The standard form of a number is also known as its scientific notation. It is a way of expressing very large or very small numbers in a concise and standardized format. Standard form of numbers are usually used in science, engineering, and mathematics.
A number in standard form is written as:
\[a \times 10^n\]
Where:
\(*\) \(a\) is a number greater than or equal to 1 and less than 10 (i.e \(1 \leq a < 10 \)).
\(*\) \(n\) is an integer (positive, negative, or zero).
In basic 7, we learnt that numbers to the power of \(0\) are equal to \(1\). We can therefore express 1 as a power of 10, as done below:
\[1 = 10^0\]
Also, every number to the power of 1 will give the number itself. Hence, we can write the number 10 as a power of 10 as below:
\[10 = 10^1\]
Every number multiplied by itself is the number to the power of two (2) in indices. That is \(a \times a = a^2\). Hence,
\[100 = 10 \times 10 = 10^2\]
It follows that,
\[1000 = 10 \times 10 \times 10 = 10^3\]
Look carefully at the number of zeros in each of the numbers on the left and the corresponding power on the 10 at the right side of the equal to sign. For instance, \(100\) has 2 zeros hence its equivalent power of 10 is raised to the power of 2.
Also, the number \(1000\) has 3 zeros, and can therefore be written as \(10^3\).
Example 1
Write the number 10,000,000,000 as a power of 10
Solution
10,000,000,000 as a power of 10
10,000,000,000 has 10 zeros
\(\Rightarrow 10,000,000,000 = 10^{10}\)
Example 2
Write the number 1,000,000,000 as a power of 10
Solution
1,000,000,000 as a power of 10
1,000,000,000 has 9 zeros
\(\Rightarrow 1,000,000,000 = 10^{9}\)
Write the equivalent powers of 10 for the numbers below.
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\(10,000\)
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\(100,000\)
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\(1,000,000\)
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\(10,000,000\)
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\(100,000,000\)
Once we know how to write \(10\), \(100\), \(1000\), etc., as powers of \(10\), we can now express other multiples of \(10\) in standard form.
Remember, that multiples of 10 are the numbers that can be divided by \(10\) without getting a remainder. Examples are 10, 20, 30, 40, etc.
Also a number in standard form is written as:
\[a \times 10^n\]
Where:
\(*\) \(a\) is a number greater than or equal to 1 and less than 10 (i.e \(1 \leq a < 10 \)).
\(*\) \(n\) is an integer (positive, negative, or zero).
Given the number 20 for instance, to write the number 20 in standard form we can express 20 as a product of any integer between 1 and 10 (1 inclusive) and a power of 10, in this case 10, as below.
\[20 = 2 \times 10\]
But we have learnt that \(10 = 10^1\). Hence to write \(20\) in standard form, it will be written as below:
\[20 = 2 \times 10^1\]
Example 3
Express \(4000\) in standard form.
Solution
\(4000\) in standard form.
\(4000 = 4 \times 1000\)
\(\Rightarrow 4000 = 4 \times 10^3\)
Example 4
Express \(80000\) in standard form.
Solution
\(80000\) in standard form.
\(80000 = 8 \times 10000\)
\(\Rightarrow 80000 = 8 \times 10^4\)
Write the following numbers in standard form:
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\(800,000\)
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\(6,000,000\)
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\(4,000\)
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\(70,000,000\)
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\(1,000,000,000\)
When the given number has only one (1) significant figure as in the examples above, it is easier to write them in standard form.
In this section, we will learn how to write any integer with one or more significant numbers in standard form.
We have already learnt about the definition of the form of numbers written in standard form as below:
\[a \times 10^n\]
Where:
\(*\) \(a\) is a number greater than or equal to 1 and less than 10 (i.e \(1 \leq a < 10 \)).
\(*\) \(n\) is an integer (positive, negative, or zero).
Remember
Every whole number can be written to a one decimal place, that is, to a decimal point of zero.
Example: \(25 = 25.0\), \(14 = 14.0\), \(3 = 3.0\) and \(1000 = 1000.0\).
Converting to Standard Form
To convert any number to standard form, follow the steps below:
1. Identify the significant digits in the number.
2. Position the decimal point such that there is only one non-zero digit to its left.
3. Count the number of places the decimal point was moved:
\(\hspace{0.2cm}*\) If moved to the left, \(n\) is positive.
\(\hspace{0.2cm}*\) If moved to the right, \(n\) is negative.
4. Write the number in the form \(a \times 10^n\).
Example 5
Convert \(56,000,000\) to standard form:
Solution
\(56,000,000\) to standard form
\(56,000,000 \ \)\(=\)\(\ 56,000,000.0\)
The significant figures are 5 and 6.
\(\Rightarrow\) To position the decimal point between 5 and 6 (that is 5.6) as indicated in step 2 above, we will need to move the decimal point 7 steps to the left.
This will make the power on the 10 positive, that is, \(10^7\).
Hence, in writing \(56,000,000\) in the form \(a \times 10^n\),
\(\Rightarrow\) \(56,000,000 =\) \(5.6 \times 10^7\)
Example 6
Write \(26\) in standard form.
Solution
\(26\) in standard form.
\(26 = 26.0\)
\(26\) in the form \(a \times 10^n\)
\(\Rightarrow 2.6 \times 10^1\)
\(\therefore 26 = 2.6 \times 10^1\)
Example 7
Write \(375\) in standard form.
Solution
\(375\) in standard form.
\(375 = 375.0\)
\(375\) in the form \(a \times 10^n\)
\(\Rightarrow 3.75 \times 10^2\)
\(\therefore 375 = 3.75 \times 10^2\)
Example 8
Write \(2474.5\) in standard form. [B.E.C.E 2002 I]
Solution
\(2474.5\) in standard form
\(\Rightarrow 2474.5\) in the form \(a \times 10^n\)
\(\Rightarrow 2.4745 \times 10^3\)
\(\therefore 2474.5 = 2.4745 \times 10^3\)
Example 9
Write \(8,765,049\) in standard form.
Solution
\(8,765,049\) in standard form.
\(8,765,049 \ \)\(=\)\(\ 8,765,049.0\)
\(8,765,049\) in the form \(a \times 10^n\)
\(\Rightarrow 8.765049 \times 10^6\)
\(\therefore 8,765,049\) \(=\) \(8.765049 \times 10^6\)
Example 10
Simplify \(0.1 \times 0.02 \times 0.003\) leaving your answer in standard form. [B.E.C.E 2002 II]
Solution
\(0.1 \times 0.02 \times 0.003\)
but
\(0.1 = 1.0 \times 10^{-1}\)
\(0.02 = 2.0 \times 10^{-2}\)
\(0.003 = 3.0 \times 10^{-3}\)
\(\Rightarrow \)\(0.1 \times 0.02 \times 0.003\) \(=\) \((1.0 \times 10^{-1})\) \(\times\) \((2.0 \times 10^{-2})\) \(\times\) \((3.0 \times 10^{-3})\)
Grouping like terms:
\(0.1 \times 0.02 \times 0.003\)
\(\Rightarrow \) \(1.0 \times 2.0 \times 3.0\) \(\times\) \(10^{-1} \times 10^{-2}\) \(\times\) \(10^{-3}\)
\(\Rightarrow \) \(6.0\) \(\times\) \(10^{-1-2-3}\)
\(\Rightarrow \) \(6.0\) \(\times\) \(10^{-6}\)
Example 11
Write \(83000\) in standard form. [B.E.C.E 2003]
Solution
\(83000\) in standard form.
\(83000 \ \)\(=\)\(\ 83000.0\)
\(83000\) in the form \(a \times 10^n\)
\(\Rightarrow 8.3 \times 10^4\)
\(\therefore 83000\) \(=\) \(8.3 \times 10^4\)
Solve the following:
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Write \(3560\) in standard form. [B.E.C.E 2011]
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If \(4956 \times 25 = 123,900\), evaluate \(495.6 \times 2.5\) leaving your answer in standard form. [B.E.C.E 2013]
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Write \(56\) in standard form.
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What is the value of \(x\) if \(10^x = 1000\). [B.E.C.E 2015]
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Write 39.9748 km in standard form.
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Express 0.0043216 in standard form. [B.E.C.E 1994]
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Write \(356.07\) in standard form. [B.E.C.E 1995]
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Write \(4687.02\) in standard form. [B.E.C.E 1991]
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Write in standard form \(1342\). [B.E.C.E 1999]
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Express \(2345\) in standard form. [B.E.C.E 2001]
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