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neurologyYou need to know

• In base 10 place value, each digit position to the left is worth 10 times more and each position to the right is worth 10 times less (ones, tenths, hundredths, thousandths).
• The decimal point separates the ones column from the tenths column, so digits keep their place value when numbers are written in a column with decimal points aligned.
• A decimal can be written with trailing zeros without changing its value (for example, `3.5 = 3.50 = 3.500`).
• When adding or subtracting decimal numbers, digits in the same place value must be combined (ones with ones, tenths with tenths, and so on).
• In addition and subtraction with decimals, you may need to exchange (regroup) across the decimal point (for example, exchanging 1 one for 10 tenths).
• Multiplication is commutative, so `a \times b = b \times a`, including for decimals.
• When multiplying by 10, 100, 1000, the digits shift one, two, or three places to the left, so the decimal point effectively moves right (for example, `4.2 \times 100 = 420`).
• When dividing by 10, 100, 1000, the digits shift one, two, or three places to the right, so the decimal point effectively moves left (for example, `4.2 \div 100 = 0.042`).
• When multiplying two decimals, initially the total number of decimal places in the product equals the sum of the decimal places in the factors, after which trailing zeros may be removed (for example, `0.3 \times 0.2 = 0.06`).
• A sensible estimate can be made by rounding numbers first, and the exact answer should be close to the estimate.
• When dividing a decimal by an integer, the quotient may be a terminating decimal (for example, `6.3 \div 9 = 0.7`) and the division can continue past the decimal point.
• Dividing by a decimal is equivalent to multiplying both the dividend and divisor by the same power of 10 so the divisor becomes an integer (for example, `4.8 \div 0.6 = 48 \div 6`).
• Multiplying or dividing both numbers in a division by the same non-zero number does not change the value of the quotient (equivalent division).
• Zero rules: `a \times 0 = 0` and `0 \div a = 0` for `a \ne 0`, but division by zero is undefined.
• The sign rules apply with decimals as with integers: positive divided by positive is positive, and positive divided by negative is negative (and similarly for multiplication).

• Add and subtract decimal numbers by writing them in columns with decimal points aligned, adding placeholder zeros where needed, and exchanging across place values when required.
• Check an addition or subtraction with decimals by estimating first (rounding to 1 significant figure or to the nearest whole number) and confirming the exact answer is reasonable.
• Multiply a decimal by an integer using a short multiplication method, then place the decimal point using place value so the product has the correct number of decimal places.
• Multiply two decimals using a written method (long multiplication or grid method), then place the decimal point so the product has the combined total number of decimal places from both factors.
• Use multiplication by powers of 10 to scale decimals quickly (for example, turning `0.6` into `6` by multiplying by 10) and explain the effect on place value.
• Divide a decimal by an integer using short division, continuing past the decimal point with zeros as needed until the division terminates or reaches the required accuracy.
• Divide by a decimal by scaling both dividend and divisor by the same power of 10 to make the divisor an integer, then perform the division accurately.
• Verify multiplication and division answers using inverse operations (for example, check `a \div b = c` by confirming `c \times b = a`) and by comparing with an estimate.
• Present final answers with appropriate place value, including using trailing zeros when needed to match required decimal places (for example, writing `2.5` as `2.50` for money).