Explore

Numerical Analysis

Significant Digits

Significant digits let us know which places have measured values

All non-zero numbers are significant

Number without a decimal:

trailing zeros are not significant, they are place holders to shift the non-zero digits to the appropriate place value

Number with a decimal:

trailing zeros are significant, they are measured.

leading zeros are insignificant, they are place holders to shift the non-zero digits to the appropriate place value.

ALL zeros between two other non-zero numbers are significant, they are measured.

Counted numbers are infinitely precise or have an infinite number of significant figures (eg. 10 books, 5 beakers, etc.), unless otherwise stated.

This is because counted quantities are discrete (not continuous). You can’t increase the amount of beakers you have by anything less than one beaker. You can’t have half a beaker, or a third of a person.  

When a number is shown in scientific notation, all shown digits are significant.

Scientific Notation

When numbers get really big, or really small, scientific notation helps to make things more compact.

eg. 32 700 000 000 000 000 = 3.27 x 1016

eg. 0.000 000 718 = 7.18 x 10-7

Example:

Convert 5 120 000 000 m into scientific notation

Example:

Convert 0.000 37 kg into scientific notation

Determine number of significant digits.

3 sig digs -> 5 1 2

Move the decimal from its original position to the position which is to the right of the first non-zero digit, then get rid of non-significant digits.

5 . 120 000 000

Count the number of spaces you moved the decimal.

5 120 000 000 -> 5 . 120 000 000 (9 spaces to the left)

If you moved the decimal to the left, the exponent is positive. If you moved the decimal to the right, the exponent is negative.

5.12 x 109 m

(alternatively, if the value of the original number is greater than 1, then your exponent is positive, if it is between 0 and 1, the exponent is negative)

Determine number of significant digits.

2 sig digs -> 3 7

Move the decimal from beside its original position to the position which is to the right of the first non-zero digit, then get rid of non-significant digits.

0 000 3 . 7

Count the number of spaces you moved the decimal.

0.000 37 -> 0 000 3.7 (4 spaces to the right)

If you moved the decimal to the left, the exponent is positive. If you moved the decimal to the right, the exponent is negative.

3.7 x 10-4 kg

(alternatively, if the value of the original number is greater than 1, then your exponent is positive, if it is between 0 and 1, the exponent is negative)

There are no rows in this table

⁠

Calculations - Respecting Sig Digs, Managing Uncertainty

Adding & Subtracting

Calculations are limited to the common decimal place with the smallest place value

Keep the lowest number of DECIMALS

0.0245 + 2.104 = 0.0245

+ 2.104 .

2.1285

2.129

The addition shows a value of 2.1285, but the last decimal is insignificant, so the value is 2.129

* only quantities which have the same units can be added *

Multiplying & Dividing

Count the number of digits which are present in each of the quantities and limit your answer to the smallest of those two numbers.

Keep the lowest number of SIGNIFICANT DIGITS

0.0245 ⨉ 2.104 =

0.0245 has 3 sig digs 2.104 has 4 sig digs

Our answer should have 3 sig digs.

0.0245 ⨉ 2.104 = 0.051548

0.0245 ⨉ 2.104 = 0.0515

* quantities with different units can be multiplied/divided *

Multiple Steps

If all calculations are done in a single step, use the rule from multiplication and division.

Count the number of digits which are present in each of the quantities and limit your answer to the smallest of those numbers.

If calculations are done in multiple steps, you must track the number of digits through each step. Using the rule which is appropriate for the operation.

Want to print your doc?
This is not the way.

Try clicking the ··· in the right corner or using a keyboard shortcut (

CtrlP

) instead.