Before we talk about our next subject, we’re going to sidebar into the topic of scientific notation.
Scientific notation is a useful shorthand for writing lengthy numbers in a concise manner. And although scientific notation may seem foreign at first, understanding scientific notation will help you understand how floating point numbers work, and more importantly, what their limitations are.
Numbers in scientific notation take the following form: significand x 10exponent. For example, in the scientific notation 1.2 x 10⁴, 1.2 is the significand and 4 is the exponent. Since 10⁴ evaluates to 10,000, 1.2 x 10⁴ evaluates to 12,000.
By convention, numbers in scientific notation are written with one digit before the decimal point, and the rest of the digits afterward.
Consider the mass of the Earth. In decimal notation, we’d write this as 5972200000000000000000000 kg. That’s a really large number (too big to fit even in an 8 byte integer). It’s also hard to read (is that 19 or 20 zeros?). Even with separators (5,972,200,000,000,000,000,000,000) the number is still hard to read.
In scientific notation, this would be written as 5.9722 x 10²⁴ kg, which is much easier to read. Scientific notation has the added benefit of making it easier to compare the magnitude of two extremely large or small numbers simply by comparing the exponent.
Because it can be hard to type or display exponents in C++, we use the letter ‘e’ (or sometimes ‘E’) to represent the “times 10 to the power of” part of the equation. For example, 1.2 x 10⁴ would be written as 1.2e4, and 5.9722 x 10²⁴ would be written as 5.9722e24.
For numbers smaller than 1, the exponent can be negative. The number 5e-2 is equivalent to 5 * 10⁻², which is 5 / 10², or 0.05. The mass of an electron is 9.1093837e-31 kg.
For Linux users
If you’re using arch linux and the superscript of 5 * 10⁻² is missing the negative sign, you may need to install a font that can display these characters. See this reddit thread for more info.
Significant digits
Let’s say you need to know the value of the mathematical constant pi for some equation, but you’ve forgotten. You ask two people. One tells you the value of pi is 3.14. The other tells you the value of pi is 3.14159. Both of these values are “correct”, but the latter is far more precise.
Here’s the most important thing to understand about scientific notation: The digits in the significand (the part before the ‘e’) are called the significant digits (or significant figures). The more significant digits, the more precise a number is.
Key insight
The more digits in the significand, the more precise a number is.
In scientific notation, we’d write 3.14 as 3.14e0. Since there are 3 numbers in the significand, this number has 3 significant digits.
3.14159 would be written as 3.14159e0. Since there are 6 numbers in the significand, this number has 6 significant digits.
How to convert decimal numbers to scientific notation
Use the following procedure:
- Your exponent starts at zero.
- If the number has no explicit decimal point (e.g.
123), it is implicitly on the right end (e.g. 123.)
- Slide the decimal point left or right so there is only one non-zero digit to the left of the decimal.
- Each place you slide the decimal point to the left increases the exponent by 1.
- Each place you slide the decimal point to the right decreases the exponent by 1.
- Trim off any leading zeros (on the left end of the significand)
- Trim off any trailing zeros (on the right end of the significand) only if the original number had no decimal point. We’re assuming they’re not significant. If you have additional information to suggest they are significant, you can keep them.
Here’s some examples:
Start with: 600.410
Slide decimal point left 2 spaces: 6.00410e2
No leading zeros to trim: 6.00410e2
Don't trim trailing zeros: 6.00410e2 (6 significant digits)
Start with: 0.0078900
Slide decimal point right 3 spaces: 0007.8900e-3
Trim leading zeros: 7.8900e-3
Don't trim trailing zeros: 7.8900e-3 (5 significant digits)
Start with: 42030 (no information to suggest the trailing zero is significant)
Slide decimal point left 4 spaces: 4.2030e4
No leading zeros to trim: 4.2030e4
Trim trailing zeros: 4.203e4 (4 significant digits)
Start with: 42030 (assuming the trailing zero is significant)
Slide decimal point left 4 spaces: 4.2030e4
No leading zeros to trim: 4.2030e4
Keep trailing zeros: 4.2030e4 (5 significant digits)
Handling trailing zeros
Consider the case where we ask two lab assistants each to weigh the same apple. One returns and says the apple weighs 87.0 grams. The other returns and says the apple weighs 87.000 grams. Let’s assume the weighing is correct. In the former case, the actual weight of the apple could be anywhere between 86.950 and 87.049 grams. Maybe the scale was only precise to the nearest tenth of a gram. Or maybe our assistant rounded a bit. In the latter case, we are confident about the actual weight of the apple to a much higher degree (it actually weighs between 86.99950 and 87.00049 grams, which has much less variability).
When converting to scientific notation, trailing zeros after a decimal point are considered significant, so we keep them:
- 87.0g = 8.70e1
- 87.000g = 8.7000e1
For numbers with no decimal point, trailing zeros are considered to be insignificant by default. Given the number 2100 (with no additional information), we assume the trailing zeroes are not significant, so we drop them:
- 2100 = 2.1e3 (trailing zeros assumed not significant)
However, if we happened to know that this number was measured precisely (or that the actual number was somewhere between 2099.5 and 2100.5), then we should instead treat those zeros as significant:
- 2100 = 2.100e3 (trailing zeros known significant)
Tip
You may occasionally see a number written with an explicit trailing decimal point. This is an indication that the preceding zeros are significant.
- 2100. = 2.100e3 (trailing zeros known significant)
One of the nice things about scientific notation is that it always makes clear how many digits are significant.
From a technical standpoint, the numbers 87.0 and 87.000 have the same value (and the same type). When C++ stores either of these numbers in memory, it will store just the value 87. And once stored, there is no way to determine from the stored value which of the two numbers was originally input.
Now that we’ve covered scientific notation, we’re ready to cover floating point numbers.
Quiz time
Question #1
Convert the following numbers to scientific notation (using an e to represent the exponent) and determine how many significant digits each has.
Remember, keep trailing zeros after a decimal point, drop trailing zeros if there is no decimal point.
a) 34.50
Show Solution
3.450e1 (4 significant digits)
b) 0.004000
Show Solution
4.000e-3 (4 significant digits)
c) 123.005
Show Solution
1.23005e2 (6 significant digits)
d) 146000
Show Solution
1.46e5 (3 significant digits). Remember, we assume trailing zeros in a whole number with no decimal are not significant.
e) 146000.001
Show Solution
1.46000001e5 (9 significant digits)
f) 0.0000000008
Show Solution
8e-10 (1 significant digit). The correct significand is 8 (which has 1 significant digit), not 8.0 (which has 2 significant digits).
g) 34500.0
Show Solution
3.45000e4 (6 significant digits). We don’t trim the trailing zeros here because the number
does have a decimal point. Even though the decimal point doesn’t affect the value of the number, it affects the precision, so it needs to be included in the significand.
If the number had been specified as 34500, then the answer would have been 3.45e4 (3 significant digits).
h) 146000 (assume you have knowledge that zeros are significant)
Show Solution
1.46000e5 (6 significant digits).
4.7 — Introduction to scientific notation
Before we talk about our next subject, we’re going to sidebar into the topic of scientific notation.
Scientific notation is a useful shorthand for writing lengthy numbers in a concise manner. And although scientific notation may seem foreign at first, understanding scientific notation will help you understand how floating point numbers work, and more importantly, what their limitations are.
Numbers in scientific notation take the following form: significand x 10exponent. For example, in the scientific notation
1.2 x 10⁴,1.2is the significand and4is the exponent. Since 10⁴ evaluates to 10,000, 1.2 x 10⁴ evaluates to 12,000.By convention, numbers in scientific notation are written with one digit before the decimal point, and the rest of the digits afterward.
Consider the mass of the Earth. In decimal notation, we’d write this as
5972200000000000000000000 kg. That’s a really large number (too big to fit even in an 8 byte integer). It’s also hard to read (is that 19 or 20 zeros?). Even with separators (5,972,200,000,000,000,000,000,000) the number is still hard to read.In scientific notation, this would be written as
5.9722 x 10²⁴ kg, which is much easier to read. Scientific notation has the added benefit of making it easier to compare the magnitude of two extremely large or small numbers simply by comparing the exponent.Because it can be hard to type or display exponents in C++, we use the letter ‘e’ (or sometimes ‘E’) to represent the “times 10 to the power of” part of the equation. For example,
1.2 x 10⁴would be written as1.2e4, and5.9722 x 10²⁴would be written as5.9722e24.For numbers smaller than 1, the exponent can be negative. The number
5e-2is equivalent to5 * 10⁻², which is5 / 10², or0.05. The mass of an electron is9.1093837e-31 kg.For Linux users
If you’re using arch linux and the superscript of
5 * 10⁻²is missing the negative sign, you may need to install a font that can display these characters. See this reddit thread for more info.Significant digits
Let’s say you need to know the value of the mathematical constant pi for some equation, but you’ve forgotten. You ask two people. One tells you the value of pi is
3.14. The other tells you the value of pi is3.14159. Both of these values are “correct”, but the latter is far more precise.Here’s the most important thing to understand about scientific notation: The digits in the significand (the part before the ‘e’) are called the significant digits (or significant figures). The more significant digits, the more precise a number is.
Key insight
The more digits in the significand, the more precise a number is.
In scientific notation, we’d write
3.14as3.14e0. Since there are 3 numbers in the significand, this number has 3 significant digits.3.14159would be written as3.14159e0. Since there are 6 numbers in the significand, this number has 6 significant digits.How to convert decimal numbers to scientific notation
Use the following procedure:
123), it is implicitly on the right end (e.g.123.)Here’s some examples:
Handling trailing zeros
Consider the case where we ask two lab assistants each to weigh the same apple. One returns and says the apple weighs 87.0 grams. The other returns and says the apple weighs 87.000 grams. Let’s assume the weighing is correct. In the former case, the actual weight of the apple could be anywhere between 86.950 and 87.049 grams. Maybe the scale was only precise to the nearest tenth of a gram. Or maybe our assistant rounded a bit. In the latter case, we are confident about the actual weight of the apple to a much higher degree (it actually weighs between 86.99950 and 87.00049 grams, which has much less variability).
When converting to scientific notation, trailing zeros after a decimal point are considered significant, so we keep them:
For numbers with no decimal point, trailing zeros are considered to be insignificant by default. Given the number 2100 (with no additional information), we assume the trailing zeroes are not significant, so we drop them:
However, if we happened to know that this number was measured precisely (or that the actual number was somewhere between 2099.5 and 2100.5), then we should instead treat those zeros as significant:
Tip
You may occasionally see a number written with an explicit trailing decimal point. This is an indication that the preceding zeros are significant.
One of the nice things about scientific notation is that it always makes clear how many digits are significant.
From a technical standpoint, the numbers 87.0 and 87.000 have the same value (and the same type). When C++ stores either of these numbers in memory, it will store just the value 87. And once stored, there is no way to determine from the stored value which of the two numbers was originally input.
Now that we’ve covered scientific notation, we’re ready to cover floating point numbers.
Quiz time
Question #1
Convert the following numbers to scientific notation (using an e to represent the exponent) and determine how many significant digits each has.
Remember, keep trailing zeros after a decimal point, drop trailing zeros if there is no decimal point.
a) 34.50
Show Solution
b) 0.004000
Show Solution
c) 123.005
Show Solution
d) 146000
Show Solution
e) 146000.001
Show Solution
f) 0.0000000008
Show Solution
g) 34500.0
Show Solution
h) 146000 (assume you have knowledge that zeros are significant)
Show Solution