 = 64)$ and multiply the index by the power $((-3) \times 2 = -6)$. This gives $64 \times 10^{-6}$. Convert $64$ to $6.4 \times 10^1$, resulting in $6.4 \times 10^1 \times 10^{-6} = 6.4 \times 10^{-5}$.

Example: Addition and Subtraction

To calculate $6 \times 10^4 + 8 \times 10^3$, there are two main methods:

Method 1: Conversion to Ordinary Numbers. Convert both to standard integers first: $60,000 + 8,000 = 68,000$. Converting back gives $6.8 \times 10^4$.

Method 2: Common Factors. Take out the common factor of $10^3$: $10^3(6 \times 10^1 + 8) = 68 \times 10^3$. Adjust to standard form: $6.8 \times 10^4$.