1-2. Significant Figures And Scientific Notation

ALL non-zero numbers (1,2,3,4,5,6,7,8,9) are ALWAYS significant. ALL zeroes which are SIMULTANEOUSLY to the right of the decimal point AND at the end of the number are ALWAYS significant. Thus, the digits 3, 5, 1, and 0 are all significant.

It takes a lot of practice get familiar with significant figure rules and determine whether a zero is significant. ALL zeroes which are to the left of a written decimal point and are in a number >= 10 are ALWAYS significant. Also, ALL zeroes between non-zero numbers are ALWAYS significant; Thus, thel zero written in the third decimal place is also significant. The leftmost zero (to the left of the decimal point) is not significant.

ALL non-zero numbers (1,2,3,4,5,6,7,8,9) are ALWAYS significant. The zero in this case is not significant.

When multiplying/dividing numbers, the answer will have the same number of significant figures as the smallest number of significant figures you multiplied/divided. In this case, we are multiplying 4.56 (3 significant figures) and 35.6 (3 significant figures). Thus, the answer will also have 3 significant figures.

The given answer, 0.00002563, is the expanded form of the number. It represents 2 hundredths, 5 thousandths, 6 ten-thousandths, and 3 hundred-thousandths.

The given answer, 5.67, is the correct way to write the number 5670000 in scientific notation. In scientific notation, a number is expressed as a product of a coefficient and a power of 10. The coefficient should be a number between 1 and 10, and in this case, it is 5.67. The power of 10 is determined by the number of zeros in the original number, which is 6 in this case. Therefore, the number 5670000 can be written as 5.67 x 10^6 in scientific notation.

When measurements are added or subtracted, the answer can contain no more decimal places than the least accurate measurement. Thus, look at each number you are adding/subtracting and see how digits are to the right of the decimal place. The answer will have the same number of digits to the right of the decimal place as the smallest number of decimal places of the number you are adding. In this case, we are adding 2.55 (2 to the right), 45.723 (3 to the right), and 0.30 (2 to the right). The smallest number of places to the right is 2 digits; thus, the answer must be rounded to 2 decimal places.

Following the order of operations, add the two numbers in the numerator first. When you add/subtract, round the answer to the small number of decimal places. In this case, we are adding 3.4 (1 to the right) and 5.678 (3 to the right); thus, the answer will have 1 digit to the right of the decimal place. The answer to that addition section is 9.1. Now, we need to divide: 9.1/235. When you multiply/divide, the answer will have the same number of significant figures as the number with the lowest number of significant figures. In this case, we are dividing 9.1 (2 significant figures) and 235 (3 significant figures); thus, the answer will have 2 significant figures. The correct answer to this calculation, with the correct number of significant figures, is 0.039 (2 significant figures).

To express a number in scientific notation, we move the decimal point to a position where there is only one non-zero digit to the left of it. In this case, the original number is 0.000466. By moving the decimal point 4 places to the right, we get 4.66. Since we moved the decimal point to the right, the exponent is negative. Therefore, the number 0.000466 in scientific notation with an exponent of -4 is 4.66 x 10^-4.

The given number is 7,860. In expanded form, this number can be expressed as 7,000 + 800 + 60.